PARTICLES, SOURCES, AND FIELDS Volume III
ADVANCED BOOKCLASSICS David Pines, Series Editor Anderson, P.W., Bask Notions of Condensed Matter Physics Quantum Mechanics, Third Edition Bethe H. and Jackiw, R., Inr Feynman, R., Photon-Hadron Interactions Feynmm, R., Quantum Ekctrodynamicr Feynman, R., S ~ t i ~ t i Mechnnics cd 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 of s 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 Schwinger, J ., Particles, Sources, and Fields, Volume III
ULIAN SCHWINGER late, University of California at Los Angeles
PERSEZIS BOOKS Reding, Massachusetts
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Copyright 43 1998, 1989, X973 by Perseus b o b Publishing, L.L.C. A11 rights reserved. No parr of this publication may be reproduced, stored in a rerrieval system, or tmnsmitted, in any fom or by any meam, electronic, mechanical, photocopying, recording, or delwise, without the prior witten pemission of the publisher. F"rinud in the United States of America. Perseus h k s is a mernlber of the Perseus b Cover desip by Suzame Heiser
1 2 3 4 5 6 1 8 9 1Q-EB-0201009998 First printing, September 1998
h Crow
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 monograph. 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 i s 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, gravitational 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~ver~ty niwzsity of C&fo&% md Profeswr of Physia at the S Los hgeles since: 19'72, was born in New York City on U~versity Febmw 12, 1918. hofesar his PhD. in physics from & l m b b 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 , schvviiager has tau@t at to t e a c b g at the U~versityof G a l i f o ~ a Professor firdue University (1941-431, suld at H w m d U~versity (2945-72). Dr. w a a R e ~ m c hAswiatr: at the U~versityaf G&EoMh Berkeley, m& ai Saff Member of the Mmsa~husf=ttsInstitute of TwboXot%ybdiatian Labaratav. fn IS165 hafessor %hknger b c m e a ca-resigient ( ~ t 1Richmd h in Physics for work in qumtm A Na~onafRese 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 ~ dSociety (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 nAcademy of Ackevement Awwd (1987).
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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 famation, 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~tionswere written in 1973, but remained in partially t m d fom for fifteen yews. I m ag& indebred ta Mr. Ronald B o b , 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, G l v n r i a April 1988
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Contents
Itvo-Particle Pnterac?im, N~n-relariui$tic Disiseussion Two-ParficleIjzter~ctiom.Xielatiuistic m e ~ r fy M-Paaicle I~ferac?ions: Relativistic me~ry11 Phatm Propagation FmcfionIl P ~ 8 i t r ~ ~Muonim im. Str~ngMapetie Fie[& Eleetr~nMagnetic:M~ment Photon Propagation Fmction 111 Photon Decay of the P~QH. A Confrontation
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I f pcan"&in 'em, beat "em*
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Par
9
Sources,
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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&~duatipropagation functions:
Ilftibing the indi~duafSfferentid 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 ( 6 1 . 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&~daap J b i c l e fiel&,
n e n , the two-pa~iclefield, defi n d under noninteraction circumstants by
The basic characte~sticof 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~ationhxnction 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 -- $7expf-- i(TI + G)@- l')] &frl- rf1)4%- F'%).
The fmction defind by
(&I.f2)
Two-particle f neermfonr. Non-rcilatlvistic d fscuasion
3
then obeys
which is a more familiar two-particle generalization of the one-pa&icle Green's function equation of Eq, (&l,(i), To examine further the re1ation between the two t y p s af propagation functions, it is convenient ta adopt a matrix notation with regard to spatial variables, while time variables are made explicit. Thus we present the Eqs, (5-1 ' 4 ) and (5-1.12, 13) as
and if) G1+%(ts
- ig(t - t') expf-
i(T1
+ I",)(t
---
g')],
(5-1 ,l@)
where the latter notation, Gx,zl emphasizes that tbe equal time version regards particles 1 and 2 as parts of a single system. Suppose, for example, that t, > t2 and t$ > >'%, Then
x expf- iT2(it\
- kf2)"j
Z'G~(tlitt~)G1+2(t%t t'lj)G2(tf1at ' ~ ) 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 ~ ~ + as ~ (ant )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 argumentsof the functions. With the analogous source definition
S1
Two-particle Entarzctfono. 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@eit in detail, let us write out the sources, using Eq. (5-1 -19):
where each propagation function can be represented as
G(t,t"
dE exp[-- iE(t -- t')] -, 2n E ia - T '
m
+
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
Chap. 5
Cl
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 f o m 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($
G(r - r', 6 - f)* = G(rt - r, t'-
- l')])E - 7(pj g),
(5-1.42)
(&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 must y 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 function 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 prduces an integral equation that deintegration sic, of #(l7)lJ"(23by #(l%') scribes 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 o b t ~ n e directly d 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, (5-135)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,
Chap. 5
CE - T, -- I",
--.
V(r1 - r%Zl#(rlr,~) = v(flrzt)*
(5-1 .sr)
The Green" function solution is
where
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
by the (more generally stated) relation Aceor&ngly, if we now define G1+% G(tltz* t'lt") tXxcl;t
-
&(I,, t,)G,,%(t,,c, tf,)6(t',, t',f,
function obeys the integraf equation
( 6 1,m)
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~zethe 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-X1 ,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 $(2) would then provide a complete description. That suggests the introduction of the field
ORcombining the differential equations (&l'8) and (6-1.46) we infer the folfowing equation for this difference field,
10
El-mdymmfict
II
Chap. S
[ ( E -- T),(E -- T ) , - iV(i2)]~(12)= iV(12)$(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~nciplewill 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 %*(l21is 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 $(l)and tlf(2). Then, we have
and
The effect of the quartic term in (6-1.79). which we are now verifying, has been to substitute, for the Green's function of two noninteracting particles, the function G(12, 1'2') that contains the full account of the interaction. In the situation under discussion, where interactions are instantaneous, the action principle can also be formulated using the equal time field +(r,rd), or rather,
The structure (5-1.68) is maintained, where we might now write
but
The X field equations are [V = V(rl - re)]
and
which are solved by
= ( d )d
) dt ( ) ( t )V
The use of either solution presents W, as
-)
G ( l ,)
(61.86)
Twouparticle inteructlons. Non~retotlvfsttcdiscussion
t3
The addition of the primitive interaction intraduces the combination (matrix natation is wed)
where the latter form involves the differential equation of (S1.W). The aiternative version of this equation,
completes the elimination of T/" ta give
=
( E - T, - T,)[G,,,(k, t') - iG,(t, tt)Gg(l,t')](E' -- T1- Td. (61.90)
The last form is written out as
As discussed before, we now use fields that obey field equations withoat interxtian. As a result,
where the added term again serves to intmduce the Green's function of the interacting system in the relevant term of the vacuum amplitude [cf. Eq. (5-l.$@)]. 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
r p1
mx
+ rM
-+ m%)
,
=%P+P, M
rz = ft - m -r,x M p2
= %P-_,
(5-1.95)
we infer that T,
m,
P
=Mm+
l M ~ *T~=-----PM f z---P 2m1
m,
,
P8
2M
I M
e
p
+ 2%, --, P
(5--1.96)
wfiich implies the kmiliar decomposition of the total kinetic energy, lt)2
p2
l"l+I"z=:-+-=Tp+T, 2M 2p
(5-1.97)
f/p = llml -I- lfma
(Sf.88)
where
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(honarma1ity 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";Zb multipged by $;#(r) and integrated. In view of the adjoint form of (5-1.104).
that gives
The solution af this Green" ffunction equation is
Use of the completeness properly, as expressed by
then supplies the eigenfunction eonstmction
and, conversely, the completeness property is recovered by eampa~sonof (5-1.l10) with tbe limiting value deduced from the differential equation and the retarded boundafy condition :
The center of mms motion can be reinstated, in accordance with Eq, (6-1.99), to produce the Green's function expansion
where
These eigenfunctions have the arthonormality property
We can also write
with
Eigenfunction expansions for the multi-time Green" functions will now be cansidered. We reed1 that ( t , > 4")
where
and the matrix notation implies integrations over all spatial coordinates, The introduction of the expansion (lE-1.115) for Gl,,(t,, 6") gives
with
Two-particfs interactions. Non-relativistic discussion
5-1
The factors of
2
?7
have been introduced so that we shall have
.As the construction (5-1.1 f 9) suggests, the multi-time functions are eigenfunctions of the homogeneous version of the Green's function equation in (61.48). To verify that, consider, fczf example (the quantum number labels are omitted),
- sf(4, - t,)($(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)]#(~1tlr2t~) (G1.126) 0 8
and a similar procedure shows that
AnaIogous operations can be exploited t o 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 eigenfunctims 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~ablez i s required. We note that
where z is r e q ~ r dto 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- T 2 , the ordinav o&lrronom&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/at),
~,=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@dtranslation of both the v&abls, which leaves z fixed. The: remnGning vac~ablesin 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 f o m 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 e a n s ,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%"'t612zF"(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 hto threedimension& 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,tf1)G2(tf,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(tztt") - 2"G2(82,tl)Gft(tlPtill
In e t i n g the latter form we have useta the vanishing of the p r d a c t 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 t e r n 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, a c c o r h g 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 f o m of (6-1 .165) is c ~ n e c[cf. t Eq. (5-1. l)],but the complete phenomenolagicaf 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 ~ effective1y 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,
Two-particle Intttractions. Non*rttlaiQlvisffcdiscussion
5-1
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) :
G(12, f2') = exp
S 6 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-mKznematzcs and Dynamics, Section 7.9.3 According to the equations illustrated in (&1.174), we have
(E -- II"),(E - T)%G(12,f 3')
+ 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')
=
l (d(1, 1')
(&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 Zg=tXI 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 .l79). 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
Next,
W
(5-1.194)
Gbl*(rlrzltrilrfZt')
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) = (Ij i ) S(rl - i $(r2 - E%),
(&L. 198)
we have
Lim i6(rlr&, 5,fZ-Z) ~ = itlE(rl l E-+#
- 2,) &(rz- g2),
The outcome far the ~ght-handside of (6-1.193) is
(6-1.1@9)
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;inningwith
the quadratic term in the expansion of exp[zWnoni,,Jis
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 eircumstances)
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~ginatein the expansion of the interaction expression
they are given a more compact and unified presentation in the notation of functional derivatives :
1
eff.
=
8
78AL(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 p h t o n propagation iunctions af a certain class of gauges, Tbat is expresse-d here by writing
30
E l W M y n a m l a 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~alledlater, 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, G+(RI - ~'I)G+ (2% -- g",). a;
(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~ptionof these processes is dven in (62.6); it is the omission crl[ f o m 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~onsreplaced 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 t q e s af deriving a differential equation
The next step involves the reanangement
which, ~ t a hsimilar statement refer~ngto Az9,lea&. to
wbere 1[1Y(.1.2)
=
- i(q3tUI1D+@I-- %)m ( v W ) z *
(6-2.16)
The differentid rebtion inferred from Eq. (&2,11),
(?U + m)bG,"(x, x') is solved by
=
w 4 A (x)GtA(x, x')),
(6-2.17)
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 ($2.2) ,] Accordingly, the right-hand side of (62.15) becomes
in adaw with (G1.185,X86). But here the resmblance to the mnrelativistic discussion ceases, The photon propagation function does not transmit an instantaneous interaction, and the paseide propagation functions da not obey retard& boundary conditions. The apparance of four padicle propqation, functions in (62.20)means that new classes of propagation fvnctims are: being introduced in the process of finding G,(x,xz, X ' ~ X ' ~ ) Thus, . the two-paPticb equation of nonrelativistic theory has no strict eaunteqart in the relativistic domain, except in the inexact sense of approximation =hemm that relate (g2.20) directly to the two-particle Green" function. An illustration of this erneon distinguishing the propqation function factor C$(t, x',)G$([', g',) from C$(xl. 6)E$(xp. E''). The former describes the propagation of the p&icles from their initial ereation re@on up to the domain of the two-photon exchmge process considered in (&2.20), while the latter represents the p&icles i5u~ngthe interaction process, It is plausible that circumstances &odd exist: where the additianaf interactions between the particles (symbolized by exp[ 3) during tbe tm-photon exchange process wuld be relatively negligible, whereas they certainly cannot be ignored throughout the previous history of the p d & s . Accepting this arpment gives the f o l i o ~ n gappraxirnate evaluation for (&2.20)8
in which: IcB)(xz.28%l$%)
-- - (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
(5-2.26)
gauge terms,
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 pdependent 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 ~ ,xtl) 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)]
I
AO(xt) = (dx')g(x - x8)JO(x8t), -
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
(a,' Thus,
i + lkll-2lk1 =p(-
ilk1
- F I ) = &(+Q - F ) .
(62.37)
(&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 t h o u 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 l e v a nteminal t ervd. The osciUlatory character of nsidtefing a very tong time 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, rough1y,
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 predominantly nonrelativistic situations, by shourl'ng two advantages, f t simplifies the problem of extracting phpically significant information, and. it improves the convergence of the sries (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 r n 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
=
z8
g
1
8
0
W .'
4-J C"
.g
C
tct"
0-".
, :
y
a-=. 6.1
l
'*
3 E .9
0 2,
II
GM
2 %
4
8 0
g5
;iia
$ 8 2 c, 3
.S $j
"C"
9 5c:
*G
U)
P 3
$2
C
cl(
3;
0
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)- = = ii 8(xi0 8(xi0-xe")ri~g0V(~i xe")ri~g0V(~i - ~~~ 222 ))) ~~~ (5-2.59) 4S(xix2)with with
where we have returned to the use of the Hennitian matrices The equivalent equivalent integral integral equation equation is is The
GglL = = GIG, GIG, 44- GlG24V*. GlG24V*. = GglL We shall work with the equal-time functions CV3
CO
0 " .
W
3
The corresponding corresponding specialization specialization of of the the integral integral equation equation is is The
Q;; @ :
Y
N
U
.-t?a-=.
k
H
*rl i-i
4
d?
U
U
F( Fi
W
ll
2
bV
U
#-+
-5
6)
$4
V
C
0.I
Z
W
S and and
We proceed to convert this into a differential equation by using the following X'), form of.the Dirac equation obeyed by G+(%- X'), X'), (- ii a,, a,, ((-
H)G+(x++H)G+(x
X') X') X')
= yO yO b(# b(# =
- fi) fi)b(x b(x - X'), X'), -
where
Thus, we have
- HI
(iat
-- He)Gi+e(~lxd, -x'lx8d')naatnt. --
(5-2.66) (5-2.66)
The value of each G,(lrh, x2t) i s computed as an average of the two limits, t & 0:
- 6'-
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 ~~l ~2' ~" t~t ' ) n o n i a t .
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
&lsrtrodynamfuIt
Chap, S
where G(x1xZ, x * ~ x t' ~ )'1 is the retarded Green" function that o h y s
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.
d(t - t")a(ylo t-. yzOf 6(x1 -- X",)
&(X,
~'~~'~~')~i.t.
.--. X"),
(g2.81)
In a representation where both yla and yso are diagonal matrices,with the e i p 1, only two possibilities aplpear, vdues
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 Using the symbols and - to indicate the common value of y,@" yza'in row and column indices, and employing threedimensional coordinate matrix notation, we write out (5-286) as the twa pairs of equations
+
G,,.
and (;at
t- trJ, -+-
(aa, -
- W,
+ P--)G"-- + P-+G,-
==
- d(t
- P+,)G+- ~r,-i=--
m
0.
--.
Before continuing we must note that
v++ = v-- = V & and that it is possible t o anan@ matters so that
--..
tf),
( k 2-891
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 contain no y5 matrices, and an individual term may .have no y@matrices, or the factor
[email protected] yO matrices are absent, for the origin of any y@is in the matrix
v--
y = iy0y,cr,
as distinguished from and no single y5 matrix survives. Since only yl@yza-+ 1 sccurs, there is no distinctton between P,+ and P--,as claimed. The submatrices P,- and P-, come from the part of P that is proportional to Y ~ ~where Y ~ 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", together with
This canstmction is e v e n symbc3licalfy by
(&2 .97)
5-2
T w ~ i c interactions. k 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,
- t')
G--(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, dtdr'q(t
+t' - (t - t')) exp(-
iEr) exp[- iE'(t
+ t' - (1- t'))] exp(-
iE"r'),
l0
which, using the variables
becomes the integral ds exp[- iE(?T + S)] exp[- iE'(T
6-t'
dT-
i [exp(- iET) E -E"
- exp(-
- (t - t'))] exp[-
iE"T)] exp[- iE'(T
i E (+T - S)]
- (t -t'))]
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 H Q [Eq. (62.94)Jwill be ctjiscussedl in an essentially non-relativistic context, That is to say, on1y 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 con1binations 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-@ * WX @ P,
(5-2.116)
and
For the latter caIculation, where V @= a l * A * a g l
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, * p,:
4.
(a1 PI.
{o~*.Pz. v,))= PI* A * PZ
+2 ~ 1
+ 2 a z * V zX A e p l
X
A * pg
+ (al X V1)(aZX ' f l e ) : I I ; (62.2 118)
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 eircumstances 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, of the viriail theorem stated in (&g, 127). To be verified is the dyadic e;~tneralization The proof folfows directly from a simple generalization of the scale transforrnatio1.1 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
8r =
l S
fr, G'j
BP
B ~ K :F,.
=
1. 7 [P,G] = -- BK * p.
The induced change in the operator H, f
&H = 7[H,C], is given by
(&2.1$1)
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 t h e hydrogen atom and muonium (p-++ e-1, where
When second and higher powers of nzlfil are neglected, the energy operator W O 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 c o r ~ c t e denergy 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
Alternative1y, one can write
50
Llectrodytsamfcsl #
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 s p c t m m 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
S1
accor&ng to Eqs, (4-11.104, 1071, and
which uses (4-1l,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 t e r n of (5-2,154), ma4/2n3,with the delta function term
The reXativisti~correction exhibited in Eq, (S2.158) describes the splitting 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 of the 2e2 degenerate levels of quantum number .rz into
+
E and get
To the extent that
=m (g
*
g=
+
(&2,1SO)
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 it 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 f o m 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 2 5 , 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 We need its eigenvalum,
+ l) still appeaxs as an oprattor.
S4
Chap. 5
Elmrodynamlcs ll
Concerning the angular momentum propedies of spin 4 particles, we know [Section 2-71 that a given total axlmlar momentum quantum n u m k r j can be r e a l i ~ dwith 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 1fr2in (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-
which, af 2% -.. 0, reduce to j (k2.1'76) are thus completed by
-+ 9, j - 3,
f ) , with
respectively. The comespondences
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 ~ )i!'(I'~ + l),
(6-2.182)
+ - (2a)T1ig--- +,
(45-2.183)
X
and t"
E
[(I
+3
The camespnding enerw fomula i s again (5-2.181), but with the integer quantity, j replaced by the integer I. In the spciaf. circumstance n = 1, j = the formula (62.181) reduces to
9,
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 a p p a r s of the form
m2
a5 -
M
m -- or (fine structure), M
The next section is devoted to evaluating effects of this type, which are contain4 in the full theory, 5-3
TWO-PAWT1CLE INTERACTIQ NS. RELATIVISTIC T W EORV E I
We begin thissection by demonstrating that there are energy shiftsofthe 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, on1y 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 (am)3,this effect is of the order of (mjM)2a(cr4m),and is $(@)l2 omitted. The comparison of interest is that between the eigenvalues of the energy opratar
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,
-+
E=E,+
1
R ( + l f -m k
+ W (p)
--
/pP/2
( k 3 . f I)
This determines the E. spectru~nfrom the known E spectrum, The eigenftxnctions of H. [cl, Eq, (5-2.136)] are approximately those of the non-relativistic system with the reduced mass p fEq. (5-2.162)], In the application which is dominated by relativistic energies m [rather than M, to (5-3.llf, 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.lyand 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 'd W(W m)%=
1 P (cosh @ +Tm W
W -- m dp W(W m)%=
cwhIf--1 X (cosh 0 11% =G
+
+
+
(&3*k7)
The rwult, stated for an s-state of p ~ n e i p a quantum l 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 & f i tm / 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~repancyis 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 devefopment 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 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 =I=
The implied differential equation for G, gresnted in symbolic notation, is
which reduces to (h2.85) when one sets xI0' = xZa" version of this equation one has
t', In the transform&
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 ( s 3 . 2 3 ) 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 intexactian, 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
- f2)
8(t2 - ttX)
- t2),
(G$.39)
the terms in (5-3.38) tha"tnvo1ve the Green's functions of different particks do not survive, as illustrated by
cl,,. ft - tl)~,,,.(t, - t ) = 0, This reduces (63.38) to
If &F', were an instantaneous interaction, its expectation value, a threedimensional coordinate integration, would estimate the energy displacement that it induces. But the actual structure involves a tirne inteeation,
and one must recognize that, when a particular state of energy E is consi&redl, G,,(f, t8)effectively differs from G,,($, t') by the phase factor
Accardingly, 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, only relativistic energies contfibute to the efhct under consideration. That enables the energy of the state to be approximated, E g M m; the which reduces the unitary transformainternal wave function is replaced by It(@), tion 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 r n is @xg[-- i2Mz], which contlributes a factor 1JMon 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 i n t s r a 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 e v e s
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 spacetime extrapoltati~nis 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, It 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 ( k 3 . 6 7 ) according ta
Chap, f
(E- W)p -.+[E - H , p] = - iVV, PIE W
~
- H)
3
[p,E
- W ] = iVY,
(G3.68)
cotnve~s C ~ it into
: 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 ~ [Eq, n (4-1 [1,M)]
which, incidentagy, is v e ~ f i das 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
X IC
kg
+ ~~p~
"'"
A log 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 with d the aid of the axial vector ]I"
A, = I
+ a,+(&
x p - g x L).
(k3.86)
68
Blccrttody~amfnt 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~zedmultiplication 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 t a 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~mponentsof vector :
a,
it 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 ~ the s &mensionless variables for llzof ($3.76) tben gives
2. The application
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 dby the twoparticle extension of (&l1.30), that dominates the non-refati~sticsituation,
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~ribingthe 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 1JMenables 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 QYYna, 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 e m i n a t d by relati.vistic effects at kO m, and by the finite momentum assaciated with the bomd state at k@ Ern, Hence, PW
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 integratian 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 are p-erfomed first :
S,
where the latter f o m is produced by partial intepation. The logarithmic depndence an, Y is a Consequence of the vy combination in (G3.fX.7)far small values ofv. 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+t2
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 expf -- 2%)C 1 + 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
E l ~ t r d y m m f ctls
74
Chap. S
The generating function (63.114) is thus found to be
and its eonsquenctl: far n = 2 it;
(6v'"22,
1 m -- a;n: M
-2L 37 4 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. ( s 2 . 1 2 )*ere, in the radiationgauge,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%rentidagerator :
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 p a ~ i c u l a reigenfunction 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')]
--.
2
f8(%---X ' )
----S(X - g)]G+&(x,n'),
(b-3.148)
which is a version of the divergence equation [cf. Eq, (W,48)]
a, C.lf;#(x)?@~"e9#(x)l m i$f ~ ) $ ~ 1 1 ( % )* Consider the situation with xQ
(5-3.149)
> 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. I50) 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 ~ @ nthe , 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~bedif 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 1521, be large compred to xO -- xQ" If while campletely including factor of (H, this interval, With tbis eEmination of end effects, the functional operator simply replaces q A Qin 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 i n t e ~ a t i o nhere 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 threedimensional 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 c o u n t e ~ a r employing a real photon. In. t 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 np r w a s 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 subsequently detected by the extended photon source, There are two p s i b G i t i a 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~atem d f i e a t i ~ n s 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 r w e s 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~atee a u 4 controh, must %we both functions,
Photon propagatfon function t E
5-4
8t
The three-particle exchange pracess is a new feature of such problems, and we discuss it 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,
B(P' - P, - P,) = &f' -- P' - p,) dmpt $MtP(2~1%
- p,
- p,). (5-4.2)
Carrying out the successive integrations for two particle systems gives
X(M, m,, m,, m,)
=c
dM 'B 2( M ' , mm,, m,) ( 2 ~ ) "dm,, dWc (j(P - P' - p,) dM* 2r(Mf,m,, 111-,)IfM, M'$ 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 t a
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?rt l
Chap, S
which evaluation uses the variable
with
l, is indeed given by (W.41, The asymptotic khavior for M 2>r (2m)2, ZIO while near the threshold, M 2 2 (2mI2, va <
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 p r f o m a n c e 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 a p p a r s as
where it is understood, that the u-parameters are positive.. The spectral restrictions are evident here, in parametric form:
~ ~ 1g , 2 9 ,
1.
(W.14)
The anticipated threshold mass
M@ = 2 mm,
(M. 15)
can be used to rewrite (M.14) as
and the possibility of attajning the indicated lower limit, with
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 . 1 4 ) . 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 d%, d%&&gefl(1 - "U, ( 4 ~ 1 ~
m-------
- 2aCb - H,) -
m M,
-
mb2
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
Chap. 5
E I u t m d p u n i a I1
For the situation of two vanishing masses we omit the final integration of (64.21),
The resulting single parameter integral,
is equivalent to ( 6 4 . 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,
- V),
=~ a ( i
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
Photon propagation luncfllon II
5-4
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 W z z 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~rvationand gauge invariance statements
86
Blmrodynomlers )l
Chap, S
- &.(--P')K,(-- $)l,"- 4.,I
= 8ePAg,(--
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
- p - p' - ~)VI,AVE% ((5-4.43)
dw, do,.dw, ( 2 ~ ) &(K '
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 k h a s a factor does not contribute to the required produet ,
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
W
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 II r
Mq W
1 - 4 -
1 [ ( W - M'?' W
Chap. 5
2 - 4paM9ln g [(M - M')' - /8j". M
(-.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*
2M'
Ikl =
1 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
Photon propagation function II
5-4
In the region M
- M'
^I
p it becomes
"
I
- ' [ l - (4m2/M7][(M-M')' - p7
(M - M')'
1 (M - Mt)[(M- M')'
-~ $ 3 ' ~ Jf - M' + [ l - (4m2/W)11"[(M - M')' - p$31fl X logM - M' - [l - (4m2/W)l1a[(M- M')" p711'
M2
U9
(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" ' wa(a)(m=X-I 12* ~t
h
2 'l2
( l ) ' ( - )
(-*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 83
(vPv~,,)that is appropriate to M - M'
p
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 m g e s from M"
22m to M"
M
p<<8iM<<m,
or, from v' = 0 Oto v' = v
- --M, where (M. 72)
- -.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 d(M' m2
-- 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 . 7 8 ) 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~iglionof the individual emission and absorption acts, since there is completenon-locality (propagation)under the energeticcircumstances 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 a n d o p e 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 ~ t a b l y 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
1
(1
vl% + v18) log 1-
9 Then, using the fact that
we have
(H,&?
The resulting coefficient of a2/f2$ in (W.86) cancels the photon mass t e r n 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 p r o p % 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 0p2+gy#z
1,
1 - vf2
dv2 -= log -= log 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), that
we first observe, through pa&iaI intwatian,
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
dar ($
+
v", - vOg(u))(l f vq lag X - v"
(M.IW)
For the remaining integrations, we perform various partial integrations, as illwtrated by
(W.10f)
we exmples of the inteuab
md note the spif:i:e reu1.t
The outcome is expressed by
This increse in the vacuum plla~zatianeffect 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 h a w stmetares eontaininf: a denominator and a logarithm which are different linear functbns of one vadabb. A standard function of this type is
+
O<%<
X:
I(%) =
" dt
-log-
f,
t
a2 It X" = l-t .=l nB'
2-
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
+
Llf prdaces
The function that effectively appears on changing the sign of
and the analogous relation for x
X
is
> 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 yielding
4
-log t = log x log(%- 1) This is given different forms dependingupon whether x 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
+ t in
(5-4.108),
+t).
(64.113)
- 1 is greater or less than 1.
- 1) + +J((X - I ) % ) , (84.114)
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 [ ,,V-v' N
+
v
-
- V' log V'
The substitution v' = vt then brings these integrals to the form
Another consequence, produced by the substitution v
- -1
v, v'
-,1
and, incidentally,
according to (64.106). Changing the denominator in (64.120). we have
- v', is
Chap. 5
-
fog "v
38 - --X -
=g
(W.12q
2
6
Also required i s
Now it i s the transfomactian I dt ,(l+a) 6
+ v'=
(1 1
+ w ) t that produe@ the f s m
-log -41
(%+@l&
Flog(t
- l)
Fadhemore, we have
-lag(1 - v"
w ~ c huses the transfomation v ' = l "v - 1 + ((1 -- v)$,
/4l+of Q
& I -log t 1-t
- (1 + v)$, arid, with the transfamation
l+"[ 28
I -
-*+- (
log- 1 ; v T
- 2log2log- ll -+vv + #(v) - l(*
-;(
1 ; ~ ) ~ log -
- 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-lwI+v
Il -+ vv
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 ascomes 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. According to the non-relativistic relation
we have
which ixlded identifies (-.l%) with the mscfiification factar of (M.13.4). Incihntally, the elastic fomfactor itself, in the non-refativistic limit, is emntiaHy identicd with the wave function for relative rmtcr-tionin 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~valentampfitude
Photon propagationfunction 11
5-4
101
we infer the effective source
-r
l t ( ~ ' Jl .A
P
4'4%- €)yAG+(%- x')YAz(x') +yA2(x)G+(x-x')yAd(x'-
€)l.
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
(64.144)
and tr, inacatef the trace, so nomalized that (M.xar
trs 1 = 1"
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 r(P + fk) 3-
l +~"yy(~'+k)+nr 1
+ R) + m The matdx factors in brackets are reduced ~ nnatriw in (W.153). Thus,
The matttrix prduGt af tM.153) then becomes
yd
1
+&) + m Y v (W.1 m)
t the h aid of the projection
We specifically note the appearance of the term
which is the expected, infra-red sensitive, radiative modification of the twoparticle exchange mecfianism, There are two t y p of terms in (54,158) that involve a pair af yk factors. One is yilykYV(~ - yp)ylykyv(-- m = r?hk[zvlvkvP
- ?P')
+ (m + r P ) y Y y i y ~ y(-y l m - yp') = 0.
(5-4.158)
which holds since yA?/kyAyk = 2(yk)% 0, =t
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 dto
the last step records the result of the trace operation. An example of a t e r n 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 ( 5 4 . 165) mduces t o (M,167)
since tr, yAyB =
- AB,
while, in ( 5 4 . l 6 6 ) ,we encounter tfn(m - yfi)yk(m - cyp" = -
trPI(y&k -t- Y ~ Y * ~ )
-- m(Pk 3-P'&
(H169) .
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
This gives the following contribution to the coefficient of
- M' > p, where
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 M" 4 d -- N 3 29133, for the factor that expresses the different form of acZ)[(M2f, where the additional factor of 4 is used to replaw ~ ~ / 1by 2 nor2/3na. ~ Thus, with the multiplicative substitutioll in Eq, (5-4.80)of vZ -- 4(3 - v%), we get the following addition t o (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, ( L 3 . 2 0 , 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 for (W.]L78),
3cx
Fx2(M2 + 2mg)+ F2M2, 2n:
- m,
This gives,
(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 2% with [cf, Eqs, ( M . 6 8 , 77)]
(W.183)
Photon propagation function 11
107
The consequent change in W U ( ~ ( W= ) fivt(3 3n
- fl
(M.186)
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, terms in (6-4.176):
- 8, gives the integrated value of
the non-singular
For the integral of (8-4.176), we observe that
The sum of the integrals of (6-4.176) and (M.176). from which $M cancels, is then
(64.191) where we have introduced the numerical values of all the integrals of type ( M .102),
dv vL~(v),n 2 1. As for the integral of (64.187), its contribution to the coefficient of aa/3# is
(84. 192) which uses the principal value integral
- Vt) - v'%
Vt(3 v2
8 --v*3
V*@
--
vY)x(v').
The sum of (64.191) and (M.192), from which the fictitious photon mass finally cancels, is
Some significant combinations for this evaluation are
and
The result is expressed by
This fractional increase is somewhat smaller than in the spin 0 situation, bat it is still roughly one percent. The effect on the added constants of the energy displacement calculation is given by
where the unit [Eq.(4-1 1.114)Jis 136.6 MHz. This represents a decrease in the 2s-level splitting of 0.24 MHz. I t alters the last estimate, of Eq. (6-3.168). to
W
Photon propagattan funetfan I t
,,
-- E2
H:
= 1067.93 MHz,
l09
(M.199)
which is strikingly close to the nominal experimental value of 1057.90 & 0.10 MHz, The usual caveat a b u t still unconsidered effects continues to apply, however. The integrations required to exhibit &(Mz)are very similar to those of the spin O situation, Such a relationship also agpears in the results, far the substitution "v +(3 -- V%), performed in all the terms of (5-4.132) that have such a factor, yields the precise spin 4 munterpads, as dispfayed below: a W g ( M % )= -v4(3 - V%) 3z
li+v I+@ -+ log -yj"-- log -6 X.-v
+
v + 6 a ~- - -2- v " +log-
2
- 4%-$-V'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 gfrom the first bracket of the brace, with its origin in the f o m factor: Indeed, as was to be expected, the multiplicative is the s m e as with spin O [Eq, (H.136)]. A simpfe interpolalion faetor of (Mew) 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 :
X
l+v 3 log - -v(& l.-v S
+
3 3-v% - 3 ~ 2 )- -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
--#vE~(Q~E'--~)--*(*z'---~)],
@
(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%isfomd 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~ngproces 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~alionfunction 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 ghere 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 t2
Eleetrdynamlcs lt
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 (psitivenegative), 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~peticinteractions, and their instabitit y only involves decay into photons. In contrast, nnaoniurn invokes the weak interactions, which intro23". duces the neutrino: p+&- -.. e+ eThe 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
+ +
=2
X -RY 2n2
6.8029
=-
P??
ev.
me states af given principal quantum number $2 =.: f, 2, 3,. . . can be further lahled by the quantum n u m k r L -- 0, 1,2,. . . of relative orbital angular momentum, the spin quantum number S == 0, I, and the total angular momentum quantum number ] = 0, 1, 2,.. . . A particular state is designated as n'S+lL,. Relativistic effects and electromagnetic i~lteractionsother than the Coulomb attraction induce a fine stmcture splitting and a byperfSne stmeture splitting, Unlike hydrogen, with its Izge mass ratio, the fine and hyperfine stwetures in positronium are of the same order of m w i t u d e . Particularly interesting is the hyperfine stmcture of the punc-l state, the splitting between the X and 1 ISo itevels. Positroniurn atoms famed in excited states wiH radiatively decay down to one of the hyperfine levels of the ground state. These atoms eventudy annihilate completely into photons, We be@n with a dixussion of the annihilation mechanism. The nature of the photon decay of psitronium i s governed by a selection rule mociated with the concept of charge reflection, In general, charge reflection (Q -. - Q) converts a 8ven state into a different one. But, far eltectrically neutrd systems, another state: of the same kind is produced and one can introduce the eigenvedars of the charge reflection operation. With two particles of opposite charge, as in posifronium, there is a symmetrical and an antisymmett-ical cambination of the two ehwge amignments, corraponding to
Now, the effect of interchanging all a t t ~ b u t e sof the tvvo padicles is controlled by the statistics of the particles, which, for F.D. particles, demands a net sign ehmge. When the spatial coordinates are interchanged in a state of orbital quantum number L, the spherical harmonic pveming the ang!e depenknce resgonds with the factor ( - 1)" [cf. Eq. ( s 7 . 2 1l]. As for the spin functions, tfiplet and singlet states are, respectively, symmetx*ieaf and antisymmet~eal,as symbolized by the factor --- (-- l)$. 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 yof an n-photon state is
Hence, if charge parity is to be maintained in time, the I. ISostate, 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 t o that of 1 Sopositronium. There is mother reflection asyzect of thew states that desewes mention, f l: refers to space parity. The space reflection m a t ~ xis [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 %Sxparticle, 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 g a u e 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 e precise combination for a pair of photons can be famed, We shall later d e ~ v the 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~entationsmust 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 1.113). 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 appearshere, a specialization of $(X)$($), is to be replaced by the two-particle field of the interacting The field of a given para positronium atom paums a rather system, )(%X'). simple*non-relativistic character. For the two spinor indices we have, effectively, yO' = 1; the charge labels in this C = 1state 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, $(r), and a wave function for center of mass motion [cf. Eqs. ( 6 1.l 13, 116)] which in the rest frame is
116
Elardyntmrfcs ll
Chap. S
since the m- of psitranium is very close to 2%. We mwt f i n d y remmk that the normalization condition for two identical padicles cont&ij,s tbe factor to avoid repetitious counting. Thus, the field iated with. a s p i f i c atom is indicatd by
+
+
f restfiCtion is implicit. Now, the a n t i s p m e t ~ c dmatrix where the yO" yOys anticommute with yO, but connmutes with the charge a d spin matrices, Accordingly, it connects equal values of yO and oppsite v d u a of chwge and spin quantum numbers, since dIt the asmciated matrices we antispmetricd. fll"his is illustrated by wy6q$ = w y S # = (--. q+pyfJ$.] fn the yO' = $- L subspace, the a n t i s ~ m e t r i c df y f Jma.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 . f 3 ) , The vacuum amplitude that descxiberj three-photon decay is @I. Eq. (%X%%)]
We exhibit the coefficient of iGi];ld~:,l,t, With the p h c f c f field taken as # e x p [ i p ~ , g $$, mB =;; 0:
-
(dx) expfi(2fio k - k"
- R"")]
.)lBi)wq
(dwkdwh.dwL.
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 = "Ipy?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
"PE
==
- 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 a p p a r s 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" - e * e""y @')l+ cycl. perm.
-+ -t-'
ef"y. e
(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
- a,+o, we encounter With the proportional identification of +wqykJIwith a pa&icular linear combination of the two stmctures given in Eqs. (5-5.11, 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 twophoton 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) (&mdcu,)llz =p@&),
(66.~)
and 4NxfPqv * y+(x) -+ *'*fi24L(Ofv
+(X).
(~5.45)
The decay rate is then inferred as
where, intrdueing unit vectors have
n,
along the photon propagation directions, we
!X,
E l e d y m m l c r It
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 him h been written: in the unsynnmetrieaI form p m i t t e d by the equivdence of a& the photons with wspct to the integration, This stwe of the calculation is
Qne @vks invharrt f o m ta 1 - n * a I
- n * n L- h a T
kk" hPk P
i k
--&g%
kk"
(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 integration gives
==
- kband h". Performing the z
Then, since dcu, dwkPe(%E)%8(K
+ k f r- PP)
;=;
1
(4%)"
the remaining integration is proporPianal to
in which 21
t;:
M2/4m2,
Successive padial integrations reduce the latter integal ta
(65.69)
Chap, 5
(5-5.62) which evduation makes use of yet another partial intepation to get f 1 dzc -log -- = @
3%
=X"
f + @
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 If
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 yaccurate 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~gondto 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. Accordingly, 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 levels: ortho positraniurn $JJstates relative to the para positronium 'lJ
+
+
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 orthopara 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~mentafvalue is
The agreement t a 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 fulfil1the 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 e n e r e , 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 slate since the an@= momentm level m = O of the %S1state is coupled with the
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
~ti-s.xo3)
f
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. t07)
128
E l a r d y n a m l a I1
Chap. 5
and
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 y,). weighted average 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 electronpositron 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 Y A(R)
W
a 11 F,(k)yA (4 -l5 Spg(k)flF(k) e
(&G. 112)
The propagation function calcuEation is immediate, The evduation for k2 = @ves
D(R)
1 - 4mg -
- 4mg
(&&. 113)
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 @venby the other factor of (&6.128), (W.X 24) Tbe complete ma&fication of the singge-lphoton annihilation part of the or-thupara 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 (Mt,27) with the momentum vctrsion
'The spaicetime stmcture of the p&icle fields is sufficiently. indicated by
#%(..l
rlrl(x) = exp(- i#$tx)tC",
=
exp(i4132x)Ib
(5--.5.129)
where, in the positronium rest frame,
PI0 == Pz@ 2
2n~.
(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 ymatrices in (6-5.134) meet this requirement, since
where the dots indieate other terms involving a single pmatrix. T h e 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
= $*" A partialar consequence
makes similar reference to the property is @ven by
To evaluate the momentum intepal of (&6,t36), we use the represntation
- is,(k -I- +$)T expf- iiiz(R - ifi)%expf-
ds, dsz dss sl, expf
OR introducing the! praprty
p% the caefficient of
-
: ,
&pp,
- i s in the expnent hconxes
The basic structure of the momentnm i n t w a l is, then,
is8(kS
+ m%)]
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
= g--
f
L
#g,, - I , ( 4 ~ 1 % m2
(w.r&)
where
On employing ( 6 5 .X 401, and intrducing 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;, c o ~ e s p n d i n gto the instabgity of the pa&icXe. n m ,
and (m,
g 2m)
For a consistency check k t h the earliea calcutilti~nthat 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~butiont-o Re I is found to b
S5
and Re I = g(1
- log 2).
This Efives, as the two-photon annihilation contribution,
amwra = - 4(f
ara - log 2) m%
z=-X--log2aa~y. 7t
(5-5.161)
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 h we have often explaited in single-pa&icle contexts, the [this is the use- of #(@)l+ 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 a n s v e r ~interact function tions of Eqs. fk3.131, 1321, but to work with the c o v a ~ a npropwatian
With the introduction of causal labefs for the pa&icle fields, the elflective source of (66.1126)&comes
The two padieles will be & t h e s h e d by s u h e ~ p t as and b, so that the vacaunr amplitucfe for two-photon exehmge is in&ieatd by
td;b
El-rodynmicr
El
Chop. 5
It suffices wain to use the simple form of the pa&icf e fields @venin 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 a m 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 s p a ~ evector of the latter result can then be r o t a t i o d y a v a w d ,
What emerges for the two terms of the vacuum amplitude (66.166)is
where, written covariantly,
(dk) I
11
(PR)'
1
=Iw(p) -r(Kp+&Y
and
(M. 172) The sum of the two terms is
(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 -pkv After the redefinition 4 in I is
= (R- &pu)r + m W
- 4pu -,K,
(66.176)
the basic momentum integral encountered
(MX.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 propagamP- ie)-l, evaluated for fie = m, is to be replaced by tion function (p' (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 dit 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
expP(iPzx)the f ree-pndicle form of the two-part,icle field We r e c o p k in $(X%) that is msociated with an emitted positroniurn atom. The field X refers similarly to a detected atom. What replaces them, in describing eq(the 'boand system, factors into a normalized center of mass function, which is removed by the spatial integration of (&5.183), and a wave function far relative motion that is evaluated at the origin : #(Q). The resulting coefficient of -- i dro in (66.183) is the desired energy shift,
with Q," CTb
=
para:
-- 3 '
This contribution to the o&-tfio-parnsplitting is, therefore,
There is one other effect to be considered at this level of description. It is the a/2a nnodjifimtion of the magnetic moment, which nruitiplieies (65.89) by
The v a ~ o weont~butionsto the ortho-para splitting of order contained in Eqs, (G6.125,f61, 186, 187) are ma& explicit in
a8
Ry, as
and the resulting modification of Eq. {&5,95) is
This represents ai decrease that is close ta Q percent, reducing the numefical value of (k5.95) to
E,,,,
-- E,,
= 2.0338 X l@MHz,
(6-6. fW )
which greatly improves the comparison with the experimental value even in. Eq, (5-5.96).
Salelrtea! Pa$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 ysimplified, Presumably one could, now go on to the next level of descfiption in much the same way I H[.: X see that you have reprduced the old KarpXus-Klein result [cf.
S. : f should think so, at least with regard to effects of relative order aslog 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~babtein 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, is replaced by
-+ I z )
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
-Bv2 T/13
m
1
+
I [(m, m , ) / 2 m(v, -jv,)
-
wherc4 the firnits of V i n t e ~ a t i o nare 3 -- m,)/Mjv-). This @-;v=
+ [(m,- m,)EWv+and - (1 i-[(m,
--.
The other neded inlegal, apparing in
They are combined in
P
--
1 [(m,- ms)/2W(V+ 2 [l [(m, n,)/&Ml(u,
-g&[ ] = -
+
-
"U-)
+ v-)l'
1
log 1
+ [(m, - m.)lWv+ + [(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 immediate1y yields
mid (M, ]Em)becomes i 2 m# log -- * ( 4 ~mBS ) ~ m, m,
(&5*2013)
-
The muanium energy c2ispfacement can be inferred from the positmnium result of
Eq,(S6.fM) 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/2nmodifiesthn 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 snff ice to give excellent agreement with 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 effectsof order asthat 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 will be useful in understanding Section 5.9. 5-6
STRONG M A G N E T I C 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,) the expmJsiCxl
= IQ = O)), conve& M into
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
Chop. S
EIoctrodymonricr II
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 SJ 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
Chap. S
Elutrodynamicr I1
of D - A,
and then by applying the a n -
A+A~=D+D~.
The main problem, the evaluation of (e-h), is now solved by devising a differential equation: i-
a ( e - ~ x= ) (e-kx)
a8
-
(c-'.x)ufl-
(c-'.xk)2ull
+ (c-'.xk2) + (e-'.x)u(m2 - e q u ~.)
W i t h 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 - 6 s ) into the form
and the d t i n g integral of (S-(eux)
is
-(
-
C det' 2:F)-1fie-".
In introducing the determinant we have made use of the differential property IEq. (4-83311 8 logdet'X
-
t f ( X - l b X ) = 8 tr'(1og X),
(66s)
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 (c-h)
- C-
1
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 complementarity w i t h 6 0. Accaadingly,the limiting structure is given by the elementary i-lim (4-8JjVI'
-
and
150
Eloctrodynomics II
The complete d
Chap.
S
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 redumto 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 -ds ds)
sin-
= 2ueqFr(s),
-
[Y,@FI -2iFy,
and the solution is
5-6
Strong magnetic fields
1S1
which is then combined w i t h 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 ) y n ] + 2(1-
u)yn
-
-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
where
- tc(yn+ m),
(6-666)
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 w i t h 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,
where the p a r t i e sign of q
-*
which quantity varie~from 1, at
rather, its inverse-as
l is irrelevant. On introducing the variable
X
-
0, to (l - u12, at
X
-
00.
The related
5-6
Strong magnetic fields
153
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
which enablee us to w
=
,
%Y2
--
iY, P
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,
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
(5-6.73)
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 edgmfunctions,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 energy eigenvalue p'
-
Accordingly, the Dirac equation referring to an 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
5-6
Strong magnetic fields
assigning to
S
-
157
Q%,
an eigenvalue, S' = f l, and, independently, introducing an eigenvalue for
The familiar one-dimdmal oeciliator problem provides the latter spactnrm,
and one i n f m 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 w i t h 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 differencesin 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 yOlprovidee 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 lIH2in 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'butionof 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,
W i t h inmiming magnetic field m g t h the total energy d e m e m m o n d c a l l y 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 e x h i i t M IEq. (6-Wl)] in the ground state, where
which makeo me of (6462). We give thia exprdon another fonn by the
5-6
Strong magnetic fields
161
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 is
U>
eH/m2.The initial tenn in the y-expadon of the brace in (5-6.119) and
-
+ l , re&atee (5-6.113). which, as the weak-fieldmagnetic moment term for The nerrt power in the expansion of the brace is displayed in
Chop. S
andhear,wenrsetan~-~I#obl~Butthe~~tid~oft '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, of
U
< U,,, the brace is w d e d in paw-
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
5-6
Strong magnetic fields
163
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~onricrII
Chop. S
The performance of the tbinbgd,undcir theee cimmm&mtm, givee
Thk leading mymptotic term is quite d&ieant to indicate that, far hmdahing 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 d t h 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 ynHin (5-6.81) u c q u h the fom pmmted by
and the ground-state combination of (S-6.116) mds
Shong magnetic fields
5-6
For the -era1
165
situation wheite IEq. (6-69Q]
the structure appearing in (66.81) becom8~
Thaq if one raer the projection matricee :(l f 5) to erpr~asthe (dependace of functions, and recalls that the function of 1multip1ying y& is to be averaged wer i b ( = *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. W 1 t h 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*
D*
-+* l
U
iu(1- u)x
- U($ - !U + u2)x2+
(5-6.149)
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 h a g h w y part of M, we have
Apart from wrnishing delta-function integra4 the three c l d y related ingrals encountered here are (X > 0)
&X
&*&ask-x2
-
r
-(l 2
- tA)()(2 - A),
whem ~ ( xis) the Heaviside unit step function
Thus, all the inbgmb that constitute ImM vanish for U > 2eH/m9.This p r o m y 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 r e d t , 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
[Notetao 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 1in the quadratic H-term. The three integrals that appear here are evaluated as real parts of cormponding complex in-
-
;: ;;1 -
Ref log-
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
5-6
Strong magnetic fields
171
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
dependence in the decay constant (S-6.1S9) is to be The &mce of 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 r e p d u c e 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' 1form 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 obwmatim 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-integralsthat appear h are of the forma
j0ao&- [ r i n A x - i n ( A - 2 ) r ]
-42-A)
(6-6.191)
X
and m&
0
[
X
A
ain X -X
(
-1
1
-(
2
- A).
(6-6.192)
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
D,
U
&(l
- u)x - U($ - f u + u2)x2
(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, + L4 ) ) . (5-6.199)
5-6
Strong magnetic fields
distinction need be made between S' and pmmt the last two terms of the brace as
in H3 terms,one can alao
Since no
175
In writing the last fanas, we have not troubled to keep poaitive p a s of U,, which we know w i 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-stateevaluation 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 will not be cansidered further. Now we proceed to the u-expadon for U < U,,. The ingredients are:
&X
+u2(7e*2k
- 2-."sinx +l-X
-e"k+e"k) &X
+
...
and
+2~(= X
sinx X
+ * g o .
(5-6203)
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: sinx &=X'sin2x coe2x-- sin 2x + 1 - coe3x + coe2x, X
X
and
also
--
Cl
sinx X
8inx coe3X-2-cosx+l, 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-
5-6
Strong magnetic fields
177
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 H2diseuasion, and we refrain from giving further details. The remaining structure in (5-6204) can be preeented as (c S l)
where 1
~,+&-~dn2~-2xeos~x--ai$2~. 2x
The X-integralsappearing here are effectively given by
and
The subt3equmt integrals wer the variable
(5-6m)
Chap. S
in which poeitive pwem of X,
-
U, are d t t e d ,
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 a d d i t i d 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 enerwhere 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), w i l l be dominated by the two expanential functions w i t h 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 w i t h 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 y, such that
variable
l#)
Chop. S
Eluirodynamicr I1
This amverts the bracket of (6-6XB) into
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
Strong magnetic fields
5-6
181
and the^
The integal that produced +rwas identified through the limit of large [,where y IKJCom- ==P=W~Y6
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)
A final property of interest, which exploits mother mcurrence relation,
The various integral evaluations are inserted in (6-6227) to give
(Gem)
Chap. S
For all pawe~tlyattainable artificial magnetic field strengths and electron emergiee,
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
L*
d€ tp-lK.(
€1
-
p-v
4
p+v
T )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 d a m i d limit the qmnestial function 4- iex(u)] of Eq.(6-64 will be rapidly oecillatory, and the major contribution
5-4
Strong magnetic fields
183
to the k-integral is umcentrated in the neighborhood of the stationary phase point,
This is not the momentum of a red photon - h 2 = -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 ,
c t i m t0 4 d u e s of U. We can now present (6-6.243) as the spectral integral
8 comhte~td
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 mIIi u
Chap. S
which appliee the integral [(6-6.!246)]
To vedfy that (5-6is 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 d t . The rotational frequency of the electron in the magnetic fie& as inferred from (6-6357') and the relation p = E v, is
5-6
Strong magnetic fields
185
This is the fundamental £requency of the c W c a l 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 velocity. Then, since
direction and the instantaneous
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 (= 1is d d e r e d negligiile in comparistm w i t h 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/oCin the terms that have an explicit u-faeta. Cancemning the quantum d o n terms that are independent of S', we note that
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 w i t h the subetituticm .ra e ~ a r s g u r o ~fgthe integral evaluati-
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 w i t h the hat two Xdepsadent terms of (5-6.277), m&Eq. (S-64, for example, by the additional factor (this procedure depends upon the circumStancex-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
-
e H w , 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-relativistict h x m s h n ~If~that ~ . were the whole story, we should arrive at (6-6.267), with the additional factor
- -
-
Electron mognetic moment
5-7
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 identificaticm of the spectral distribution leading to the power calculation of Eq. (6-6274) iscorrect,
This already wer10ng section w i 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 eapanaion of the el-
magnetic moment,
(a/2n)'
in a poaerdes
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 multiplicativecomction 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 dctmtribution, 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 i s t n i a n of which is dedbed by the weight fsctor CEq. (4-3.105)l
The replaceme~tof kg by k2 + M ' altas the function ~ ( uo 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~tlyaccurate for our
1
we get
- ( M2mM - m)'
l
+
+ ( M+ m)2 2mM
yn - M
1,
(5-7.14)
192
~octrodynomiuII
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'butionto 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~countermomentum 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
(4:-
-U
-
t, we get
1 -~ld(;1twG
1
- l)
- -(;
- l),
(5-7B)
again invoking (6-4.107). The complete contribution associated w i t h 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 w i t h 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 r a m 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. causal dhgmm here is also a rectangle, with virtual-photon Knee forming the top and bottom, and real-ellines 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 m e 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 m m g n b 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 into n s a single straight line, with the two photon graphic symboIs traced as intersecting arc&)
me
196
aechodycramicr l 4
Chap. S
m or of performing one n m - c a d calculation. It is the latter strategy that will be adopted her& What has just been d88~1'bedcan 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 Thus, we have the option of evaluating three distinct c a d
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 of th- particles are electron propagation function. The 80-
and iJ:(r')n(x)rOl,-
in which
+&,(X)
-
~l(~)Y~sqY"~(x+?P
(5-733)
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 roperator amtnian characterizing this two-photon exchange p C8g&
5-7
Electron magnetic moment
197
Them is a cl- r a l a t i d p with the structure of M [Eq.(6-6.3)], which dBBQibtssingle 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 8iU-j--
("') (24'
eWx~(kt) h') &4(k3 81M 8A( k') '
we can preesat (6-7.39) aB
(W
4M M , ( L ~m
1
1
1
+ m Yp + c
iesIm fin f- ~ L ) + m w y v y ( n- L -
~ Y ~ ,
(5-7.42)
or, interchanging k and K', m
v4 &Ap(-C)
-ies/-
("9
-
, l ( 2 r ) 4 y kn fill
1
1
- k - k 3 + mQPICfill- i t ) + m Y. + S . Y " *
198
Chop. S
Elutrodynamitr II
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. y,, structure in M@ mpmmts the exchange of a photon with The y" 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):
M m as a whole are without effect in this non-causalsituation, 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~tumk. That is descri'bed by the combination (5-7&), which requires the following contact term in MQ: (Nde that contact temil, added to
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 But the y' y, combination in no contact team appears for such the same first term has the form of (5-7.43), which demands the contact tenn
5-7
Electron mognetic moment
1W
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
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): h 0 8 the Single M structure d
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 mII i~
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
+A-(M)[
[
-(M-rn)+a+m
( M + m)'
-M
1
I
+ ( M + m ) + yn + m
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
&A,(K')
8A,( k')
-m---
&?MO
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
Chop. S
we immediately get (without contact terms)
whw
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
w h e e the symbol X' indicates that the subetituticm (67.71) hrs been made in
5-7
Electron magnetic moment
203
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 --eq &AV(h')
-
J~ 8 d ~ , ~ ( ~ ~ e - h ~ ) ~ ~
k 2
do
-e2~&82*Typ([m
According to the structure of X [Eq.(5-6.4)], appears on the right side is 1
4x
the functional derivative that
- ~ [ (-n + ( m-
p
eq &AV(k')
l+o
- fin- k)]m[-bTx]
h)'
k)'
+ ik!aAv].
(5-7.75)
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
w i t h 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 in our situation steam generalization of (6-7.79).] The relevance of this thh m the remark that the commutator [X, X'], being explicitly linear in the h 0 m 0 g ~ 8 0field, ~ ~ 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, will
through to the final calculation,
5-7
Electron magnetic moment
205
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. Accordingly, 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)]
can be replaced by -m2 and the OF term d t t a d , owing to the Whenewer preeence elsewhere of an electmmqpetic field factor, this combination will appear as
l+o
-X
where
2
1-0 + -X' 2
+
+ D',
206
Chop. S
fl0CtrOdyRCrrni~rfiI
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 amaider the gauge ~ o a m a t i i o n
in the various fields. Now
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
5-7
Electron magnetic moment
207
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 = 1. On noticing that
a s p d a h t i o n of (5-7.90) to the situation +(l- v )
of the aimplificati~~) asmchted with the use of %"(h') in and availing on& (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~getdto 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) :
a
u)eqoF*[mu - ( 2
M[ -r - - / ' & u ( l 2~
0
1
- u)yk8]-D{
(6-7.108)
The list of ccmtnWo118 to $(h') lads off with the explicitly fielddepadent tarns produced by the rearrangements o f (6-7.76, 77),
+ r( lI - k ) ] [8u2(1+ O) ep( m)'
X ([2m
5-7
Electron magnetic moment
209
The transformation
combined with the basic integrals
(4-14-78, (4-8.531
then yields
In connection w i t h the teat provided by (5-7.101), we aleo record that
The next mntn'butim involvee the
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 commutator
210
~ ~ o d y n c miIl u
Chop. S
and the trandormati011(6-7.110) effectively 0
xcq(PK)'
0 1 1 it ~ into
l
+ [(l - u)(2n - k') - uvk']veqyRk'
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~tiemof 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
211
Hare,we have introduced the symbol (omitting the prime) X
-
u ( 1 - u ) ( P + m2),
(6-7.120)
+
(6-7.l21)
a distingukhed from X- ~ ( 1 U - ) ( . @ m2 - e q t r ~ ) .
-
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 ( g 2 + X' + m2u2)]( 1 + ieqtrF)r
(6-7.123)
-
(6-7.124)
The obeervations that y
2
y'yCoFy,--2uFyC
The w e n t i a l function involving the sum %' + X.' am be decompomd into a product of exponentialswith 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 w i l l cancel the additional comutatars,
Then, as we have remarked before [cf. the dimmion preceding (4-16.13)], the and k in the reprdred integration can only appear in terms quadratic in F, which giva for (5-7.125)
distinction between
In the latter version we have reintroduced X', with the appropriate aF anmctian
212
Chap. S
Noctrodymomicr II
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
- -2~1 1o ~ U 4 V ~ ~ p m 2 U+*,S 2
(5-7.128)
and [Eq.(5-7.99)]
With the qh tQam of (6-7.75) still set aside, and conscim of the rearrangemen* already introduced, we find that the &dual form of the second tenn in (5-7.74) lmdf3
X
([m
- r(n- &)]((U- k)'.Ip[-i.(TXl + v
+m[-i.(Fx+
(D'
1-0
+ TXt)]
-
First,we & i t 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~gneticmoment
213
change, there to be replaced by egaF. This givts
y:( 1'),1
--
2ie2/ c&
8'
cl^
du
u2(l
l+v - U )qaF 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,
m(1 + U )
As in (5-7.118),
1-0
+ U-+')2
[(l
- u ) ( 2 n - k') - U V ~ ' ]D~"- + yp-D'
the product with K; contains
and we obeerve that
cEo
a
+ fU/ & z u e y a ~ - av
m(1- U)
- -U 1 - 0 yk' D'
2
,
(5-7.136)
Chop.
S
or, with a partial integration,
the end-paint8VBJuatiom have introdwed
Aftes reamwing the spin termefn#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
D', w h i c h i s m by useof P.Inview ofthe istobedistin-h a b c e of spin matricee in this e q m d o n , the implied form of (6-7.130) can be simplified to
+-[-it(z2 + v ](n- c)')), (5-7.140) which has already invoked the p o a a i t y of omitting yII + m when it stands entirely to the left. Cancaaing the integration symbolized by ( ), we first note that, as in the dimmio11based 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 0
1 )
28
and then use the appmximate rearrangements
S
ru'n + is-
l-W, [k2, n 2
] ~ ~( 6~7 '. 1,~ )
But the latter commutator term disappears after the ~~inbgmticm, from which we learn that
Finally, we require an evaluaticm for
(Q, exp[ -Y K + 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
5-7
Electron magnetic moment
217
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 Apart fmm the distinction between D; and
-
We now pesform the 8-integratiaaein 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 u m t r i i o n 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 mplrecieely with the anticipated eXpreeaian (6-7.101). This may occamon 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- d n - 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 Umatrices, is itself a linear combination of U-matricqand is therefore annulled by yp operation. Accordingly, the yC
Electron mognatic moment
5-7
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 p o m i i t y 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 w i t h previous calculations. The simplest of those is
S7
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 d t t e d , 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/m2u2in (5-7.101).
it &tee the asymptotic form of k,y:(k) for 1k21 to Mi2) is
, m2. This initial contnion
We perform the momentum integral in our usual manner, l
k2 (l?
-
+ m2
-
-ic'/dss&(exp(-&[(l-
-
u)k2 + u(k2 2 k l I ) I ) )
5-7
Woetron magnetic moment
223
If we were! now to combine this s-integral with the very nimillrr expmsion for 'l t would indeed be finite. There is, however, the danger
[Eq.(&7.51)], the d
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 l e c t r o w n e t i d y neutral photon:
(The preemce of additional factors other than s8is 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 lq 2
=--
2a8
+8)ql-
28)
r(3-8)
S
"(L + f), 2 r 28
(6-7.187)
where the last form is a sufficient appmximation for 8 a 1. The analogous modification in the calculation of (5-7.185) is
8)nl- 4~a -8l nl +r(2 - 8)
I
28) SE
a --(L+ f). 2~ 28
md the value thus d g n d to the backet in (S-7.184) is :(a/Zr). the^ firat contribution to M{@ is obtained as
with
(67.188)
Acccdhgly,
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-integralsthen encountered in (S-7.191) are elementary,w i t h the exti- 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
[the combined in
Chap. S
r e d t fm 8 = 0 is already given in (S-7.18S)], and they are
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 ) -
1
- v2 l - U
+(l - 0 + 2m) log
2
{(l
1
- u ) ( l + 0)lW-l + v - u ( l + v ) l ~U -
2 -(2-u+m)log l-0+2w
2-u+w
},
(C'7.210)
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 w i t h the contact terms that isolate
228
EIoctrodynomicr II
Chap. S
the slplidt aF dspcmdenm They are f d in the part. of M P 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
IY k2 (n-k)'
+ m2- e q a ~
(6-7.212)
The surrounding I&factoxshave been retained in order to emphasize the following property. This object is dimdonlIf 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 t e r m 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 f h t 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
[
ZdequF f - m 2 f ' - m 2
(i: --
1
(h2 tin)'
To find an eqmdan for f', which is only requhd for F (5-7316) with reepect to D', and then multiply by +W:
)l.
(5-7218)
-
0, we differentiate
0,
fi + 'm
That leavet us with
Now we return to f, which is (S-721% evaluated at F
-
-
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~twith (6-7.214). To indicate the caution w i t h 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 the e 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
5-7
Electron magnetic moment
231
-
The presence of this factar indimtee that the WCintegral 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 i m d o n l e e 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-integralthat 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
5-7
Electron magnetic moment
233
Another type of integral m p i r h g 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
(
+4 yk-
l
(k2 - 2 k ~ ) ~
In both of the integmki that compote Lp we can make the effective subetitutia
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
5-7
Electron magnetic moment
235
We have now illustrated the various laborgaving devi- that can be used to reduce the remainder of M P 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
+ / $ $ [ - l - 2u + u(2 + u ) ( l
Chap. S
- v)]&
- V )- U(! + 2(1- 0 ) + * ( l + u2(l - v)(+ + ?(l - v ) ) + 3uY1 - 0)(1 - ! ( l - v ) ) ]Cs + / u s [ - $ + ! ( l - v ) - ! ( l -v)'+ u(1- v ) X($ + $ ( l - v ) + % ( l - v ) ~) u2(1- v)2(! + $ ( l - V ) ) 4 ]
+/U'
[!(l
0)')
liV
+ 1U 2 - [ - ! + u ( y + y ( l - v ) ) +u'(3
- $(l- v ) - +(l- v ) 2 ) + u8(l - v)(-3 + +(l- v))]E2
+/ U ( I - .)(l + ~ ) [ 4 +q 2u24 - 3u2d2]+ / [ - 2 u + 6~l + (5-7W
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
...
S7
Electron maglletic moment
237
The 0 t h functions ~~ of the capital class can now be found by combining direct intagration w i t h 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 farm [ W = f(1 - v)]
are the integrals of the
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 p o a i i t i e e here, for K, I > 0, are set out as
OnealsDrneetsfamsoftheseintegrdbinwhichI< O,specifidy, l- -1, -2. To deal w i t h 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
5-7
Electron magnetic moment
241
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,
-2f 1-0
1
(2n + ')l
II
3 w2 -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
-
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
As a axwqumce,
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
-1
& ' l 1 1 1 / " ~ - / ( 1 - v ) 2 ~h [ i - 7 1 0 a+G log l - U
2U
Chap. 5
Eloctrodyncwniu II
where the ,last term, which lacke a v-singularity, has already been integrated over v. The U-integralthat 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
(0
and
0
l
)
-
:R319
5-7
Electron magnetic moment
247
which then yields
It a h d d also be remarked that the core of the structure which producee 5(3) and r2log 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
-
$(l+ v ) .
On remarking that
we evaluate the zu-integral:
from which we deduce such integrals as
Another set of inbgds is i n f d from the relation
Electron magnetic moment
249
w i t h its cometpence
J1&wrB-
1
--~
1
1
~
log -; 1
-
~
an example is
The complete contribution of the r, class is - 2a(2.l 2
lr2
10g2-!3(3))
-(3+3+:++)-
6
+7+$-:.
The remainder of the integrals contain L, and various powers of Examples of these that can be inferred from known results are
(6-7330) U
and
W.
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
The other piecm of c, listed in Eqa (5-7.54,190,197,201,264)
Chap. S
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 stnl~constant
which is in exceptional agmeme~twith a rece~tmeaaurwnent,
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
5-8
Photon propagation function 111
251
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 still 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'ha l y ~ , , ~ + A b
+ 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 + w~ith 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. of this formula state the effect of We recognize that the firet two 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 mechanirnn. 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 electmmagnetic 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 p r o m 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 contributiane 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 detemination of the contact tenn in that fonnula:
In addition to this "internalw contact term, there are, of come, "externalw contact termswhich exploit the noa-overlappingarrangement of the fields A, and Ab to satisfy the physical requhments of the theory.
254
Chop. 5
Moctrodynomicr II
A useful analysis of the structure of tion of M into two pnrts [Eq.(5-7.!50)]
follows from the known decoxnpoei-
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-particlepropagation function that is given by [cf.Eqa (5-7.55, W1
where A*(M')
-
zL'(14 r M' $)'(l
'
(M' T m)'
and we have indicated the n d t y for an infra-red m&cation cal factor near the threshold at M' = m:
*
)
(6-8.11)
of the kinemati-
The use of the individual ppqptian functions (yn M')" in (5-8.6) reducee the problem to the known one of two-particle exchange without interaction, but w i t h the mass substitution m -,M' (the algebraic sign of M' is without effect). Thus, we get a first mtrihtion to the modified weight function of the photon propagation functioa, a(M2),by combining the elementary result [Eq.(S-4.186), for example, w i t h m -,Mq and the spectral distribution of the mass M' that is i n f d from (S-8.14),
Photon propagation function 111
5-8
255
where we have hincluded the coafxibution aamdated with [cf. Eq. (5-7.52)], the infra-red-dtive part of the amtact term. That has the advantage of removing all reference to the fictitious mass p at this initial stage of the calculation. The latter remark is verified by isolating the part of the spectral integd in which M' ranges from m + p to m + 8M,where
(
a2
ID
l
-
(l+2$)lbMd(M'-m)
[(M'
- m)2 - J]ID ( 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
Chap. 5
Eloctrodynomics I1
The outcame is
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
l-t?
givee the d
2 "-l 1 2n+lk,,2k+1'
E-
t
although it is potsi'bly h p l e r 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
5-8
Photon propagation function 111
257
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 &transformationdevice 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 d t t e d , and the k-integration perf m e d by the usual formula
the d t obtained for (5-8.37) is p d s e l y K,as presented in Eq.(5-6.54). We therefore confine our attention to the explicit linear F-termsthat 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 ll
Chop. 5
or p", r m p s ~ v d ywhme , pf - p"
'T"
p*
(b8.45)
we find that
9
w h e the an D es t h t U' replam U h the of D. Anoaw way of ~ t a ~ g gaug*htr&mt fam mager?ran a d o p ~ ga hrmtz:gauge, so
tbt
Photon propagation function l11
5-8
261
The result of (5-8.48), then appears as
--4ar u(l s-
do I - e q y ~ ( e - * ~ a-")(ypN 2
-
+ m),
(5-8.62)
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-integralsthat 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
e*
cEo
dss2mtu-yp([m-
2
~ ( n - k ) ] ~ - " ~ t ( l +9 Y.~])e s- bq ~[ !~( l~- ~ ) )
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" D
- u 2 q i - U(1 - U)(p" - p'") d
l-
0. (6-8.67)
where [we again rer#rt to the integration sign simplification of (6-7.2S4)]
while
and
S8
Photon propagation function 111
263
This decompoeitian evidemtly catalogueethe 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
-
G,( p")
-
-
u2(l. U) 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-causalevaluation 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 W o r m a t i o n on A&), for example, when combined with the
Chop. S
identity (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-timeextrapolation. 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
Photon propagation function 111
5-8
265
by the translation
The net tmmdormatian,
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
[
x ,&p,
- g,,,
(
1-02 p2 + m2 + -h')4
1-02
- +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~lnot 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
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, w i t h one exception. That is the last contribution to M, [Eq. (5-8.59)], which has in its numerator the factor etructure that the
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 (5-7.187) now reads
a
= -ll&(l 2~ o
U) - u ) l q m2u2+m2*(lu2
S
:(log; 2r
X
3' given in
+ f), (68.85)
266
Chap. S
EIectrodynomia II
where the last form refers to the situation X S m. The advantage of the mbstitution (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 d h m s h g (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 "reguMzation" 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 b
-
1 + 0' w-(p"+ 2
m')
l - 0' + w-(p" 2
+ m2)+ ( l - W ) u(lD- U )
and
is the appropriate replacement when D&occur& The introduction of the variable P, agto p'-p+ik,
p"=p-ik,
ccmverts A into
in which we have employed the symbol 5 - too'+ ( l
- w)o.
The trmmlatian p +p
- f5k,
(5-894)
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-gaugesimplification of the gauge-invariant coupling. Thus, the action expmsicm of Eq. (5-8.89) has become
with
In the latter form, the following symbols are used:
The S-integrationis now performed to give
5-8
Photon (nopogafiom function 111
which anticipatee that 8'; doe^ not contribute in the limit X cmly role of X is to appear m the integral
-r
271
m. Indeed, the
where the farm of the right-hand side exploits the fact that 4 is domb&d by w)/u, exoept for values of u that are very clme to unity, when it is m to retain the term having the factor (l U)-'. We shall separate this expmsion into two parts:
A2(1 n
-
-
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 - 8 a ) , 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), &mthat
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 w i t h 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,;
M 4 oo
(6-8.114)
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
Chap. S
Electrodynamics II
and
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-pN+fk.
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, w i t h 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):
log
+m2
8, l-d + -k2 8, 4
-W) l-U
bog
a+ -+-8+
~ ( 1 -W ) l-U
8,
l-U
l - v2 k2 ~ ( 1 -W ) 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 hthe various pithat 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? 2r2
db
clo'
W 1 l-ui~+z(l-w)c? l-u
where
When modified to incorporate a m a l l 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 (k212in 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-causalv d m ) . I think that the empshould 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 w i t h 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 thingswhich would make this behavior plamile, at least, without detailed calculations? S.: I'm glad you arPked that, ainde I intend to clue this section w i t h a diemidon of just such an i n d g a t i o n 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 ( M 2 )by the dimdonleee combination M2a(M2). And we make evident the dynamical nature of this function by exhiiting a factor of e Z ,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 acmmpanieethe appearance of a new dimdonle88 variable, m2fi2. It is precisely here that a plausible Simplification 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/A2a 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 in a power series when eM2 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 c01lveyedby
where
-
If m e extmpolatae them definitions down to M' m2, w h m x eamtially reduus to l/e2, the'followhgapproximate 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 M 24 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:
Em k 2 e q +(k) = ew2 , &=+m
where, indeed, c', expreeeea by
is the limit of e:
as A
+
m. In
this situation, which is
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), m q u h 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 extrapolating from the initial asymptotic form of the weight function, given in (5-8.163), to the ultimate asymptotic farm as M 2-+ 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 stronginteraction aspects of the procee& Rather, the emphasis still 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 m11i a
Sup-
now, that field and source are redefined by the I
Chap. S
d transformation
although, despite the appearance of the exponential function,only terms at mast linear in 9 are of intan& in the p a e n t dimusion. The anof y5 and its anticmnmutativity w i t h yOm b i n e 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
Chap. 5
flutrodynomicr II
Than,applying the d t s of Section 4-3 that were cited in the firstparagraph 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 isused (6-9.10):
-
g m -yll +in-ivy6a+ 2m p n2+m2-eqa~'
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 m e 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
fa-,
It is the preearce of the commutator that introducesan additional field-strength 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 momentum 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 w i 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~tsthe 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 ecare become6 apparent on inserting the intinitemomentum-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-bylight 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 ohstiolrs 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 m e encountered in the pmudoecalar calculation [Eq.(S-9.14)J. Now that we have brought to the surface the cimmatmce of initial non-overlap 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 a n d i l a t e 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)]
Photon decay of the pion
5-9
293
The vacuum amplitude d m i i n g the two-particle exchange, as inferred from (4-8.4), is then
which also f o U m from (5-9.10) on insorting the expansion of G+". The causal situation under consideration is conveyed by the propagation function fotms G+(Xt
- rtt)
-
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 a l l particles and photons have the energy $M, and the integration reduces to one over a scatteakg angle of coeine z:
in which
5-9
Photon decoy of the pion
295
Having devised the vacuum amplitude (5-9.46) for c a d circumstances, we now make that space-timearrangement 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 + m 2 = p R + m 2 = 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 w i t h 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) w i l l be replaced by
l
do,, eiK(x-x') a'+(xt)
2m2log 1+ v -
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 it 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 r e d t 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 n q u b 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 pionphoton 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 with 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 calmlati0118 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 calculationshave 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 transparent 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", w i t h the pion field supplying the momentum
Chop. S
and 1 (p'
-
-
1 -ita + m2 ( p M- k)' + d k2
1
824'
ull
?j hC
- ~ X(&gm) ,
-1
with
which hincoqmratae a finite photon m ~ n In a 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
M,-
a & - -2i r/ & ~ l k u ~ e - ~ ~
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 normdbtion condition, in the fonn
-
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 effectively applim to any momentum transfer such that
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 ,,
Photon decay of the pion
363
is given by mc m.
-1 ,
S-.,
(S-9.96)
in which [Eq.(5-6S)J m, m
a ds -1&(l + 2w s
-I1 2~ a
u)e-umauyl
du ( l + U) log
0
- e-~Aa(l-u)
1
m2u2
If we also decompoee f, into high- and low-msssaensitive parts,
(b.+l;,
(p".-
(5-g#)
we have m,
a
X
(5-9.99)
and a
m
w
C
l" P". = {G*= - - log -. 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
Chap. 5
Elutrodynomics I1
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 (6-9.106) -mc I,(Y~ + m).
-
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 q u a d i n g about the evaluation of inte2p.als. E r m 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 y x a + team that is ale0 produced in anticommuting y n w i t h y6+ has bear d W d , as justified by the n d surface integral to which it
5-9
Photon decoy of the pion
307
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 w i l l indeed be maintained at this dynamical level if the first term of (5-9.116) vdea:
We now verify this by explicit calculation. The procedure of Eq. (5-9.118) convert8 this statement into
The structure of MAfor 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-strengthcombinations of the fonn F'%,, since only ie of interest to us. Accordingly, all that survives of (5-6.43) is
%'cp
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
has been treated as a very large quantity in whem A2, correctly for them 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-integrationsare 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 contributionsto 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 ) ] y P+ 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
Chap. S
Eloctrodynsmics II
The first of the tana on the right bearmar a multiple of 9yv ypnand yields identical contributionsto 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 m a s s l e ~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,
C,114. See (3eo Charge parity. CP, 114
C a d arrangement, for two-particle interaction d d p t i o n , 30 C a d control in twbparticle exchange, 91 and unstable particles, 9f Causal Diagram, for pion decay into pho296 tons, 292 in para poeitmnium, 138 -8 parity of n-photon state, l13 W2' in poaitronium, 113 modified by single virtual photon exCharge reflection, eigenvectors of, 1 s change, W4),252 and a selection rule, 112 non-causal evaluation of, Completeness, of eigenfunctians, 16 WC4) Compoeite particle, and statistics, 24 contact term in, Contact terms. structural analysis of, ZM fiction associated with, 152 Action principle, 25. Sae a380 Action. locality of, 191 for two-particle interacting non-relativfor M('),198 istic system, 9 for pseudoscalar coupling, 301 &g equal time fields, 12 for pseudovector Adler, S., 298 coupling, 302 Amplitudee, of mixed ortho-para poaitrodynamical modification, NS nium states, 127 relationship with electromagnetic Analogy, with time development, 146 parameter, 305 Angular momentum, and spin $ particle% for strong field calculation, 160 W Adhihtion coupling, in ortho poaitro- Convergence factor, 302 Coordinates nium, 126 center of mass, 13 Anomaly, in pion decay into photons, 298 relative, 13 k a l vector? 67 Current, transveate, symbolic construction of, 36 Cumnt algebra, 291,298,806 Beeeel function, of h m g h r y argument,
K,,180 Bound state, as a oompoeite particle, B Breit interaction energy, 38
Decay constant for magnetic dipole radiation, l7l,174
312
Index
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 De R-4 L 299 non-relativistic, of positrmium, 123 Derivative, functional, U of pRra poeitronium, 122 Determinant, differential property of, 149 positronium, submatrix of, 127 a w e t i c term,in operator, in Divergence equation, 77 rest frame, of poeitronium, 122 tweparticle, approximately relativistic, Divergences, 140 46 Double commutator, identity for, 216 with large mass ratio, 48 in reet frame,48 Effectivecharge, in esymptotic photon Energy ahift. See also Energy displace propagation hction, 280 ment. Eigenfunctions, 14. See crlso Wave hmcfor instantaneous coulmb interaction, tion. 63 including center of maas motion, 16 for non-local intaraction,61 completenas property of, l6 of order m/M, one-photon exchange, differential equation for, 16 78 and Green's function construction,16 for 2p level, 79 multi-time, 17 for 2s level, 79 differential equations for, 17 unit for, 79 and orthonormality, l8 technique for evaluating, 76 physical interpretation of, 18 orthononnality of, 15 Energy spectrum in magnetic field, 167 Electrodynamics, narrow sense of, 112 with a/2w magnetic moment, 169 Energy one-particle, approximately relativistic, of ground atate in strong fields, 164 61 kinetic, non-relativistic decomposition relativistic, for E@ 0 particle, M of, 14 single particle, from second order Dirac Energy displacement. See cJsa Eaergy equation, 63 ahift; Mase, displacement of. total momentum dependence of, 48 additional, in ortho poeitronium, 126 two-particle, expectation values for, 49 apparent discrepancy in, 68 first relativistic corrections to, 44 effect of modified vacuum poldzation with large mats ratio, M) oa, 10s and reduced mass, 82 high frequency, of order m/M, W,g9 Energy splitting, ortho-para, modification low frequency, of order m/M, 86.74 of, 128 of order a(m/M), for Coulomb interacEquations of motion tion, 6s non-relativistic, 68 relative to fine structure, 66 solution for constant field, 146 of order m/M, d i m a t e for, n Equivalence, between peeudoecalar and f t 2p ~ 1 e ~ g9 4 pseudovector couplings, 286 for 2s level, 87
Decay canstant (cont.) in magnetic field, at high energy, 1 a of para poeitronium state, 133 of state in a magnetic field, l68 Decay rate, of mixed orthepara poaitronium atate, 127
Index
313
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 Factor, modifying pion-photon coupling, Green's function. See a&o Propagation function. 299.30s advanced, 42 Feynman diagram, see Non-causal diaapproximate equal time equation for, l.!rf'm* with non-local interaction, 60 Field, difference, 9 change of order m/M, through single multi-time, equal time speciali211:tionof, photon exchange, 77 2 composition property for, 77 single particle, non-relativistic, 2 effective single particle, 77 source of, 10 eigenfunction construction of, 15 two-particle, differential equation for, equal time, differential equation for, 3 7 symbolic construction of, 26 two-particle, equal time, 4 factorized, 14 diff erential equation for, 4.8 functional derivative of, 24,27,32, 75 two-particle non-interacting, non-relaidentical particle symmdry of, 28 tivistic, 2 for immobile particle, and photon Fine structure, M) source, 33 constant value of, 250 multi-time, eigenfunction expansiunsof, in positrcmium, 112 16 and reduced mass, 52 integral equations for, 20 Form factor symbolic construction of, 24 and causal control, 9 l unified expression for, 20 causal diagraxn for, 195 multi-time and equal time, relation of, S charge and magnetic, for scattering, as non-relativistic version of, 76 checks, 220 relativistic, with instantaneous interaceffective, in ortho-para splitting, 129 tion, 38 in neutral pion coupling to photons, 296 retarded, 40 non-relativistic, as wave function, 100 single particle, relativistic, equal time in photon weight function, 111 limit of, 39 for p s e u d d a r coupling, 301 spin i, 145 spin i , in vacuum polarization calculatransformed, boundary condition for, 43 tion, 105 differentialequation for, 41 in three-photon decay of positronium, sets of equations for, 42 l18 two-particle, differentialequation for, 7 Functional derivative, 24 equal time, differential equation for, of Green's function, 76 8 as symbolic photon source, 29 Euclidean metric,in momentum space, m,289, m,291, m,308 Euler's dilogarithm, 96 applications of, 91 Exponentials, theorem on combination of, 204
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
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 Harold on the asymptotic behavim of the photon spectral weight function, 278 Karplus, R,140 on comparison of causal and non-causal Klein, A., 140 methods, 278 K'ung Fu-tzu, 268 on a comparison with experiment, 144 on a conceptual and computational imLorentz gauge, 260,2488,270, e73 provement, 140 on the existence of an integral, 138 on the expectation value of yO, 168 Magnetic field on magnetic moment calculations, % l energy spectrum and eigenfunctioas in, on the quantum correction to synchroIs6 tron radiation, l87 -g, 144 on "mgularhtion", 268 Magnetic Moment on surface integrals, 289 causal diagrams for, 1921; on the vacuum polarization calculation, comparison with d e n t , !B0 110 contact term, contribution to. on Ward's identity, 207 (c!+, 199 Heaviside step function,l67 (c2 )a, 233 Heavy photons, 140 (c2),3, 225 Hyperfine structure (%lr, 226 interaction for, 48 (%)$P 238 of muonium, 140 (cz)'b, 249 in positronium, 112 induced, 163 labor saving devices for, 2!H Infinite momentum frame, and three-parmodified propagation fimction contrii ticle kinematid integral, 82 utor t o (%)C, 196 Infra-red cutoffs, 140 non-causal diagram for, 1921; Infra-red problem, mathematicalorigin of, numerical coefficient of ( u / ~ u )260 ~, 162 strong field modification of, l78 Interaction, additional, of order m/M, 70 three particle exchange process in, 194 non-local, Green's function treatment vacuum polarization correction to, of, 68 ( ~ 2 ) " . pd.9 1909 IQ1 phenomenological, for p a ~ apoeitronium Magnetic polarization, 168 decay, 116 strong field modification of, 178 Interaction energy, additional, of order Mesa,displacement of, in ortho positronium, 126 m/M, 64 in para positrdum, lS3,lSS Coulomb, 34
Index
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 contactterms,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-relativisticformula for, 143 two-photon exchange in, 140 and weak interactions, 112 Neutrino, in muonium decay, l12 Newtan, R, 178 Non-causal diagnun for magnetic moment, 196 Non-overlap, in n o n a d calculatian, 291
35
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 contribution to, 129 Ortho pogitronium, 114 Orthonormality, of multi-time eigenfmctions, 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 fermiSS pseudovector coupling to fermions, 286 two-photon coupling of, 287
36
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, ll9,lm ortho-para splitting of, l l 9 ground level, l25 para, decay Fate of, 114 s field ppertiee of, 1 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 exchange 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 twophotan exchange, 78 modiiied, lm multi-time, equal time s p e c h b t i m of, 2 non-relativistic, phdd pictur% for, 3 multi-time, and equal time, datiun of,
c,
8
non-relativistic, dj&t form of, S photon, and a ElRWr of pug* 29 ~ c t i o of n, 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, nmrelativistic, 2 two-ptuficle non-relativistic, differemtial equation for, 2 two-particle, relativistic, with akeletoa interactions, 30 Propagator. See Propqation function. Ptmdoecalar coupling, dynamical modification 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
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
317
Tensor spin-spin coupling, dgenvalue of,
124 in positdum, 124 Three-particle exchange, kinematid integral for, 81 Thre&old behavior, of modified photon weight function, 99 masees,= Time,relative, and orthonormality, 18 Skeletal interaction theory, 28 Soft photon emission, in pair creation, 87 Total angular momentum quantum number, for poeitronium, 112 Sommerfield, C., 261 Tramverse interaction, approximate Source, compoeite particle, and contreatment of, 74 -cy test, 23 'Mangle diagram, for pion decay into multi-time construction of, !44 photons, 298 and statistics,24 TBai, W. Y., 299 effective, 29 for photon, two spin particle emis- 'Pwo-particle equation, machinery of, l40 relativistic, 32 sion, 101 relativistic and non-relativbtic, comtwo-particle non-relativistic, 5 pared, 32 for two-particle, photon d o n , 86 7bo-photon exchange, non-relativistic,89 for two-photon, composite particle emiseion, 70 for two-photon d o n , 135 Unitary transformation, on Green's funceffective photon, 39 tions, 40 and functional derivatives, 29 infinitaind, for generalid virial thetwo-particle, equal time, 4 orem, 47 time locality condition for, 5 simplified, 44 Source theory, what it's all about, 306 Source theory requirement, of internal Vacuum amplitude -cy, 299 for partial Compton scattering,198 space parity, in p o e i e u m , 113 for photon, two spin particle exSpence function, see Euler's dilogarithm. change, 101 Spin-orbit coupling, in poe&xmium, 123 for photon exchange by a ampoeite Spin quantum n u m k , for positronium, particle, &d 112 for photon proceseea, 29 States, of a two-particlesystem,22 for pseudwector description of pion deStatistics, of a composite particle, 24 cay, 297,298 F d - D i r a c , in positmmium, l13 symbolic factor in, representing multiSurface integral, in momentum space, photon exchange, 30 289 for threparticle exchange, 88 Sutherland, D, 396 non-relativistic, 70 Synchrotron radiation, 182 for three photon decay, 116 angular distribution of, 186 for two non-interacting particlee, nonclassical power spectrum of, 183 relativistic, 1 Dopples effect in, l85 for two non-interactingspin 3 particles, first quantum c o d o n to, 186 30 Symmetry, crossing, identical particle, of for two-particle exchange in pion decay, Green's function, 28 !m
a
+
38
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
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 Ward's identity, and gauge invariance, 201 threshold behavior of, 99 Wave function. See also Eigenfunctions. of photon propagation function,89 short distance behavia, calculational rule for, 138 two-particle, change in, 92,107
