Advanced Book Classics

Volume I1

ADVANCED BOOK CLASSICS David Pines, Series Editor

Anderson, P. W., Basic Norionr of Cotldensed Matter Physics Bethe H. and Jackiw, R., Inte diate Qwntttm Mechanics, T"hird Editim Feynmm, R., Photon-Hadrm Inreracrim

Feynman, R., Quantum Ekcnodynnmics Fe ynman, R., Statistical Mechanics Feynman, R., The Theory of Fa td Processes

Negele, J. W. and Orland, H., Quantum Many-Partick Systems

Nozieres, E?, Theory of lnreracting Femi S ystemr

Parisi, G., Staristied Field Theory Pines, D., The Many-Body Problem Quigg, C., Gauge TheoTies of rhe S n a g , Weak, and Ekc Schwinger, J . , Panicks , Sources, and Fields, Volume I Schwinger, I., Parcicks , Sources, a d Fields, Volume II Schwinger, l., Parricks, Sources, and Fields, Volume III

3 0 U R C E S , AND

ULLAN SCHWINGER late, University of California at Los Angeles

P E R S E U S BOOKS R e d i ~ g , M C I S S ~ ~ U S C B S

Many of the desipations used by manufacturers and sellers to distinguish their products are claimed as ~ademarks. m e r e those designations appear in this b k and krseus Books was aware of a trademark claim, the desipations have b e n printed in initial capital letters.

Library of Congress Catalog Card Number: 98-87896

Copyright @ 1998, 1989, 1913 by Perseus boks Publishing, L.L.C.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior mitten permission of the publisher. Printed in the Uniced States of America.

Perseus bob is a member of the Persew h o k s Group

G v e r design, by Suzanne Heiser

1 2 3 4 5 6 7 8 9 10-EBc0201009998 First printing, Scpternber 1998

Find us on the World Wide Web at ktq: aw.corn/gb[abp/

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 physicswithout having to devote the time and energy required to prepare a formal review or mono- graph. Indeed, throughout iu 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 informal accounts would be replaced by more formal counterpaw-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 Frmrienr in Physics or its sister series, Lecture Notes a d Suppk ts 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 cost to the reader.

Tnese lecture notes by Julian Schwinger, one of the most distinguished theoretical physicists of this century, provide both beginning graduate students and experienced researchers with an invaluable introduction to the authois perspective on quanmm 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 panicle physics, electrodynamics, gravita- tional theory, and many-body theory, the notes serve as both a textbook on source heory and an idomal historical record of the author's approach to many of the central problems in physics. I am most pleased that will make rhese volumes readily accessible to a new generation of readers.

David Pines Aspen, Colorado

July 1998

Julian Schwinger rli~versity Professor, University of Califoda, and Professor of Physics at the UBiversity of Galifo~a, Los Angeles since 1972, was born in New York City on Februw 12, 1918. Professor %h obt~ned[ his Ph.B, in physics from Columbia U~vergty in 1939. He has also rwkvd honorw dwtcrraks in sience from four institutions: hrdue Univasity (1951), H w w d Waiversity (19621, Brmdeh fldversity (19732, and Gust dolphus Qilege (1975)- h addition

Professor % h ~ a g e r has taugfit at hrdue University (fW1-43), snd ward U~vasi ty (1945-72). Dr.

ger was s Reswcb Assdate at the Uniwrsity of Califoda, Berkeley, Staff Mexniber of the Massachusetts Institute of Tshofogy Radiation

fesm Schwinger &me a co-rwipient (with R i c h ~ d Tamonaga) of the Nobd Prize in Physics for waxk in

(1970), R~fessor the C. L. Mayer Nature of Li@t Awwd (1949); the First &stein (1951); a J. W. a b b s H o n o r ~ Leeturer af the h e ~ c m Matha (19SO); the N a ~ o n d Medd of Science Awwd for Physics (19641; a IXumboIdt Award (1981); the h e ~ o Citta di Casti@me dt Siciga (1986); the Monie A, Ferst Sima X Award (1986); and the h e h c a n Aeadmy of Aekevement Awmd (1987).

Isax Newton used his 13ewly invented methd of f lu~aus (the cdeulus) to compare the implications of the inverse square law of pavitation with Kqter's empificd laws of planetq motion. Yet, when the time mme to write the Pri~cQrh., he resorted entirely ta geometicd demomtrations. Should we conclude that cdculus is saperfluous?

Sowce rheory-to wkch the mncept of r a o m h a t i o n is &reign-md rmormalized operatorfield lrhesry have both been found to yield the same answers

c problem (wkelz disappoints some pmple wha would prefer prduce new-and wrong-answms), Should we conelude

that m r c e theory is thus superfluous? Boa quwtians m k t the s m e response: the skpler, more intuitive forma-

tion, is preferable. This d t i o n of Partielef, Sowces, and Field is mare extensive than the

ofi@xlaX, tvvo votumm af 1970 and 1973, It now ins four rtdditiod =lions that finish the chapter entitled, " Elwtrdyn IX." "ex sections were w ~ t t e n h 1973, but ned in partially w e d r fiftmn years, X m a g ~ n indebted to Mr, Rondd B o h , who managed to decipher my fading s ~ ~ b b X e s and czomptetd the t w e s ~ p t . Particular attention should be directed to Seetion 5-9, where, in a mntext somewhat Xwger than elwtrdynadcs, a disavmment betwen sourer: theoq and operator field thmy findly does appear,

Raders m&ng their first acquaintance with smrw theory shauld consult the Appenhix In Vdume I, 'This Appm&ix contains suggestions for threading one's waiy thou& the somethes cluttered pages,

b s Angeles, C~lgornia April 19188

This page intentionally left blank

The writing of this d volme has sprmd over two y m s md two continats. A substantid portion wm p e m d in Tokyo where, to a sabbaticd leave from filmad University md additional support from the Gu~eAeim. Founda- tion, I spent the first ei&t months of 1970. S m e day, when not prwcupied with the d t i n g of a bmh 1 shall. return to Japm and fufXy savor its deE&ts. The bmk was mmpletd d u ~ n g the 2971-19315 p e ~ o d when X wm Visiting Professor at the U~vmsity of Galifofnia in Los Angeles.

This volume is &most exclusively eomemed with qumtum eleetrsdynam- ics, As s u ~ h it is retrospstive in its subject matt=. But the wnwntration 0x1 this relatively simple dyxla~cal situation is &ire~t& tawwd the exploration and elaboration of v i e ~ o h c s and teh.&ques that should be viable in the damkns of strong and weak interactions, And it is intended that the self-mnt~ned source theory development, with its significant wnwptuill and wmputatimal simplifications, shall be the face: of quantum deetrodynami~s for future generations. No longer need suweyars of tjhe subject maple =static remabs about the awuracy of the t h w q with rumblin$s a h t its uasatisktory wnmptual bmh.

Perhaps a word af tion is n e d d for the histohcalfy o ~ e n t d vipettes that masionagy in these pages. mey are not priority clkms, After dl these years? But X do wish to place on r m r d some aspem af my personal rwlleetion of various evmts, which are not likely to be for thca~ag from any other source,

One etemat of the orgmization of the book, or the lack of it, rqujses nt. A topie that ;has apparmtly been conduded is soxnetinnes taken up

again to explore some paint in geater detail. This represents the h;istori.cd evolution of the subjst, in w&ch various subtleties becme e f a ~ f i d ody after mme time. The alternative to the plan actually followed w s to g e ~ t e various sections as p a t e r understaxlding became available, But, sinee muGh experience had taught that a progam of constant rewfiting led to no bmk at dl, this alternative was rejwtd.

Finally, I must again thank devoted and talented typists, Mrs. Susan Wagenwil at Hawad and Mr. Ronald Bohn at UCLA,

h 8 Angeles, CalVor~ia Mareh 1972


Contents

charged Psrticlk Propagarion Fwdions A Mapetic Moment GaIeuktion Photon Propagatian Function Fam F~ctors 1. Sc~ttering F ~ r n F&etars II: Single and D d I e SpeefraI Form Fom Factors III. Spin $ Fom Faet~rs Il? n e Datero~ W t e ~ n g of Light by fight 1, b w F r e ~ n c i e s Scattering o f fight by I;i&ht II, Fomard Scattedng Se~tter* of fight by Ughi 111. Do&!@ Spectral H-Particle Enerw Di8placements, Nonrt;%at;ii~t:ic Bismgion A Relatiuistic Scattering Calculation Phatsn- Charged PmticIe Scatteri~g N ~ a - C a m ~ l Methods H-Particle Energv Bisplacemnts; Spin Q Relatiuisfic meaty H-Particle E:nerw Displacemmts. Spin 5 Relaiuhtic" Theory I H-Pmticle E~etgy Dzsplacewnts, Spin j Relativistic Theory II

Particles, Sources,


Gaudity and space-time unifomity are the creative principles of source theory. Unifomity in space-time afso has a cornplexnentary momentum-energy implication, Xt is illustrated by the extendd murce canept. Not only for that special bailance of eneru and momentum involved in the enlission or absorption of a single particle is the source defined and meaningful. Given a sufficient escws of. energy over momentum-an excess of mas-several particles can be emitted or absorbed. The dmcription of the coupling bettveen sources must include an account of such multi-particle exchange acts,

The addi,tional singb photon emission, or absorption, that can accompany the working of a charged pa&iele source has already been considered. It is dmeribed by the primitive interaction. That sane primitive interaction also permits an extended photon soure to emit, or alrsorb, a. neutral pair of charged particles, Them two-padicke processes are the simplest exarnplrss sf the multi-particle exchanges that supplement single-partide exchange, with com~pndting additions to the appropriate propagation function. The replacement of the initial singtct- pa~icle propagation functions by the modified ones is of universal applicability bat does not exhaust the implications of multi-particle exchange, The propagation function mdification giva an account of what is common to all realiistic sourcm of a specified type, but does not represent their individual characterh- tics, To illustrate this additional mpect, let us consider the emission of a pair of charged particles by an extended photon source. This process can be reprmentex2 as the conversion of a virtual photon into a pair of real pa&icles, The description provided by the primitive interaction refers to circumstances in which the two particles do not interact. The subsequent interaction of the padicles must be invoked to supply the mechanism that introduces the modifid propagation function far the virtual photon. As we have seen in Sections 3-12 and 3-13, where g a ~ i d e s of spin O and 4 were considered, particle-antiparticle scattering involves two distinct mechanisms. In one, an ordinary scattering act that alsa occurs in particle-particle scattering, the particles persist while exchanging a space-like virtual photon (a photon that c a r~es sptrce-like momentum), The other is an zlnnihilation of the particle-antipartide pair, producing a time-like widual photon, which quickly dways back into particles. It is the fatter process that supplies the modification of the photon propagation function, for, connecting the final pair of chwged particles vvith the extended photon source is a chain of events in which a virtual photon produces a pair of particles that eventually recombine to produce

i

a *d photon. Here is just the additional coupling that operates between extend4 photon sources, But the introduction of the mdifiied photon propqation function does not describe the effect of particle interactions completely since the simple scattering process of space-like photon exchange is not included.

The p~mitive interaction can be characterized as a fwal coupling of the fields. Two charged padicicle fields and the electromagnet ic vector p t e n t ial, a11 refehng to the same space-time point, are multiplied together. The introduction of modified propagation functions alters the quantitative meaning of the fields, but does not change the local character of the field coupling. This complete lw&ty disappears when we consider the ordinary scattering of the oppositely charged particles, which can t&e place at some distance (in time and space) from the p i n t at which the virtual photon decays into the pafticle pair, The final particles" are described by fields referring to the redon in which these particles come into being, as distineirjhed from the coordinate values in the electromagnetic potential vector, which reprewnts the initial particle creation act. This nsn- icxxl mdifieation of the pdmitive interaction alters the electrornwnetic properties wsiped to the charged particles. New couplings can appear, and the non- lmdity of a11 couplings is the intrduction of farm factors, which convey the effective space-tirne distribution of the particle" eeletmmagnetie properties. It is the quantitative discussion of such refinements, apgearing at the dynamical level that incorporates two-padicle exchange, to which the developing techniques of Source neo ry will now be applied,

4-1 CHARGE0 PARTICLE PROPAGATION FUNGTtQNS

The descriptian of the emission ar absorption of a charged particle and a photon by an extended particle source involves the choi~e of an ekctrarnagnetic mdel for the source. As &scussed in Sections 3-10 and 3-1 1, the most natural, covaniant, souxe mdef suppresses charge acceleration radiation. With this choice, photons do not: accompany spin O charged particles and there is no corresponding modification in the propqation function, Radiation does aceur for spin particles, howver, and we shall examine that situation.

The primitive interactian incorporated in (3-X0.63) contains an arbitrary wornwe t i c ratio g, For the particular value g .= 2, which i s very nearly that exfibited by the ekctmn and the muon, the effective photon-partiele emission wurce is [Eq. (3-1 1,79)]

where we have omitted the charge acceleraticlm term

since it does not eontfibute to the rdat ion :

&i::;.k, = 0. (4-1 .a) The probability amplitude for the twepadicle emisdon process, obtained as

is

which utilizes the equivafent forms

The derivation collfd be repaled for the two-particle absorption process, Tbe required probability amplitude can h obtained areetly, hawever, as the negative complex con jugate of the emission amplitude :

The basis for this is the same as for single-particle emission-the maintenance of orthogmality between the vacuum state md the class of states into which the weak wurces emit and from which tbey absor;b.

We now consider a causal anangement of extended pa&icEe sources, and evaluate the contribution to the vacuum amplitude of two-particle exchange:

in which we have introducd the expression af campteteness far the various patfieicfe states of spdfied momentum,

The algebraic pr0pr.l y

impties that

yk(m - y*)yk = 2kM, while

4 Elcctrodynamlcs I Chop, I

Then, i f we exercise the option of using real polarization vectors, for which

we get

Since the sources refer only to the total momentum P, one must integrate over all partitions of this momentum between the particle and the photon, That i s facilitated by introducing the total momentum as an integralion variabfe, the kinematical relation with the momenta J> and k being conveyed by the appropriate delta function. Thus, we insert tile uxlit factor

wllere

itt12 = - P2 (4-1.16)

replaces the energy as art integration variable in accordance with

2 P d P = dM2. (4-1 -17)

The vacuum amplitude contribution now appears as

1 dMZ d w ~ M , - m% 9 I (P) *r;fy"rr (allu (P),

Only the vector P , is available to supply a, preferred direction for K, in this integral, and k , can be replaced by its projection onto P,:

Accordingly,

4-1

where

defines a scalar, a function of - P == M2 2 m2, A reXated integral has been encountered, far arbitrar-~r masws m, and m,, in

the discussion of scattering cross sections, Qn completing the angular inteeat ion in (3-112.76) and dividing by 2n;, we get

Some special examples are

and

This evaluation was performed in the rest frame of the time-llike vector .PU. In the interests of exploring various computation. techniques, we shall repeat the calculation, using a quite different cooranate system, with which we also have had some experience. Xt is one in which the speed associated with the vector P" approaches that of iight. Such coordinate systems are often known as infinite momentum frames. All moment a are decomposed into longitudinal (L) and transverse (I") components relative to the direction provided by the vector P. Denoting the magnitude of this vector by P&, we state the associated energy as

where indicates statements that become exact; asymptotically, as P, -m. For the individual particles, t-z and b, we shall write, for example,

P a t "" PI,@,,

The invariant momen"rum space meuures appear as

while the four-dimensional delta function becomes :

The econd vesion involves the recopition that the energy and longitudinal momentum factors are incompatible, unles jsa and .ec, are positive, Also used is the delta function property

Since the in t e~a t ion over the var;iabEes of one particle is immediate, we? now get

where the step function contains the conditions for the nonvanishing of the i n t e ~ a l , In general (m,, m, # 0). the a rpment of the step function is negative a t both intepation limits, a d becomes positive in the intedor of the interval only if M 3 m, + m,, The value af the g-intetgraf is the range of positiveness, which is found by subtracting the two roots of the quadratic equation

The resdt is (4-1 -231, of course. We return to the vacuum amplitude cont~bution (Q-1.18) and use the evalua-

tion (4-1.25). On introducing

this contribution becomes

4-1 6 h q d prtlcla pro

The ca-1 space-time anrangemexxt i s made? explicit by wfiting

which gives the spce-time coupling:

WC: recowize, in the form appropriate to the causal order xO > xo", the invariant propagation function

vvfitjre the variable m m M has k e n made explieit. The adation of this couI>ling to the one assmiatd with sineilfe-partide exchange is reprewntd by the madified pmopagation function

which possesses the required F.D. antkymmetry:

The ;nomenturn. space tran~ription is

(4-1 -41)

The latter form makes explicit that the spectral mass integrat begins at M = m,

and can be extended tentatively to large mass vdues since, as is more e ~ d e n t in. the first form, the integration is quite convergent,

An alternative presentation of this, and related calculations, begins with the term in the vacuum amplitude for noninteraeti~lg particles that repre~nts the exchange of one photon and one charged partide:

Xnta that term,

one introduces the effective source [Eq, (3-X1,66), with g = 21

which is equally applicable to emission and abso~ptisn, The prefemd fom far the latter process, however, is

The momentum space versions are:

First, let us pick out the contribution that does not involve photons emittrscf or absorbed by the sources, and therefore has no reference to the electramapetic source model;, Inserting in (4-1.43) the propagation function farms appropritate to the causal situation, we find

The last evaluation differs from that of I,(P) only in using the effective projec- t ion

which also foflaws fmrn (&1,2Q), since p = P - k. The m a t ~ x F(P) appars in the combination

w11ich exhibits the evaluations

This prt ion of the vacuum amplitude thus becomes:

The matrix that occurs in the second line is simplified dgebraicalty by irttrduc- ing the projection matrices for yP relative to its eigenvalues jc M,

Note that the r au l thg wurce coupling reduces to (4-1 35 ) for M > m, The space-time extrapolation pm as hfore, and sin= the factors are fixed by the asymptotic (M m) aweement with the earGer cdcdation, we can immediately write the momentum space f a m of the incomplete mdf ied propvation function :

Chop, 4

Xn contrat with (&l.(al), there is now an infrared &vergence, a koga1-ith~c sinpl&ty at the l m r limit of the spctral inLe@&. It woakd h premature, however, to accept the quantitative details of this phenomenon, since fhe above propqation function is incomplete. Note also that, again in contrast with (&1,4t), the weight factor of (yp + M)-l is not dways psitive.

Tkis calculation enables us to emphazjize physical. requirements that have r e n n h d implieit thus far. Suppw the space-time extrapolation bad been b w d on the first, field eoupIing, verslron of (4-1.62) rather than the second, source coupling. The addition& action term obtained in that way would be

Two things are wrof~g here, First, tbe modified propagation function that now emergs, on expressing the fields in tems of sources, is

Since M(yp) dses not vanish for yfi + m = 0, the behavior of the propagation fun~tion is drasticdty madified in the neighborfioad: + m 0. This contra- &cts the phenomenolo@cail basis of the theay, as we dis~ussed extensively in Sctiorr 3-43, under the heading of m a s normalization, Second, the space-time extraplation that W= p d o m e d in reaching (4-1.55) is invalid, since the intwal of (6f .M) does not exist when atended up to indefinitely farge M vdues.

S u m couplings that are inferred thrwgh space-time extraplations af causal mangemen ts can &ways be supplemented by con tact interactions, UnXms adstiond physicaJi considerations can be adduced, the contact terms rem~n. mbitraq and may Ise omitted. But, Wl;hen fields replace sources such local interaction tems do have physical content ; their existence must be recomizd and related to the accompan~ng physical requirements. Since contact coupfings in coorsnate space appea as pofynomid functions of rnomenta in momentum space, the camwt fom of M(?@) supplements (8-1.56) by a polynomial in ye +- m.

atie and higher pwers of this convenient combination mdifly the propagation hnction (4-1.67) by constant or polynonniaf functions of momenta, We are

not interested in contact additions to the propagation function, and it suffices to use constant and linear terms in yp -+- m. These are fixed by the physical requirement that the scond term of (4-1.57) does not have singularities in the neighbor- haad of yp + m = 0, as stated by :

This is achieved, for each value of :M, by the contact modifications

With this corrected interpretation,

and the prapagation function that now emerges from (4-1.57') is just (4-1.M). The partial propagation function we have been discussing has its canterpart

for spin 0. The momentum space presentation of the effective sources is [Eq. (3-13.16))

i j$(- k ) K l ( - p) err* = [+l(- P)(p" + P") - X,(-- P) i lLfkfIq , (4-1.62)

When the fu terms are omitted in the tws-particle excllange coupling,

( d t ) * (dxf)iJf(E)K, ( X ) l,,ffD+(f - F ' M +(% - x')iJ2,(t')K,(xf) I,,,, . (41 .63)

the resulting vacuum amplitude contribution is

M" (- P)Xg(') (M" m m)

$2 P I C ~ ~ ~ m m l c r t Chap. 4

The incomplete modified pmpqation function implied by space-time extrapolation of this coupling is expressd by

where the M2 integration can be extended up to infinity, but has an infrared sinmlarity as M2 -+ me.

Let us use the simpler situation of spin 0 first, in order to indicate the structure of a mwlified propagation function that refers to an electromagnetic source model for which charge acceleration radiation does occur, When inseded in (6--1,62), the choice [Eq. (3-10.44))

hd i f i e s the vacuum amplitude calculation of (4-X,M) in the manner given by

The invew pwem of nk must be averaged aver all partitions of the total, ntomen- tum P, as indicated by the notation

I m2 do, dok 6 ( f - p -- k ) (%h)-"' --- I - * ((nk)-'s2). (4-1.68)

(44%

In the rest frame of the vector I-"@ the only significant variable is x , the cosine of the angle between the relative momenturn vector of the pa"icles and n. Thus,

which also exhibits the photon energy in the P@ rest frame, as inferred from

This integral depends anXy upon the combination

and

The integral that evaluates ((nk)-9 in the rest frame of BB contkns however. "This quantity is presented covariantly as

Although this second inteeal is also an elmentary one, we prefer lio leave it in integral form, as given by

The substitution of (41 .67) now appears effectively as

M2 + m2 zr - 2 -8- . (4-1.76) (M" mm212 (M l + (Q%/M%)(I - 2%)

Tlre result thus deduced from the incomplete propqation function (4-l.%) is

where q2 is formed fram the arbitrary vector pH in the manner of (4--1.74j. Note that q" 0 continues to be valid since nU is a time-like vector, Xn contrast with (4-1.65), the weight factor of each (p2 i. M2 -. &?)-l is now a positive number. The M' integral in (61.77) converges rapidly with increasing M', but has an infrared sinmlarity as M2 __+ m2. There is another presentation of (4-1.71) that uses the variable

l 4 Ctsctrdynrmlcr; I Chap, I

structure^ such as (4-1.41) and (&1.65), desfibing the mass spctnxm of one- and two-pa~icle excttations, are sin@@ spectral forms, In contrmt, (4-1.79) contains a dou"be sgectral form, But, to what kind of excitation the mass M" refers cannot be clear from the purely mathematical origin of this structure.

Since spectral forms represient the possibilit y of excitations wit h variable masses, the weight factor of a standard propagation function, (p5 + M 2 - rie)-l for example, must be a measure of the effectiveness of the source, a relative probability, for that kind of excitation, The unphysical nature of partial propagation functions, illustrated by (4-L.=), is thus manifest in the appearance of negative weight factors. As a check of this intuitive probability interpretation, we sltall extract a knawn result by examining the propagation function (4-1.77) for M2 m2, mrrespnding to a particle accompanied by a soft photon:

The quantity k"< m) is the photan energy in the rest frame of the particle. This discusion is restricted, for simplicity, to the situation

which, interpreted in the rest frame ctf the unit vector nU,

corresponds to considering slowly moving charged particles. Then, the portion of the propagation function that is selected by (4-1 -80) and (4-I .SX) reads

X L d+(p, g) - ------ - p" mnz" - it3 - (4-1.83) p% -f- (m" 2192KO) - 2"e

Since multiple photon emission has been neglected in arriving at this result, the differential probability exhibit& here, (2a/3z)v2(dk0[k0), states the average number of photons emitted. As such, i t i s in complete aeeement with the calculation of (3-1 l .W). The discussion Efiven in connection with the latter equation emphasizes that the mathematical s i n ~ l a ~ t y at k@ O is spurious. The integration in (k1.83) must be stopped at such a value of KO that m2 and m2 -+ 2mk0 are no longer experimentally distinpishable, The eantfibution of all smaller valam of k@ is included already in the description of the emission and absorption of the charged particie without a detectable accompanying photon. The infrared situation is of general occurrence and will be commented on from time to time in the course of various applications.

To perform the analogous calculation referring to spin $ we return to the effective sources (4-1.46) and insert; the e1g;etronzagxletic model function (6L.M).

The change prduced in the vaeuum amplitude (4-1.52) is indicated by

where the notation ( ) is that introduced in (P-f .68). When the right-hand side is written out and some known htegrals inserted, it reads:

We have &ready evaluated [cf. Eq. (4-1.75)]

which u s s the abbreviation

Also required here are (yli/f?ak)L*2) where y enters just as an arbitrary constant vector, After integration only the combinations yn and ?P appear, and the% stmctures are fixed by the results obtzzined when y is replaced by E and by P. The fatter are all known and no further inteeations are needed. This leads to the evaluations

The algebraic reduction of the various terms in (41.86) brings about the introduction of the space-like vector

QU = P"^ + n"rtP, .nQ = 0. (4-1 .S@)

It is appropriately designated since

QBQ, = (%P)% - @P = QZ.

Alternative presentations of the result are given by

On setting Q 5 0 this reduces to yP/W, as exhibited in (4-1.35). The modified propagation function that now appears is a mixture of single and double spectral forms :

The infrared singularity is concentrated in one term. When we retain only the portion of the spectral integral for which

and also introduce the simplifying restriction of (4-1.81, 82), where

the propagation function becomes [the soft photon regime is KO << (rf)lf2 << m]

1 1 G+(P. q) ,,$ + m - ia yp + (m + KO) - ie

(4-1.98)

As expected, one finds the same differential probability for soft photon emission as in the spin 0 result of Eq. (4-1.83).

The spectral presentation of Eq. (61.92) obscures one aspect of the propagation characteristics exhibited by the two-particle excitations, that of parity. The basic spin d: propagation function (yp + M - is)-' differs from (* + m - &)-l

4-i Charged particle propagation functions 17

in mass only. Both represent excitations whicll, in the appropriate rest frame, are characterized by y a k + -t- and thus have the same parity. The propagation function (?p - M + zr)-I describes spin + excitations wit11 the opposite parity, This remark explains, incidentaily, why terms of the Eatter type are missing in the restricted propagation function of (4-1.95). There must be flexibility in distinguishing between the description of an electron accompanied by detectable and by undeteetable photons, which is possibIe only when both situations have a common paritjr. But, tire last term of (4-1.92) does not Itave the parity classification of the preceding terms. kVe tlrerefore seek anotller presentation in which the two excitation types are more in evidence.

For this purpose let us return to (4-1 -91) and introduce the space-like vector Q?

Q" Q Q PP(Q2/'%f2), Q" = 0,

After multiplying (4-1.91) by A I , to make it dimensionless, it reads

Now, yB and yQ" anticammute, since Q h n d E) are ortfrogonal, wflic!r means that the square of the linear combination of yP and y Q 7 s a multiple of the unit matrix. MTe shall call that number NZ, and

The weak inequality

z(Q2iMa) .=: 4 suffices to show that

The yQ9erm can be removed by a matrix Lorentz transformation, This is expressed by

Chap, 4

where

LTy6L -- p. The transfomation matrix has Ihe fallawiq realization,

L = @ 4- (b/Mg)ypyQF,

in which the parameters given by

The positiveness of N is implicit in this realization and the product ab must also be psitirre, which permits us to give positive vafues to bath a and b.

These observations enable us to W t e (61.97) in the Isrm

P - M 2&f

where

and the inequality /&1,100) provides the assurance that both N , and N - are p s i l i v e quantities, Since (M yPf /2M are projection matriees for yP, one can give (P-l.106) another appearance, based on the structures

Chargod partide propagation functions 19

= PLi.

They also obey

The propagation function now emerges in the desired form:

where

Both the positive numbers A, and the matrices L, are functions of M and of the vector

which are produced by substituting q for Q. The normalization relation (4-1.112) continues to apply with P replaced by the arbitrary p, since it involves onlv the

property

The F.D. antisymmetry property here reads

where the sign reversal of q is implied by that of p; it is explicitly needed to restore L+ and L, which have been interconverted by transposition [Eq. (4-1.1 ll)]. The presence of the additional transformation matrices L, does not alter the interpretation of (4-1.113) as a superposition of excitations having variable mass and parity, with dMA, supplying relative probability measures. This is also consistent with the soft photon Iimit where, simplified by qa/me << l ,

restates the soft photon emission probability.

4-Z A MACNETEC MOMENT CALCULATION

In the course of the preceding section we- have intraduced a methd of cafcufa- tion in which fields play the role of wurces. It forced us to use explicitly the physical requirements that accompany the consideration of multi-paxlicie ex- cfiange processes- that the initial phe~~omenological &scriI)tion of single particle processes must not be altered. It is not too sson to recognize that this requirement refers to the behavior of free paftieles. When motion in applied dectromagnetic fields is considered, deviations from the primitive elc?ctroma@etir: interaction can appar. The season is that the primitive electmmagnetic interaction refers specifically to an elernentaw process such that no subsequent interaction takes place. When, in different experimental a~angernents, interactions do occur, they imply modifications in the effective electromwnetic coupling. This will be made explicit in latex considerations. But even now we can draw an experimentally significant consequence of this fact by a simple modification in t fie caliculation of Section (4-3.

First let us consider some features of the spin 4 propagation function G$ ( X , X ' )

that refer ta a weak, homogeneous eleclromagnetie field, When we use the kind of matfix notation that was introduced in Section 3-12, the cXef ining differential equation [ E q . (3-12,2)] appears as

Now we shall write

and deduce that

This involves the algebraic properties

W

6 t; '2

4-3

G

S g g

Q, c:

rU

a

c,,

%;;+ 0

-W

-W

m

m

*j . C

I

25 W

G 'W

,CL,

C"4r

'W W a

T

tF I t-r -!$a I

%.%& F

-?

.--. H

6. .?a,," 's)

U

C - d

a$ .E 2 $6-

4 +

U

*$l

Q,

5-,

dg

k

s==

5 'E:

h

B E

W

where

[ 1 7 ~ . 1 7 ~ = [PI, - q A , ] - [P,, - q A J = isqF,,, (4-2.6)

and we have introduced the simplified notation (note the factor of +)

The matrix viewpoint also makes clear that the construction of ( 62 .3 ) can be presented as

for A$ is a function of the matrix y17 only. The defining equation for A$ is a familiar one. I t differs from the spin 0

Green's function equation of (3-11.40) merely in,the appearance of the - -F term. And, if F,, is a homogeneous field, the additional term is a constant matrix that can be grouped with ma. We also know that after the phase transformation of (3-1 1.37) is performed, the restriction to a homogeneous field [Eq. (3- 11.46)] thatis weak, permitting the neglect of quadratic field terms in (341.62). reduces the transformed propagation function to the free particle form. Thus, the construction of A$ for a weak, homogeneous electromagnetic field is given by

with

denoting a straight line path integral. Turning to G:, we encounter

which expresses the gauge transformation that produces the vector potential (3-11.46). Another such relation is obtained by interchanging x and X' while revexsing the sign of F,,:

where we write a* as an indicator of differentiation to the left with an associated minus sign. If we use the average of the two constructions given in ( 62 .3 ) and (&2.8), the yYF,,(x - X')' term, appearing with opposite signs in the applications

of (4-2, l l ) and (&2.12), will fail to cancel only because of noncommutativity with @F, But this cont~bution is quadratic in the field strength and will bc: omitted. Accordingly, we get

where symmetrization in multiplication of the two matrix factors is understood, We now return to the two-pa&icle exchange coupling of (41.43) and ask how

it is m&fied by the presence of a weak, homogeneous electromapetic field. The change in physical circumstances is canveyed by the intrduction of the charged particle propagation function that we have just constmcted. As in (4-1,47), we shall omit reference to photons emitted or absorbed directly by the sources, which gives the vacuum amplitude coupling term

Using an arxangement that is more convenient for our present purposes, we summarize some significant aspects of the ealiet cajculation in

with

and its space-time extratvlation

+ contact terms, (62.17)

The structure of (62.13) shows that the introduction of G: is produced by supplgng the p h a s factor

@ == mp[.iq?(%, X ' ) ] , (&Z.lS)

repjacing mvby mrc2 - q g F , and thereby M2 by M2 - q g F , while gositioning yg at the extremities of all products in a symmetried way:

D,(% - %')G$ ( X , x') = @

+ contact terms, (4-2. lt 9)

Nate that we have not troubled to isolate the y matrices; the sense af their multi- piicatian should be kept in mind.

Since attention is restricted to weak fields, one can replace (P-2-19) by the equivalent version

+ contact terns. (4-2-20)

Before we can apply the physical considerations that fix the contact terns, the cdculatian must h completed by supplying the yU factors that appear in (4-2, 14). A basic property of the Dirac matrices is the fotiawing,

for two of the y matrjces commute and two anticommute with any a matrix, "This a l s ~ implies that

yRA,(x - x', M2 - e p F ) y , =- - 4A,tx - x f , M", (4-2.22)

in view of the limitation to linear field effects. The other needed combinations are

We shall, also want to restore the M8 - eqaF combination, in accordance with

24 Eierdynanrics l Chap. 4

which can be followed by partial integation on the variable M2; there are no contributions at the integration boundafies, mZ, and infidty, When this has been done we reinstate the pmpagation function (4-2.9), now refenkg to the variable mass M%:

+ contact terns, (4-2.243)

The formal matrix construction

enables us to present (k2.26) as

+ contact terms, (62.28)

with syrnmetrized matrix m~lltipfieation still understood, The physical requirements that determine the contact terns apply in. the absence of the electramag- netic field. They are exp~ssed by the substitutions of ($--I .59, W) although we retain the gaage eovadant symbol LT, which a n be identified as P in any fixed gauge. The result is to coneet the first term of (k2.28) :

S s much, and RO more, is required by the: phenomenolo@eal constraints. The additional action tern that i s now obtained, replacing (4-1-56), is

with

This supplements the initial action expression

Let us confine our attention to the motion of a particle far from its source, where, in the absence of the weak electromagnetic field, 1+4 obeys

( y n + m)# = 0. (4-2.33)

The weak field limitation justifies the use of (4-2.33) in simplifying (4-2.30,31), which leads to

The effective action under these circumstances, combining (4-2.32) and (4-2.30, 34) :

identifies the additional spin magnetic moment of a/% magnetons. In terms of the g factor introduced in (3-10.63). we have

The fine structure constant value

gives

and values of #g that are in remarkable accord with those measured for the electron [1.0011696] and the muon [1.001166]. The tiny discrepancies, which are real, are reasonably assigned to the as yet unconsidered exchange processes involving more than two particles. Our conclusion concerning the electron and

the muon is that the g factor of the primitive interaction does not equal the measured value, but rather, gpdm, = 2, exactly.

The inference that the small deviation of +g from unity is dynamical in origin, and should not be assigned to the local primitive interaction, is reinforced by the following additional consideration. The action expression (4-2,30,32) relates the field $ to its sourGe by means of a modified propalyation function,

where

continues the use of space-time coordinate matrix notation, in tltis more general form, the electromagnetic field dependent part of M(E"j, the second term of (62.31 j, is nonlocal since it is constructed from the propagation functions [yLf -& f (M - i(i.)3-1. When this distributed magnetic moment interaction is examined over very short time intervals, as represented by the complementary limit yIi! 3. -* m, it hcomes asymptotically

and thus vanishes without a residual local internaction. There is another way of presenting the magnetic moment calculation which

is still more phenomenolo@cal ; the initial electromagnetic interaction now contains a deviation of *g from unity as observed. The additional hypothesis that the extra spin mqnetic moment is a dynamical effect, which is suppressed over very short time intervals, then determines it, The modified Green" function equation (42.40) now contains, added to m, the mapetic moment interaction

The dynamical term M ( F ) is certainly altered by the changed electromagnetic propedies, but if one only retains effects of order ar/2n, such changes can be neglected; as of higher order, This time, however, we must impose a phenomenologi- cdit n~malization requirement on M(F) , according to the assumption that g is the observed g factor, One must supplement M ( F ) By a contact term designed to remove the electromagnetic field depndence under the conditions symbolized by yU $- m == 0. The necessary modifications are:

X 1 "F - 1 S=: F- y n 4 - m 1 y n h (M - i4 j M F m M ~ F ~ Y ~ & ( M - - ~ @ ) '

Thus the compkte mqnetic moment term becomes

The asymptotic limit of the factor in braces, for very short time intervals (#l -+ m), is

If this is required to vanish, as the sign of a dynaxnieal origin for l ~ g -- 1, the successful result (62.36) is recowred, The viewpaint just discuwd has the advantage of suggesting that =me of the small corrections of order (a/2n)' are aspects of self-consistency; the g value is computed in terns of dynamical procesws that are eharxterized in part by the value of g. But we shall not attempt such an improved calculation now,

One point demands a comment, hwewr, Ma mention has been made of photon emission or absorption directly from the sowrees, although we know such processes to be imprtant for the pbysical consistency of the descriptiort. The remn is, of @ours, that the mqnetic moment calcalation is unaffected by the= processes, with their elements of arbitrariness. In the latter, one or both fields $ is replaced by the wurce g, or effectively (yn+ mf+, Thus, such effects do not cont~bute in the physicd circumstances of a magnetic moment mewarement, as expressed by the condition yl2 + m = O. They will alter the detailed structure of c,(F), however, dthoagh consislen~y demands that the &art time behhor of the mwet ic interaction remain qufitatively as before,

4-3 PHOTON PROPAGATION FUNCTION

The primitive electromagnetic interaction for spin 4 ehzged padides identifies the efectrornwetic vector potential as an effective extended sQarce for the two- particle emission or absorption of oppositely chargd pasticles, provided the mass threshold at 2% is exceeded,

Since the physicd context is that of nonirrterwting padicfes, we compare the vacuum amplitude for the elementav proces:

18 Elsef rdynamica I Chap. 4

with the vacuum amplitude that refers to the noninteracting propagation of two part ides :

The latter is the quadratic term in the expansion of

with qjs) designated as the source of interest in emission or absorption while $ ( X ) is the associated particle field. The comparison of (4-3.2) and (4-3,3j supplies the matrix

Note that the antisymmetry of the left side for an interchange of all indices, expressing F.D. statistics, i s matcfied s n the dght-hand side, specifically by the antisymmetry of the charge matrix.

The coupling between two-particle emission and absorption sources is described by the quadratic term sf (4-3.4), first written as

and then rearranged entirely in matrix form:

This uses the possibility of writing

where, as in the situation of interest, M commutes with the anticommuting sources, Further rearrangements are made in accordance with the cyclic propedy of the trace. The introduction of the two effective sources, related through the vector patentials A;,% to the extended photon sources Jf,,, then gives the vacuum amplitude representing two-particle exchange between. extended photon sources :

Supplying the appmpria te causal forms of the propqation functions conwrts this into

where

k = p + p t

is the momentum exchanged between the extended photon sources. With the insertion of the unit factor

the vacuuxn ampbtude coupling te rn &comes

in which

Note carefully that the trace of (63.14) refers only to the four-dimensional space of the Dirae matrices, The ad&tional multiplicily of 2 associated vvith the charge space has been made explicit.

The tensor I,,(K) has two important properties. It is symmetrical in y and v, and it obeys

krlrllv(k) --. 0. (&3.X6)

Concerning the symmetry of the tensor, we observe that

the interchange of p and $?in the integal comptetes the verification:

Chap, 4

To confirm (4-3-15), we note the relation

and that each term is annufled in the trace, by the factors m - and -- m --.- @', respectively. These properties are necessary on physical eounds, In particular, f&3.f5) expresses the gauge invariance of the coupling. The explicit tensor form that satisfies these requirements,

is a. reminder that a. massive excitation emitted by a vector source acts like a unit spin particle. The scalar function I ( M 3 can be computed h m the trace of

t U Y ( k ) Y

3 1 ( M Z ) = guvIl, ,(k)

where the Dirac matrix trace is simply

The remaining integration in (4-3-20) is just (&l.%), and

IfM2) == - - (M" +m%) 3 ( 4 ~ ) ~

In view of the flaug;er invariance of the coupling, the vector potential can be chosen as

and the continued requirement of vanishing source diwrgence,

h, J"(k) = 0, (4-3.24)

enables the vaicuum amplitude to be presented as follows:

We recognize, in

the descfiptian of the causd exchange of m excitation with mas% M between the murees, The intraductisn of the propqation fundim B,(% -- x" Ma) prduces the necwav SW=-time extrapjation, Adding this coupling to the one reprwnt - ing photon exchange, we get the'modified photon propagation function

The momentum space transc~ption is

Skce the Mqntegd is quite convwgent, it has been =tended from the threshold at ( 2 m ) b p to infinity.

We bave prformed this calculation for spin # particles since the irnprtant physicd app~cation refers to the charged pa&icEe of snraffet mass, the electron, Nevedheless, it is interating ta we ai pardlef calculation far chaqed spin O p i Note that the va~ous kinds of charged padicles give additive contributions to b , (k ) when just two-particle exchange is considered. The spin 0 pr;imitive interaction supplies the vacuum atmpGtude

which is to be compmed )TTifh

n a t gives

i1K:(%)K(~') l~ = MA*) + A"(n'))(il$)aB8(x - n'f, (4-3.31)

where the B.E. spmetry of the left side matches that of the right side, The dw&ption of two-pdcfe exchange is cantczined in

where the trace refers to charge spaee, and the effective sources of (k3.31) are. to be inserted. For simplicity of writing, we present this coupling in the moqen- tum spaee form appropriate to the causal situation, using the momentum equiva- lents of (14-3.31)

which make explicit the reference to oppositely charged particles, The vacuum amplitude i s

..

where, now,

is evidently symmetrical in p and v, and hias the necessary gauge invariance property (4-3.15) since

The scalar function of (4-3.19) is evaluated iits

since

(+ - = 121% - 4m2,

The modified photon propagation function, in which only spin O charged partictes are taken into account, is, therefore,

Several direct: uses of the modified photon propagation function can be made. The first is a dynamical application of the vacuum amplitude

yielding the probability that the vacuum is &turbd, as the complement of the vacuum prsistence probability. Far a weak extended source this eves the probability of single pair ernision lcsy the source:

where, using the spin 4 stmcture,

The pair emission probability thus obtained,

is also the result produced by &rect calculation from (43-2)" The quadratic saurce combination that appears here is psitive, incidentally,

as one sees from

wt.lich also indicates that the two particles have unit angular mmentum in the k rest frame. That gives some insight into the difference between the spin & and spin O results, For spin 4 padicles created nearly at rest, the unit an@ar mamentum can be reafized with zero orbital angular momentum, in a state. The transition probability new threshold will vary as the relative ve1wity of the particles, - (M - With spinfess particles, on the other hand, the unit anplar momentum forces the particles into the P state of orbital an@= momentum, and the threshold behavior contains two additional pwem of r e l a t h momentum, - (M - 2112)813-.

The introduction of the modified photon propqation function implies a change in the interaction between static charges--the Codomb potential. Replacing it is [Eq. (%$.92)]

34 E l ~ ~ r o r f y n ~ r m i ~ I Chap. 4

where the last version uses the variable

In writing (4-3.45) we have recognized that time integration reduces the four- dimensional Green's function to a three-dimensional one, specificauy of the differential operator --- 67% -+ M2. Its form can be obtained, for example, from the integral (3-114.35) by placing pU = 0. At distances such that

the Coulomb potential is not significantly altered. In the other limit,

is the Eulerian constant. The additional logarithmic behavior under these circumstances comes from the interval sE M integration such that

where the integral of (4-3.45) reduces to 2 l d M / M . The evaluation of (43.48) is obtained by partitioning the integral at some value of M that satisfies the in- equalities of (4-3.50).

The effect we are discussing, usually referred to as vacuum polarbation, increases the strength of the Coulomb interaction with diminishing distance, The increase is quite small, however, at any realizable distance. Thus, with 2rnIrl

which represents a distance af roughly 110-U cm when the electron mass is used, it is approximately one percent. In view of the logarithmic dependence on distance, this order of magnitude cannot he changed significantly by any con- ceivable impmvement in experirnentaf prowess; a ten-percent increase in interaction strength requires dropping to a distance - cm! And long before such distances could be approached, the situation would change qualitatively through the growing impdance of particles that are heavier than the electron. Despite their smallness, vacuum plarization effects are measurable at the present level of experimental technique. The most elementary situation is that of hydrogenie atoms where the. strengthened attraction between electron and nucleus depresses

the energy values of zero orbital angular momentum states, these being the ones in which the electron spends appreciable time near the nucleus,

Simple perturbation theory can be applied to the change in interaction energy,

where & 9 ( x ) represents the difference between g ( x ) and

In a state with nonrelativistic wave function +(X), appropriate to the restrktion Za << l , we have

(dx ) dV(x) /$(X) 1%

which uses the fact that the perturbaiion is significant only over distances that are small compared with atomic dimensions. The integration that appears here i s equivalent to evaluating the zero momenturn limit of Gl>,(Rf, and

Only S-states need be cansidered. For principal quantum number n,

and

the latter giving a conpariwn with the Bohr energy values. More details will not be supplied now since, as will be seen later, this effect is rather minor compared to

another that displaces the s-states in the opposite sense. The existence of the vacuum palarizrition effect must be inferred from the quantitative camparison with experiment; in its absence a small but significant discrepancy with experimen t would remain,

II is particularly interesting, then, that observations exist which point directly to the existence of the vacuum polarization phenomena, They are the g-factor measurements of electron and muon, which disclose the small discrepancy

The displacement of #g from unity is produced by the exchange of photons and charged particles, each described by the appropriate propagation function. The substitution of D, for D+ introduces one aspect of the more complicated exchange processes which, in this example, is the consideration of three-particle exchange. An asymmetry between electron and muon is created through the domination of the vacuum polarization processes in 6LZ, by the light electron. The mass of each particle sets the scale for the signilicant exchange acts, The he&vy muon is therefore more influenced by the vacuum polarization phenomena, To the extent that the massless photon and electron-positron. pairs of mass M << m, are not significantly different kinematically, the modified photon pmpagation function in the muan application is, approximately,

The integration limits indicate the range of M values for which the simple treatment, resembling the discussion of (&3.48), is applicable. The factor that appears here,

measures the increase of qg, relative to gge,

cx 2% m, S& - 4ge 2 log ----

2 ~ 3 3 ~ M,

in qualitative agreement with (4-3.58). Later in this section we shall give a mare precise treatment, which will further s u p p r t the interpretation of the difference between g, and ge as a vacuum polarization phenomenon.

But first let us return to the vacuum amplitude coupling term (4-3.13, 191,

Fhcton propagaelon function 37

dM2 f (M%) dw, A"; (-- k)

and proceed to make its gauge invariance explicit by introducing fieEd strengths

FBIP(k) ikrA ,(k) - ik,A ,(K), (4-3.63)

according to the relation

The resulting f o m of (k3.621,

leads directly to a; space-time extrapolation which we present as an action expression :

dMz M2a (Mz) ( - 16) (dx)(d%")Fl"9fx)[Lt.(x - xi , NZ) + cant. termIl",,(x",

where

the last version refers to the spin example. The contact term indicated here is required to maintain a gh ysical normalization candition- the action appropriate to photons (K2 3=: 0) must not be altered, This is achieved by the conlbination

or, written in momentum space,

Thus, a more complete action for the electromagnetic field is given by

The locality necessary to define a Lagrange function no longer exists, in general. But, if we consider fields that vary sloMIIy over the interval l / M < 1 [(Zm), one can simplify (4-3,70) by substituting x for x'iin the field stmcture, Then, using

together with [Eq. f&3.W)]

we can replace the last term of (63.70) with

In. this limit there is a Lagrange function:

It implies the modified Maxwell field equation

An exact solution of this equation would have no meaning, since it is restricted to the circumstances indicated by a2 m , A~eeodingXy, we d t e the approximate solution as

apart from a g a u e term. The explicit expression far W is now given by

or 1 1 (d~) (s l~ ' )J" (x )L)+f~ - xf))J,(x" ) - -- 15nm2 2 (d:)]" ( X ) ,,I, (4 (4-3977)

which implies the modified interaction e m q y of two static &we-cunrent distributions :

4-3 Photon prowgatlon functfon 39

ar, f (dr) (dx')J:(x)B(x -- x6)],,(x') - ---- - 15% mZ ( a g ) JZ(x)J,, (X). (4-3-78)

The additional~ontact energy term restates what we have already seen in (4-3.53, G@, which is the example of the two charge distributions Ze b(x) and - e Xt is the slow variation of the latter charge density that validates the present approach.

In general, the field equations derived from (4-3.70) imply a modified propagation function,

such that

Thus we now get

to be compared with the previous result [Eq. (4-3,28)]

The latter would be reproduced by retaining only the first terms in an expansion of the inverse expression that appears in (4-3.81). The use of the action p~nc ip l e to derive modified field equations has given an improved account of the altered propagation characteristics that are implied by the vacuum polarisation phenomena, I t is in this context, incidentally, that such terminology acquires meaning; the modified field equations can be presented in Maxwellian form, with an additional current flowing in reeons where the field is rapidly changing. Accor&ng to the underlying physical picture it should Be possible to exhibit the propagation function (llP3.81) in the form (k3.821, but with a different positive weight factor:

The function A(M2) can be determined by a cornparkon of imaginary parts for - k2 = M2? using the relation

This gives

which is a positive quantity. As a sinnplificirtion of this situation, which is not very drastic, let us treat

(2m)2/M2 as negligible almost everywhere, in a(M2) :

That permits the elementary evaluations

and

In applying (4-3.87j to negative values of k2 + 4m2, they should be reached from the lower half of the complex plane, according to the instruction contained in k2 -- ia. Tbat supplies the logarithm with an imaginary term of - ni, in agreement with the imaginary part of the left side, ni(-- M"ja(lM2). This simplified version gives

and

Ate the* two expressions really equivalent ? This mathematical problem Is best approached by regarding K2 as a complex variable* The function (43.89) has a pole at k2 = 0 and a branch point at k2 = - 4m2, both of which are correctly represented in (4-3.90). But (4-3.89) also possesses a pole at

which is not represented in (4-3-90), since it is an unphpical singularity at a

space-like value of k. This is a mathematical failure--it is not a significant physical one. As in the discussion of (P3.48), the values of momenta, or distances, that: are required to have the logarithm overcome the smallness of a/3n are utterly beyond the level of any caneeivable physicaf relevance. For all physical purposes, (P3.89), or its more precise version (&3.81), is correct, within the limitations of the phycslical proces*s considered. And there is no rewon at this point to believe that the formal failure is other than physical incompleteness becoming apparent under the conditions of an outrageous extrapolatian. Encidentalb, the photon propae;alion function of (4-3,895 indicates that a more general statement of the g-factor a y m m e t ~ y between electmn and muon is given by the repfacement

This would indeed represent the most important aspect of the higher-order effects if the Eogasthn were a large number; however, l~g (m~/m, ) == 5.3.

The last remark brings us back to the question of improving the estimate (68.61) by removing the sversimplifieation inherent in (&3,59), where the masses of electron-positron pairs were ignored. We must repeat the calculation of Section 4-2, =placing the photon propagation function B,(% - x") by d+fx - x', M'%), and then intepate over the spectral distribution of M'% that is contained in the modified propagation function D,(x - X'). The immediate kinematical changes in (62.17) are indicated by

and

which uses (4-1.23,42). The replacement of m% by m2 - eqaF is again cornbind with the rjubstitution M2 -+ M2 - eq&. But, instead of the explicit change of d, - (m2/M2) that is evident in (&2,X9), we must now use (43.94) and the weak fierfd expansion

42 ElarodynsmJes 1 Chap, 4

This replaces (62.20) with

-- i(4n)P@-ld +(X -- X'. M%)G$ (x , z')

where, for simplicity of presentation, some general factors have keen transferred and the common argument of the step function q and the delta function is left unwritten. The pattern of calculation that gave (4-2.26) now produces

where only the explicitly field dependent terms are exhibited. The step function that agpars here can be converted into a delta function by performing a partial integration :

in which the integrated expression vanishes at both Xirnits. The effective value of the field dependent coupling that is appropriate to a +

obeying (4-2.33) is alttrii;ned through the substitutions

The delta function will be used to determine &l2 as ;t function of the parameter u,

We obtain: in this way the falfowing expression for the ewffieient of - (al2s) X (Il2m)eqo"F in the effective action :

where

Only algebraic rearrangements have b e n used to arrive at this form. But now Ive also recognize that the second integrand is a total differential, af the function

which vanishes at both endpoints, Here* then, is what replaces unity, for the photon, when an excitation of mass M' is considered:

In the approximate treatment that gave (4-3.W, 61) this factor was crudefy represented by unity far M" m, and zero for M> m. Now we combine (4-3.104) with the weight factor of (4-3.281,

where

and m' is the m s s of the charged particle contributing to the vacuum polarization, The resulting cdouble integral i s

We must note here that the electron vacuum polarization contribution to ge and the rnuon vacuum poladzation contribution to g, are equal (m'== mm). The asymmetry is t fiat between the electron vacuum plarization contribution to g, (m1 = me, m = m,) and the muon vacuum polarization contribution to g, (mf m,, m 2=. me)- The large m a s ratio pernits the i n t ~ d in the fatter situation to be sinnplifid :

44 Elaradynotmica l Chap, 4

This is a v v small effect. For an appwimate treatment of the situation m'jm -- m,jm, <( l , the integral of (63.107) is rearranged as

and (2m,jmuj2 neglected in all but the first of the three terms. Performing the v integrations, we get

and (4-3.61) is amended to

It is desirable and possible to improve this estirnate by retaining terms of order -/m,. They appear since f2me/m,)2u[(l -- u)% ceases ta be very small for u values near unity, within a range - m,[m,.. One procedure for their evaluation begins by perfoming the v integration in (4-3.109) to produce

where

The 21 integral is then divided, into two parts, in the first of which X is small everyhere compared to unity while, in the second one, x can assume large values but u is suaiciently near unity to validate the differential relation

After rearran@l;ing the two terns to remove the arbitrary junction point, one obtains the following addition to (43.1 10) :

Partial integration converts this into

and, on restoring the v parametrization, we get as the factor of (4a/n)(m,/m,f:

This improved version of (4-3. l f l) is

The use of the numerical value

changes the estimate (4-3.61) to

The general agreement with a recent experimental value, (0.66 & 0.03) x IQeg, i s striking confirmation that \7acuum plarizatisn gives the principal contribution ta this effect, As to whether or not the residual difference is reaf, we remark that, in addition to various small corrections such as the one indicated in (4-3.92) [perhaps we shall return to this topic in later chapters], there are pliysical processes yet: to be cansidered which involve par"ticles other than electron and muon, The n-mesan, for example, is not much heavier than the rnuan and might make a significant contribution to the asymmetry, This effect can be estimated by changing the weight function of (43.J105) to accord with the spin O spctral form (4-3-39) :

Tlre relevant integral, replacing (4-3. f Q?), is then

46 E l ~ r d y m r i c r I Chirp, 4

Even this upper limit wodd barely change the last significant I i p re in (4-3.120). The calculation makes no reference to the strong interaction properties of the n-meson, however, A pair of oppsitely charged R-mesons is strongly coupled to the neutral p-meson, which has the same quantum numbers as the photon, I t is the direct coupling of the photon to p@ (and o, and 4) that may be most significant in vacuum polarization phenomena that are outside. the eIectron-muon framework, Although we are not ready to give a quantitative discussion of this question, the stage can be set by noting the simplification of (4-3.104) for such massive padicles :

The strong interaction suppfernemt to4he photon effect is the weighted average of (43.123) over the heavy particle contribution in the modified photon propqation function. We shall make explicit an electromagnetic factor, aln, in defining an average inverse squared mass, This gives the additional strong interaction contribution as

or 2a (*g& fge)strong int. = - 2z 3n:

The use of the p-mass in an estimate indicates a substantial contribution to the last significant t i p re in (4-3.120),

The final application of the modified photon propagation function that will be discussed in this section also lies somewhat outside the realm of photon- charged lepton phenomena. The neutral n-meson decays primarily into two photons, A small fraction of the decay procesws involves a photon and an electmn- positron pair; in a wnsiderabIy smaller fraction of the events, two electron-psi- tron pairs are emitted, A description of the relations among these processes is provided by the modified photon propagation function.

The effective coupling for the two-photon decay of the 0- pion resembles (3-13,75), which is a description of the two-photon annihilation of an electron- positron pair; the pseudo-scalar pion field +(%) replaces the quadratic field combination ++(x)y0y6#(x) :

where f is a suitable coupling constant. The role of the field + ( X ) as an effective two-photon soume is made expIicit by comparing iT;r/+-,, with

This gives

i JY(x) Jq(x') ldf. = f e~**~a~a l (d (x - x ' ) ~ ( x ) ) . (4-3.127)

Two such sources, causally arranged, have a two-particle exchange coupling that is given by

t [ i 5 (dx)(dxl)J:(x)D+(x - x 7 ~ ~ . ( ~ ' ) 7 I at.

which uses the relation

- f ~ ~ " * ~ & , , ~ , , = d:dj - 68;. (4-3.129)

The combination in brackets is more usefully presented in the form

)(aP)*(D+(x - X'))* - @(D+(% - xl)@D+(x - X')) + @ + ( X - X ' )

The space-time extrapolation of this coupling, performed with due regard for mass normalization requirements, supplements the quadratic field structure of the action. When one considers a field far from its source, and essentially governed by

(- @ + m34(x) = 0. (4-3.131)

the effective Lagrange function becomes

where &imny4* is the reduced form of the additional term. (A change of m: is excluded by mass normalization.) The latter describes the instability of the particle. For weak instability

y << m,: m: - im,y (m, - iiy)', (44.133)

and the time dependence of the field associated with a particle in its rest frame is such that

lexpC-- i ( m ~ - ~4W"l l* = exp(- Y#), (4-3.134)

identifying y, the reciprocal mean life time, with the total decay rate. The calculation of y, with the aid of the modified propagation fnnction D+, makes explicit

@ Elardynmmics I Chap, 4

the contributions of the various processes involving two photons, one photon, or no photons.

Let us illustrate this procedure by evaluating the two-photon decay rate. Here, only D+ is considered and, under the causal conditions assumed in (k3.128, f30), a2D, = 0, The coupling thus reduces to

where the second step anticipates the kind of field [Eq, (43,13133 to which it will be applied. Now observe that

dw, dwxe(2n)a 8(k + k' - P)~-,~_~z

according to the integal (4-I,%), with m, = m, =r 0. When the space-time extrapolation is performed, the structure becomes $H, and, after removing the factor of i , we get the additional action expression, The restriction to a field obeying (4-3. f 31) implies the replacement

Using the imaginary part of this factor, nis(MZ - m3, we pick out the addition to the action that is of interest,

which gives

4-3 Photon promaefon function 49

A quick indication of the rates for the single and double pair pr infemed from the kind of simplification illustrated in (&3,59), with rra, appeahng as the upper limit to electron-positron pair mwses. Since two propqation functions are multipked together, the decay rate acquires the factor

which displqs the relative rates of sin@e and double pair processes,

rwpectively A mare accurate evaluation of y"y will now be given. Af-ler inserLing the

general propagation function f o m (&3.83), we must extract contfibutions with one B,(% -- x' ) function and one d+(x - X ' , Mt2f function. Having in mind that m: will replace the iP operations that can be transferred to the fields, the effective fom of (4-3. E30) for this groeess becomes

Thew propagation function products are

which uses the i n t e r d (4-1 .B), Comparing this vvitb the earlier calculation, we infer the ratio

The integral will be evaluated, with sufllEicient precision, for the wekht function [Eq. (43.67)]

Ica Etetrodynamtcs l Chap. 4

This is done by decomposing the integral at a value of M'% = M: such that

which. gives the simplifications exhibited in

The parametrization indicated in (4-3.105) can be used to evaluate the first integral, and

consistent with the estimate of (4-3.141). The numerical result obtained in this

is in excellent agreement with the measured value, (1.17 -& @,Q41 x 10-2. Con- cerning y " 3 e shall only remark that the improved caleufation essentially rnain- tains the simple relation indicated in (4-3.l41),

4-4 FORM FACTORS l. SCA-ERINC

At the beginning af this chapter, we oberved that the introduction of modified propagation functions is an essential but incomplete characterization of multi- padicle exchange procesws. In dixussing the primitive interaction and the interaction processes that produce the modified photon propwation function, it was indjcated that these interaction processes have other consequences, which are represented by modified electramapetic couplings. We now begin an extended consideration of these matters.

The spin 0 primitive interaction portrays the electromwnetic vector potential

as an effective two-particle emission source [Eq. (G3.33)

provided the m a s threshold is exceeded,

These oppositely charged particles are produced un&r noninteraction conditions. In the course of time, however, the particles can interact, in a way described by the scattering calculation of Section 3-12, We shall; write the vacuum amplitude discussed there [Eq. (3-I2,83)] in a more convenient form that restores the multi- component marces in charge space:

Its vatidity depends upon the selection of appositely charged particles that is perfomed by (44.2). On inseeing the fatter this coupling becomes

K,( - @l)cq($l - p; )'K,( -- P; )I,J'(k) S

where

do,, dw,; ( 2 ~ ) ~ 8($p f - k) (P1 + PP)&; + P;_! (po - p;), (4-4.6) ( P I - Ps)e

and

The instant evaiuation of the last inteval conveys the recognition that it has already been encountered in the spin O vacuum polarization &scusion [Eqs, (k3.36, 19, 3713. Let us also note that the cunent vector

evduatd in a; re@on that is prior to the action of the detection source K Z , is

32 @lsctrdyamics 8 Chap. 4

given by

Accordingly, the second term of (44.5) is

whicll is the recognition of the spin U form of the modified photon propagation function [Eq. (4-3.39)]. Here is the anticipated mechanism that substitutes D, for D, in the vector potential of the primitive interaction,

Our attention is now concentraked on tlte first term of (4-4,5), and the integral (4-4.6). This vector must be a linear combination of the final momenta p:, p;". Now, the interchange of these momenta, combined with that of P , and P;, reverses the sign of IY [recall that (p; - = (/ll - p p ) 2 . \\'e conclude that

(P, + P,)($; + P ; ) ( f , p , ' ( P 2 - pi dw,, dw,; ( 2 ~ ) 3 &(p2 + P; - k ) - -------- - - (P, - P,)% (P, -- p; t2

1 - - -- (P3 i P , l ( p +p;! (p_, pap2 - p;) ( 4~1% (91 - P d 2 BP! - 4ezZ

Xn the rest frame of the vector K , at1 four particle energies equal +4Tl , and the integration that evaluates (44.12) is extended over the scattering angle 8, that between p, = - p; and pp = -

At this point, we must acknowledge that the angular integral has a logafithmic singularity a t e) = 0. Et is a consequence of the unlimited forward scattering that the long-range Coulomb potential produces. Were is the reminder by the for- malism that the primitive interaction context of noninteracting particles cannot be entirely realized for char@ particles, We are encountering another aspect of

the infrared problem, since the z;ero mass of the photon is responsible for the unlimited range of the Coulomb potential. This suggests that the &fficulty is only suprficial and will &apvar when additional soft photon processes are con- siclered. For the moment, however, we choose to bypass the problem by imagining that the photon haus a very small but finite mms p. Recalling the wigin of ( p , - $ 4 2 in the structure of the p'haton propagation function, we recognize that the singular integral of (44.13) becomes

d(cos 8) i - f M% - &E%

2 (M% -- 4mP) sinB$@ + pe - MP -- 4 2 log p% (M. 14)

and

Inspecting the farm of the first term, in (44.6), that results from (44.11) we see that the vector structure of the current (44.9) is again present, permitting this te rn to be displayed as

dwk exp[ik(x - X ' ) ] (M. 16)

But, prior to performing the space-time extrapolation that now lies before us, we must notice one p i n t concerning the gauge invariance of this expression. I t is gauge invariant, indeed, under the causal conditions being mnsidered, since

Being rooted in the kinematics of free particles, however, this property will not be maintained after the space-time extrapolation is perfomed. Accordiingly, we must rewite (44.16) in zt way that is without consequence for the causal situation but assures its gauge invariance in general. Retuming to the momentum space for a moment, we obsewe that

differs only by a gauge transformation from A ,(R), and can replace it in (4-4.XB). This is the substitution

Chap, 4

and the resulting space-time extrapolation of (44.16) is

tvhere we have defined

The action term that combines the primitive interaction with tfze two interaction-induced modifications that flave just been evaluated can be preserrted as

aftthough this goes some\vtlat beyond the explicit calculations in its uniform use o f the modified field

The four-dinsensional n~onnexlturn space equivalent of (4-4,23) is

dM2 /(MS) fuv (k ) = g,, -i- (k,kv - g,,K2) - - -------.

M2 k%+M%-zg

X f tlre vector potential is restricted to a Lorentz gauge, and (4-4.24) is so written, the k,k, or - a,a, terms disappear and

I,v(k) = g,,F(k),

in whit 11

The corresponding farm of the interaction (44.22) i s

the latter way of writing it empt~asizes that the current effective in interaction

4-4 form farctora E, *eerfng $S

with the vector potential at a given p i n t is a wighted average, of the local field structure p, over ail space-time, The tern "form factor'bapplied to F(x - --,X')), or F(k) , describes the role of this function in prducing an additional distribution ox shaw of the charge represented by jB(x). It should be kept in mind that the single form factor F(k ) is a simplified way of presenting the tensor form factor f,,(kf.

Before discussing physical implications of the form factor, Ielt us repeat the calculation for spin 4 padicles. The effective two-particle emission source associated wit h the electromagnetic vector potential [Eq, (4-3,5)] is conveyed by

Since it is now quite clear that the annihilation mechanism of scattering prduces the modified photon propagation function, we shall consider only the vacuum amplitude for the Coulomb deflection of the particles. To present this in a form uefrxl for our pumst;ts, it is simplest to return ts the interaction expression

and introduce the causal field structures

and

This @ves the following vacuum amplitude term :

The insertion of the effective souree (6-4.29) then prduces

UC tl&rsdynamicr I Chap. 4

with

The g a u e invariance of this coupling, as expressed by

is a consequence of the algebraic properties of the projection matrices m - ypp and -- nt - yp; :

A reduction of the matrices in IU is effected with the aid of the pfojeetian matrices that appear in (41.34). m - ypl and -- m - ?p;. They imply the equivaleaee of m with a matrix - y$, that stands an the left of IY, and of m with a matrix yp; that occurs on the right. This is combined with the algebraic properties of the matrices in the following:

where we have also used the relations

P; - P; = - (p l - p,), yvr"n = gr" (44.39)

Further reductions are indicated by

4 4 Form t.ctors I. w r i n g 57

- ~ ( $ 1 - P~)@;Y' - ~ ( $ 1 - P~)YP;Y' + (P1 - Pz)?" - my(P1 - P2)yP + ~ P ? Y ( P ~ - Pe) + (P1 - P214Y. (4-4.41)

These lead to a form suitable for integration,

yV(m - YPS)Y'(- m - @Jyrd - 2 ( ~ ' - 2m2)yP + 2(pl - PZ)Y - 4m(P1- Pe)" + 4(P; - P ~ ) " Y ( P ~ - Pe). (4-4.42)

Using the notation indicated in (44.12) we now have

First, let us observe that

because the interchange of pl and p;, combined with that of p2 and pi, reverses the sign of this vector. Multiplication of both sides with 2plP gives

a = 1

W - M ' (M-&) since

2Pl(Pl - PE) = (P1 -

2p,(p1 - p;) = (p1 - p;)% = Ma - 4m3.

Then, using the reductions

( P i - P;)'' = - HyP1 yPi - YP;) - - w ( y k - 2m) - #(- yk - 2m)yp

= 2myP + &PVk,, (4-4.47)

we get

A related contribution can be separated from the last term of (44.43):

Chap, 4

Now consider

where the right-hand side gives the general form of a symmetricag tensor that is unaltered by interchanging p, and p;, and vanishes on multiplication with k,. Contracting indices on both sides supl?lies one relatian between b ;and c,

The other is produced by multiplication with (p1 - p;), = (2pl - k),,

which involves the obsenration that

(P;) = &(PI + P;)'. The result,

is displayed ER

m --. - ir@ + -- (2%~"" + Satrvk,),

M2 - 47122 (M. 55)

which also contains the remark :

yk = ( Y P ~ + m) + (ye; - m) (44.56)

Putting these evaluations together eves

As in the spin O discussion, the remaining integral is altered to

M2 - 4m2 log ---

P%

and we now get

where

and

Despite the apparent detailed dependence of IY [Eq. (44.35)] on PE and 9;. the kinematicat reductions have produced a form [Eq. f44.59)] in which just the total momentum R is in evidence, Accordingly, the reference to the final particles in (H.34) occurs only in the two local fieid earnbinations

I I I ( - P;) =P[- + l 4- $;)xlP (44.62)

and

4lbl (~)P@P@~~#X (4 = 4 do,,dw,,*rl,(-- P1)yO(m - ?$l)e~@~'(- 9% - ye;)

The coupling ( H . 3 4 ) thus appars as

60 EElaetroclynamiu I Chap. 4

where gauge invariance has been made explicit by the substituti0:on (M.19). The space-time extrapolation of the additional coupling is combined with the pfidtive interaction to @ve the action term

in which

In a Lorentz gauge, this tensor can be repla~ed by

f ,,(a) == g,vF,(k)

with the charge f o m factor

Notice that

which states that the primitive interaction is not modified far slowly varying << m2). We shall give the same normalization to the second fom

factor,

According to the observation that

this is achieved with

and then

As the notation indicates, the last result is a rederivation of the additioxlal magnetic moment appropriate to homogeneaus fields [Eq, (4-2,3(1)]. Tlie dj.nami- cai origin of this magnetic moment is evidenced here in the vanisfling of the magnetic moment form factor F 2 ( k ) for k;2 --+m, Pin asymptotic statement is

2m2 K2 k2 >> m2: Fz(k) ..J - log -

k2 @ l 2 " (44.74)

it follows from the general form

For - R2 > 4m8, incidentally, the phase of the logarithm is appropriately chosen to give the imaginary part of the integral, nzfz(- k2). The situation seems different for the charge form factor, where

a I k2 k2 - -- log - fog - , 2 z k 2 m2 p2

tvhich exhibits only the dominant logarl thmic terms. Since this function is multiplied by - kg in F , ( k ) , the addition to the primitive interaction does not vanish as K2 + W . But another view of the ma"cer appears on reinstating the tensor form factor f , , (k) , for

62 E8mrdynarnio 1 Chap. 4

contaxns two different coupgngs. One is the pt-imitive interaction, modified by vacuum palarization effects, and the other is a dynamically induced coupling wlth the field a9P,,, which dses vanish as k" m.

The latter remark of pfinciple notwithstanding, the practical ccznequenee of the charge form factor stmcture [Ea. fM.68)] is a progressive weakening of the corresponding interaction with increaing P. This will be manifefted in energy bvel displaeernents and in scattering cross section alterations, As a first indication of these effects, consider slowly varying fields for which one can simplify

F, (h) to

The integral that appears here, according to (M ,7V , is

and

This effect can be earnpared to and combined with, the vacuum polarization increase in coupling, which is described, in (4-3.76), for these cirerxrnstances by

Thus,

which swgests that the form factor reduction quite overpwers the vacuum pcalarizaition strengthening, although the effective replacement for the transi- tional param&er y must be clarified. Xn discussing energy l e d displacernents, for example, it is quite plausible that some atornic excitatisn energy dE stands in the piace of p. The replacement - 6. -+ log(mfAE), which ipores additive constants, converts the vacuum palarizrrrtion formula (k3.67) into

dEa 8 2 % ~ ~ m P''&--.---- log - '

1 &am 3% n AE 2 ng

This is indeed a correct indication, but we shall defer the more precise discussion, turning now to the consideration of scattering mo&ficatbicms.

4-4 Form #mars; C, Swttering 63

We begin with spin O particles that are scattered by a Coulomb field. In deflecting a particle from momentum p2 to momentum p,, the Coulomb field supplies the momentum

The effects we have been discussing modify the differential crass section by the factor

which contains only the spin O contribution to vacuum polarization; correction terms not linear in a have been discarded. To this elastic cross section must be added, at least, the cross section for scattering with emission of EL soft photon having any energy less than k;,,,, the minimum detectable photon energy in the experimental arrangeme~~t. This produces the physically meaningful cross section for essentially elastic scattering, which can always be supplemented by the cross section for radiative scattering with any degree of inelasticity.

In evaluating the soft-photon addition we must be consistent. The introduction of a photon mass requires that the photon he treated as a spin one particle, with its tllree polarization states. In practice, however, the only significant point is the altered relation between photon energy and momentum,

The transition matrix elernent tor soft-photon emission during deflection in a Coulomb field appears irt Eq. (3-14.61) as a product of two factors. One of these describes the elastic scattering process; the other is

To be accurate, the denominators should be altered slig-htly, since

However, the corrilctions are not larger than the ratio between p and a partide

64 Electrodynizmico f Chap. 4

energy, which is completely negligible, h r we certainly require that

One may be given pause on noting that the polarization vector summation

contains the factor l /p2, far, wit h the altered denominators, the effective photon source given in (44.88) is not conserved, Happily,

has enough powers of p in the numerator to save the day. And, of course, (44.88) is an approximation to an expression for which conservation holds exactly.

The relative soft photon emission probability is now obtained as

which uses the combinatorial device introduced in (3-14.1 16). After canying out the angular integrations this becomes

where p2 = p12 = pz2.

The remaining integral has the form (k:,,. >> p)

Form fadera; I, Sattcring 6;f

and the resulting evaluation of fM.95) is given by

X log * (44,9&)

Although the v parameters of Eqs, (4-4,861 and (4-4.98) were introduced in very diflerent ways, the two expressions can be added, and the fictitious mass y does indeed cancel comglctely . Let us consider first the nonrelativistic situation, where q% mm2, p2 m%, g m. The elementary evaluations of the two terms are exhibited in

Also available is the nonrelativistic differential cross section for all inelastic processes, comprising photon energies that range from kz,,. up to the kinetic energy T of the incident particle [Eq, (3-14-70)]. Presented as a fraction of the elastic differential cross section, this inelastic contribution is

cos 8 1 - (n - 0) tan 4-0- cos2BBlog3

Tile essentially elastic and inelastic cross sections combine to give the differential cross section for scattering without regard to the final energy af the particle. It is the fslowing fraction of the Rutherford cross section:

V"

6s-

" 11

E ..--. 25~:

Ur(

*+ir "C

"

2a qs m 7 1 --- cos9 3n , [log + + (n - B) tan 49 + cos d. log&] sxn +e . (44.101)

These dynamical modifications of scattering cross sections remain quite small until we penetrate into the relativistic region. To facilitate the consideration of high energies, we introduce the symbols

and aIso make the scattering angle 6 explicit in

As W grows from 0 to 1, the new variable f changes from 8 cos 49 to 8. In the special situation of back scattering, 6 = n, we have 5(v) = DV. With this notation the complicated v integral of the second term in (4-4.98) appears as

which also uses the relation

We now observe that

where the right-hand side is only weakly singular as f -, 1. A similar statement, with replaced by B, is also useful. After defining the quantity

one can present (44.98) in the following manner,

(4-4. l OS)

This is not the final version, however, for the Iast term of (44.108) has a more relevant fnrm, which is the content of an identity :

i t i s stated for space-like g. The proof' begins with this identity,

which is verified by performing the u integration. Now let us relabel the u yaram- eter of the first term on the right side as v" and define

The new v variable also ranges from 0 to 1. The transformation implies that

h i c h emphasizes the restriction u 2 1 - v2, for an assigned a. The new form of (M.1 tO) is

and the identity (4-4.109) incorporates 'the value of the u integral, log(4ve/l -- v2). Now we can effectively combine (4-4.108, 109) with (4-4.86) to produce the

dynamical correction factor

The complexity of this structure resides entirely in the integral L, Eq. (P4.107), with its somewhat involved relation between the integration variable v and

There are some simple limiting situations, however. At low energies,

from which we recover (4-4.99). In the high energy limit, P -c 1, and

1 Po>>m: L =-

sin (812) +(cm P ) , (4-4.1 17)

where

The integral can be performed analytically for scattering at the angle 8 5 x,

which also continues to be quite accurate even at t3 PS 4 2 . Close to the forward direction,

+(COS 48 N 1) N 4 sin 48 log [ 1 2(1 - COS 48)

+l ] . (4-4.1M)

The accompanying high energy form of (44.1 14) is

(4-4.121)

which uses the high energy limits of the various integrals,

The scattering of spin .4; particles introduces a new feature, for here the magnetic moment interaction also plays a role, Compared to the discussion of Section 3-14, the dynamical corrections have altered the coupling, as indicated by [Eqs, ( H . 6 5 , 67)]

When only the s d a r ptential appars, descr;ibing the modified Coulomb potential, this reduces to

The transition matrix element, replacing (3-14.1 X ) , contains

When helicity states are used, as in (3-14.f3), we now encounter

~ ~ ~ n ~ ' i y ~ ~ (PI - ~ e ) ~ p ~ . , = !P! - ~ 2 ) ~ p T , , ~ @ i ~ s ~ ~ , ~ , . (4-4*126)

where, it is recalled [Eq, (2-&.W)],

Aecordingl y,

and

which states that the magnetic moment interaction contributes only to transitions in which the helicity changes. On refemng to Eq. (3-14.13). we recognize that the scattering intensity will now contain the factor

[($)l cosPt8Fl (q)P + sina@ ( F ~ (q) - 2n m8 FP(q))l A@(q)'. (46.130)

Relative to the differential cross section given in (3-14.14). this implies the following dynamical modification:

where only terms linear in a have been retained. The spin 0 discussion can be transferred, with minor changes, to spin 4. In

addition to the explicit magnetic moment term, it is only necessary to account for the slight difference between the charge form factors,

and to substitute the spin description of vacuum polarization,

The combination that appears in (46.131) is altered by

which gives a rapidly converging contribution to the spectral integral. This is significant in high energy behavior for it means that aII terms which increase logarithmically with energy are identical in the two situations. Indeed, apart from the last term of (4-4.131), the difference between that expression and (4-4.86) is

Since the additional magnetic moment effect is negligible a t high energies, owing to the (m/$OjB factor, the spin 4 analague of (44.121) is immediately obtained:

401 1 -- 17 -+ -- + g sin #@+(cos g@)

2% 72

In the nonrelativistic limit the value obtained from (44,13Bf, - ( 2 ~ / 3 n ; ) X (q21wn") 3, added to the magnetic moment contrjbution, - f2c~j3z) (q2/m2)$, converts (44.99) into

and the analogue of (4-4.101) is, correspondingly,

+ (JE - 0) tan 4s + The ene ra l spin 8 formula, written in the notation of (&4,122), is

Concerning expenmental verification of these spin Q predictions in electron scattering, we shall only remark that such agreement exists, and that the dynamical effects we have discussed are now routinely included in analyzing the very high energy electron scat tex-ing measurements that are exploring the inner structure of the proton,

With an historical gleam in his eye, Haroitd asks a question,

W. : I know that the spin & results you have been describing were first derived by you many years ago, My question refers to a comment made shortty after and occasionally repeated since, that you had made two mistakes in that early calculation, wllich somehow managed to cancel and give the fight answer. Would you comment on this remarkable achievement ?

S.: I tllink 1 have been given too much credit. No one is clever enougll to make two suclt complicated and exactly compensating errors. The real situation is far less spectacular. Et is artificial to divide the essentially elastic scattering cross section into an elastic part and a soft photon addition. Neither component has physical significance, and alternative methods of calculation will give different expressions for the separate parts while apeeing on the sum. fn particwtar, one can perform rearrangements in the two component terns, which may or mag. not be justified individually, but find their validity in the complete structure, Thus, there are several opportunities to exhibit different forms for the two pieces, white retaining a fixed answer for their sum. Perhaps X should confess that I was in no doubt about the correctness of the result because X had performed the calculation in two different ways, with concordant answers, I was, a t the time, very insistent an working only with the physical massless photon because the operator method, then in use, alters discontinuously as the number of degrees of freedom changes with the introduction of a photon mass. Nevertheless, f carried out the calculation using a finite photon mass, but did not report i t since it violated my self- imposed ground rules, Na such inhibitions about the transitory use of a photon mass hppear in tl-tese gages because we know that the description of a massive spin 1 particle with conserved sources is connected continuously with that of the photon, But, surely, the proper conclusion from all this history is that a better method must exist, one that gives the pflysicaf cross section directly, without the intervention of nonphysical distinctions. Mre shall describe such a method later.

4-5 FORM FACTORS #l, St NCLE AND DOUBLE SPECTRAL FORMS

The primitive interactions we liave been discussing, which are linear in the photon field and quadratic in charged particle fields, have two different applications where just one source operates in the extended sense to emit two noninteracting particles. The subsequent interaction of these particles modifies the effective ability of the source to emit, or absorb, the two particles, That has been considered for extended photon sources emitting a pair of charged particles. Now we turn to a particle source that emits a charged particle with an accompanying photon, and examine the modification produced by the scattering interaction of the particle and photon. The description of a charged particle source involves a choice of electromagnetic model. The simplest of these is the covariant model in which no charge acceleration radiation occurs. But, then we must pass over spin 0 and proceed directly to spin 4, since no radiation appears in this spin 0 situation.

We recall the effective two-particle source

';r;"Z.l?z(P) l&*, 'lirpyuqlrzfl"") - ')11~2(')1 J (4-6.1) where

C S Form facton II. Single and double spectra1 forms 73

Tl~e subsequent scattering process is described by the vacuum amplitude [Eq. (3-14.137)]

i f du,, dwk, do,, dok,(2n)4&Pl + k1- P 2 - k2) I

Here, using the notation

1 d o k , d w , , ( 2 n ) 3 d ( k 2 + p 2 - P ) ( ~ ~ ~ ) = - ( 1 - ~ ) ( ~ ~ ~ ) , ( 4 ~ ) ~ (4-5.5)

we have defined

1 1 r ( P 2 - kl) + m

and

The conservation requirement on the effective photon emission sources is satisfied in the following way,

74 Electrodynamics I Chap, 4

owing to the explicit presence of the projection matrix m - ypz and the appearance of nz -- ypl in (4-5.4). We begin the calculation by intrducing rearrangements that are desiped to satisfy this requirement explicitly:

where expressions containing yPz + m on the left have been discarded. The three terms appearing in the final version of Eq, (4-5.8) satisfy the conservation condition individually, since

Tfiere is, incidentally, no significant difference between y ,ykl and - ykl y ,, for any multiple of kl, can be omitted when applied in (&5,4). Both of these matrix products can be replaced by - io.,,ki.

The integrations involved in evaluating ( . * ) are illustrated by the rest frame calculation of (l f (-- fi2kl)) ,

where z is the cosine of the scattering angle. Other such integrals contain the additional factors p2# and p"lf12v. The oppsrtunity to omit multiples of K,, simplifies these calculations, Thus,

where a possible multiple of pl, does not appear since the left-hand side vanishes on multiplication with k:. A similar reduction gives

in which

4-5 Farm faetera It , Slnglrs: and double spectral forms 75

This psitive function varies from a value of Q at tW2 == m2 to unity, for MO >) m2. The outeome of these calculations is given by

The quantities we need have additional factors of yv or P' on the right-hand side. 8nly short reductions are required to groduce the results:

The combination that appears in ( 6 5 . 6 ) is

Since the only reference to the individual momenta of the final particles is in the matrix %,hi, all aspects of these particles in (4-5.4) can be combined into a 10a1 field prduct,

When attention is confined to the term that involves particle fields explicitly, the vacuum amplitude (66.4) becomes

76 E I ~ l c r I Chap. 4

Its space-time extrapolation is stated as

where #,L specifically distinguishes the field of the extended source. This asymmetry must be removed, of course. Accordingly, let us note that

which uses the rearrangement

and the integral value

When applied to the field #-,,, the factor m + yp inpa + m2 = (m - yp)(m + yp) introduces the corresponding source. Weshall group this with the other explicitly source dependent terms. Then the remaining field structure, stated as an action expression with a unified particle field, is

a eq 1 1 G G I I P " ( ~ ) # ( ~ ) ; (44.24)

we have derived again the additional spin magnetic moment of a/2n magnetons. Let us consider now a causal arrangement involving one extended particle

source and one extended photon source, the latter providing space-like momenta. This time we shall work with spin 0 particles. The extended particle source K2 emits a charged particle and a photon. The two particles travel to the vicinity of the extended photon source where a scattering process occurs, involving a virtual photon associated with the extended source, and the scattered charged particle goes on to be detected by the simple source K,. The vacuum amplitude for the scattering process is given by [Eq. (3-12.92), but without the use of a Lorentz kwgel

4-5 Form fseers I!, Single and double spectral forms 17

where

and

, f f , = ~ ~ [ ~ ~ ~ - ~ z > ~ + ~ f ~ ~ - ~ J ~ t ~ , ~ f r T , f ~ ) l ~ (4-5*27)

We shall only consider the 96% term in the effective source, which represents radiation by the charged particle rather than from the source. Harold interrupts.

H. : This question may be embarrassing, but 1 think it should be asked. X have heard that in your lectures an Source Theory you make frequent reference to space-time pictures of physical processes, which you call causal diagrams. Art: these the same as the famous Feynman diagrams ! And why have eausal diagrams not been used in this book P

S. : The utility of a diagram as an instructional aid varies with the circumstances, Xn a lecture, where constant attention to the subject matter is required, such visual aids can be indispensable. Reading a book is a different activity, however, The reader has ample opportunity to supply his own additional material. and should do so whenever this is helpful, T have p re fe r~d to put no dierams in the text, both to emphasize that the analytical structure of the theory has priority, and to keep down the cost of the book, But perhaps X should use the: causal situation now unclter discussion to illustrate causal diagrams. Thr-ee of them are needed to represent: the three terms of the vacuum amplitude (4-5.25, 26, 27).

78 EIsctrodynamics t Chap, 4

The following conventions are used: time is read vertically; circles represent sources; a thin straight line indicates the causal propagation of a, real particle; a thin zigzag line correspondingly refers to a real photon ; heavy straight and zigzag lines represent the noncausal propagation of a virtual particle and a virtual photon, respecti\7eXy. The various lines can be labeled by the fields or propagation functions which tEley symbolize or, as in tXlese pictures, by tile momenta of the several real and virtual particles,

Causal diagrams are not Feynrnan diagrams. The latter do not involve a distinction between real and virtual particles; 1;eynrnan diagrams are noncausal,

Returning to the vacuum amplitude (4-5.25, 26, 271, we observe that the tensor V,, has the following gauge invariance properties

which uses the momenturn relations

@B + kg =; $ l -l- k l , er - kn = p% - kl, (4-5.29)

and similarly

V& = 0. (4-5.30)

The vacuum amplitude contribution we are concerned wit h nstv appears as

with

where

The kinernatical relations that have been used in prducing (4-5.33) are

4-5 Form f a a r s 11. Single! a d doulbga S P ~ F I ~ I forms

All such relations are based on the physical photon property, k i = 0. Again using expectation value notdion Xor integrals of the. type of (4-5,32), we prewnt V, in the following m y :

The expectation value that occurs here has the vectorirzl form

Then, since

klfP, - *h,) = kzPlt

we learn that

I = - (M2 - m2)a + R12b.

A second relation can be produced by multiplication with p,, using

which gives us

The solution for a is expressed by

where

A = (M2 - m%)% + %(M2 + mB)kI2 + (h1%)%

= (M2 + m2 + k12)2 - 41MkZ, (66.42)

Conveniently evaluated in the P rest frame, the expectation value that appears in (&5,41) is

80 Etarodynamice Chap, 4

The vector V, is exhibited as a finear combination of the vectors 2pl -+ kl and K,. Apart from the factor (~z)-~[X - (%/M)%], the coefficient of 2pf + a, i s

where

After same rearrangement, I can be presented in the following way,

The coefficient sf kl is completely determined by the gauge invariance property

kl""V, ..; 0, (4-6.47)

wfiich specifies the combination

- klaj2P, 3- K,), 4- X.,,kl(V, + K , ) = - (k,2g,, -- k,,k1,)(2-P, + k,)"' (4-5.48)

We can ROW recognize that (k5.31) has the field structure

where

4-S Form fwe~rr C B, SImaEe and double spectral forms 95

and 1" appears as a spatially nonloeal but temporally localized opctration on the electrornag.netic field stll-enehs, A, space-time extrapolation af (4-5.89) is pra- duced by intrducing d,(x - X ' , M2) in place of its causal form for X@ > xo", But, as in an earlier disewsion of this section, a decomposition nrust be introduced [Eq. (4-5.221) in order to give a unified presentation of the coupling that involves the fields and 4%. In cambination with the primitive interaction, the resulting action t e rn can be d t t e n as

where

and

a f,,(k) = g,, f (k,kv - g,$%) 2;( dx I (MP, kg).

+tnf (M.53)

The f a m factor arrived at in this way should be identical with (44.25, 21). Or rather, it si~ould be had we intrduced a photon mass to avoid the infrared sinelarity in the integrd of (4-5.63). We shall be quite content to verify that the logarithmic infrared sinwlarities of the two expressions are identical, and omit the additional considerations associated with a nonzero photon mass. The sinplar term in (4-6.46) is the one with M2 - m2 in the denominator. This contribution to the integral in (&5.53) is

where the introductim of the photon mass in the lower limit of intagration is an essential but incomplete statement of the modifications that are required; it suffices to indicate the photon mass dependence, however. What we arc? being asked to eonfim is the following identity:

The two similar integrations can be performed, and one arrives at the common exprwsion [in the notation of ( 4 4 , l 22)j

8% Elwerdynamlu I Chap. 4

I"rH this is not very elegant, of course. Mfe should prefer to find a virtuow transformation of integration variables that would directly interconnect the two equivalent expressions. That ideal will be approached more closely in our next exercise.

Continuing with the discussion of spin 0 particles, let us consider a causal anangement of two extended particle sources, K , and K,. Causally 1oeaLed between them is an extended photon source, which is capable of transmitting spaee-like momenta. The extended particle sources emit, or absorb, a. photon and a charged particle. The photon propagates undisturbed between the two extend& sources; the charged particle is deflected by the extended photon source, M7hat we are describing is a generalization of the arrmgement used in Section 4-2 for the first derivation of the additional spin mqnetic moment of spin 4 particles. The homogeneous magnetic field has been replaced by an arbitrary field. The vacuum amplitude that describes the history of the t w particles is

where the effective sources are illustrated in (65.27). Only the term linear in A will be retained of the expansion [Eq. (%12,28)7

And, when just the field-dependent parts of the effective sources are used, the resulting vacuum amplitude term is

where K -- P1 - Pz

and

PI. P i 4- P, = 13, + K ,

It is convenient to intraduce the* kinematicaj relations by writing (the positiveness of all energies i s understood)

which converts (&6.5Bf into

x f2K2 + M," + M2' + flm?#~f - P ~ ) ~ ( P I + Pz - 2k)A (K)+n(Pel.

f4-.5.ti3)

The basic invariant integral that appears here,

can be evaluated in the rest frame of either momentum. Using that of PI gives us

where z is the cosine of the angle between the vectors k and P%, The inva~an t meaning of the PE components is supplied by

Xf the z integral is not to vanish, it is necessary that

M1" mm" KZ $- M,% + - - P- P

M , BM, (4-5.88)

I

Under thew conditions, the value of the intqraf (65 .65) is stated as

It is useful to satisfy the inequality (66.69) by intraducing new variables, x and v, according to

61 I l u t r o d y ~ r n i e s I Chap. 4

The inequality now reads

vZ C 1; (44.72)

the domain of x extends from 0 to a. We also find that

A = Ke(Kn + 4m3)(1 + 2% + X%?, (4-5.73)

while

dMladMz8 = #(KZ + 4m3)[Ka(Ke + 4ma)]l/f dx dv. (4-5.74)

The effective replacement for P1 + Pz - 2k in (4-8.63) is of the form

(p1 + pz - 2k) = (p1 + pz)a + (p1 - Pz)S. (4-5.76)

Multiplication by Pl + P2 and P1 - PE, rbpectively, implies the equations

KZ + Mls + MzZ + 2mz = (Ke + 2MI2 + 2Mz2)a + (M1' - M3Z)p,

01:- (Mlg - Mz9a + P P . (46.76)

The result is

where

K'+M1'+Me8+2me 1 1 + x A

3 -

K81 + 2 x + x V S (46.78)

We also note that

2KZ + MIZ + Mea + 2m8 = Ke + (K8 + 4m8)(1 + X ) . (4-5.79)

All these relations are combined in the vacuum amplitude

X +l(- Pl)eqdle(P~)(Pl + P e W A (K) - K K A (K)) . (4-8.80)

We now observe that

4 - 5 Form famm It. Singta and doubts spacerat forms I1J

and t heref ore

in .tvhicl~ the causal forms of two propagation functions are evident, fn carrying out the space-time extrapolations it is convenient Ca introduce the substitutions

and then return to four-dimensional momentum space, as in

where

One must not forget the pofsibiIity of adding contact terms in each of the two spectral foms. Keeping this in mind we mi t e the space-time exbaplalion of (4-5.86) as the dsubfe spectral form

W Blscf rdmanzlirs l Chap. 4

where, for simplicity, we have adopted a Larentz gauge,

kA (K) = 0, (4-&,g?)

The most elementary situation to which this vacuum amplitude can be applied is one where no field is prment. Xn a Lorentz gauge, the praportianality of the vectors A. and k demands that k2 = 0. Note that (4-5.86) does not vanish for this circumstance, owing to the term containing l fk" there is a potential-dependent vacuum amplitude, According ta (&&.71), with .K2 -+ K2, we have

The v integration in (4-5.86) can then be perfarmed,

and (4-5.86) ~duces. to

M2 - m2 - 1 + M% - it: where we have also indicated. the contact terms that are necessav physically, They sem ta repfaee fields with sources, according to

which converts (4-5.90) into

The conectness of this procedun: will be evident on mmparison with (4-1-86), which gives the correspondin'g modified propagation function in the absence of an electromagnetic fieM, and in the gauge A = 0. A more general description of the fieid free situation is produced by the matrix substitution p -+p - eqA:

4-3 Form factors C l , Sfngls md dazrbta rpwtnt forms 87

The additional tern, linear in A , for the coupling of two sources is precisely (4-5.92). This is what the normalization requirements of the phenornenolsgieal theory demand.

Now let us return to (4-6.88) and extract the particle field terms, rejecting all explicit source terms, by means of the rearrangements illustrated in

Efectively, this is the substitution

1 -- 1 1 k2 + 4mf4 m% + ( k g p B i T ) 2 ' (M. 96)

In addition, we shall write

and omit the l/k2 term since, as we have just shown, its tunetion is to introduce the gauge covariant combination p - eqA in the modified propagation function, The result can be presented as a field dependent action term. Combined with the pnlmitive interaction, it is

with

We have met the inevitable Iagilrithmic singularity at the lower integration limit, This time it is wor"th while to introduce the finite photon mass p for a detailed comparison with earlier results, The significant effect of the photon mass

appears in the inequality (4-5.68) which now reads (kg appears in place of Kg)

It suffices to introduce the following lower limit in the integal of (65.99) :

Alter performing the x: integration we find

But there is an identity, somewhat reminiscent of (P4.109) :

which can be verified by algebraic rearrangement, or by regarding -- ka/4mP as a complex variable. In the latter approach, both sides of (65.104) represent functions of a complex variable that vanish at infinity and have branchlines extending from 1 to co along the real axis. On observing that the two imaginary parts are equal on that interval, we confirm (65,IOB). Now wt?, have, effectively,

which conectly identifies f ( v ) with the f ( M 3 af Eq. ( M . 2 1 ) . There are some subtleties in the treatment of finite photon mass that would

bear further examination, but this will be deferred until the corresponding spin f &cussion i s before us.

Farm Mon Ill. Spin 89

4-6 FORM FACTORS Ill. SPIN 4

The spin f counterpart of these considerations uses the two-particle amplitude

retaining only the field dependent part of, the source:

iJ"(€)q(~)lctr. = eq - E)YJl(x) + , (44 .2 )

and considers the linear field term of the expansion

G", G+ + G+eqyAG+ + . a . . (4-6.3)

The corresponding vacuum amplitude is

X eqyA ( K ) (m - ~Pr)~vJle(Pe), ( 44 .4 )

where

$ I = P - k , $2 = P2 - k . ( 45 .5 )

We shall first carry out an algebraic simplification of the matrix

M" = yq(m - Y # I ) Y ( ~ - Y P z ) Y V , (4-6.6)

based only on the kinematical relations

- p,* e - pzZ = m%. (44 .7 )

It is designed to make explicit the gauge invariance property

K,MB = 0, W . 8 )

which is a consequence of the equality

The first stage of the simplification is displayed in

The next step involves a systematic use of the connection

90 Eterodrynamics f Chap. 4

as illustrated by

(m + ~ib1)3/iPz = (772 + I~'PI)(YPI - y K ) = m(m + yplf - (m + ypx)yr';-, (M.x~)

which leads finally to

There are four sets of terms here that indivi<lually obey the proprty (66.8). Two of them involve identities and the other two depend upon the otthogonality property

The intwation process in (4-6.4) implies an average value of the vector

PI + which, according to (4-e).14f, must have a form that is indicated in matrix notation, as

The algebraic property

(PI "4- PZZ(P1 + Pz) = + p,j2 - 2k(Px. + 1"2)

= -- (K" MN12 + M22 + 2m2)1 (M. 16)

csmbined with the recognition that

a = K2(K2 + + IM2% + 21132)

A t (4-8.18)

which repeats the result of (4-6.77). The other expectation value that we need is @veq by the symmetrical matrix

The trace of this matrix and the result af multiplication on both sides with the vector PI + Pz supply the necasary information ta deternine a and b:

Form facers Ill, Spin 4 9f

The solution is

When the variables of (4-6.71) are introduced, the three parameters E, a, b acquire the following form :

which uses the abbreviation

6 = t + 2x: -+- vgx2.

Note tha t onIy a involves Kg, and in a linear way, Now that just the vectors P I , E)% and tlreir difference K appear in (&I,), we

proceed to the final rearrangement. f 4 r x t i l i ~ s identities, such as

and the relatively complicated one

which are designed to express (Mu) in terms of the basic matrix vectors yu, kBVK, , together with possible factors of yPt on the left and yP, on the right. In doing this we make the one simplification of introducing a Lorentz gauge, so that

- (KPyKIKq is replaced by v . This is no loss of generality since the projection matrix 1 - (KKIXB) induces a gauge transformation on the vector potential that puts it into a Lorentz gauge. The final result is

I t enters the vacuum amplitude (4-6.4), which is now written as

The space-time extrapolation of this vacuum amplitude, performed without regard to contact terms as in (@.M), is given by the double spectral form

X 1 1

#I%+ M l l - k p g z + M ~ Z - ~ E '

where we recall that

The electromagnetic field-free situation is considered first. This circumstance is conveyed in (4-0.26) by omitting all aYvk, and kg terms, which leaves

(MP) 2m'BV' + 4m(~P1Y" + v~#g) + 2by#rvyYyP*. (4-0.30)

where

The following v integrals are required :

They suppX y these evaluations,

f f a 1 -dv- m - 1 p l + % -&I-- P

-1 2 d1I2 1 + 2% " -1 2 (11"- + %2x "

Accardingly, the effective value of (M') is

which introduces the single spectral mass parameter

Apart from a factor of (m"/2M4), the result we have just obtained can also be written as

[(M -- - 2mMI(M - yp,)yY (M - ~$2) f [(M f m)% + 2 n M ]

This enables the unified vemion of the vacuum amplitude (P6.28) to be displayed in the following fom :

A completely local contact tern has also been added, in order to satisfy the physical normalization conditions. They are implied by the gauge covariant generalization of the modified propagation function. It is most convenient, and

natural, to use the stmcture described in the action contribution (kli.56, 56) as amended by the contact modifications (61.59, W), without the algebraic re- combination~ tha t are a b stated in, the latter equation. The generdization even by ( A ia is omitted)

Xeads immediately to ( M . 3 7 ) . Xncidentdly, we might have proceded similarly with spin 0, retsaining fiel&

and ad&ng a suitable contact interaction in (4-1 .@Q) to obtain the vacuum amplitude

The gauge coyadant substitution then implies the additional caupling tern

Xt reproduce the double spectral form of (&B.W), but replaces the aid&tional contact t ems appearing there, which involve single spectral foms, by one corn- ptetely local contact tern. The two expresrsians difter only in their treatment of couplings that depnd explicitly upon murces, rathtsr thm fields, What. we have now done for both spins 0 and 4 has the advantqe of indicating that no single spectral foms are atzeded.

Let us return to the doable spctral fom ( M . 2 8 ) and consider the mwet ic moment tems separately, We shall apply them to three situations where single spctral foms are already known. One of t h e , described in Eg. (&&.20), dves the field dependence for coupling of an extended particle source with a simple patdicle wurce and a simple photon source (k" 0). Ctomespon&ngly, t e rw that are explicitly dependent u p n sources are discarded in (M.2-6,281 by the replacements yP1 -+ - m, -+ - m and, if is identified as the field of the simple

particle source, by (P: + M?-1 -. (M2 -- me)-'. The resulting coefficient of -- 2mioYVR, in (P8.28) is a - b, which enters the integral [cf. Eq. (P6.33)]

r (E - 6) X m m% nn"2-rn2 -dg!-.----- =.L

- , 2 dlfi ( ~ - t . & ) b 2 ~ 4 '

The defived single spectral fom,

(M.43) is the momentum space equivalent of Eq. (65.20).

The single spectral form stated in Eqs. (44.65, 73.15) refers entirety to simple particle sources. The denominators of the double spectral form now reduce to

This time it is the z integration that must be performed. Using the integrals

we get

The derived singk spctral form,

is the expected one. The third application is that discussed in Section 4-23. The electromagnetic

field here is a uniform one, so that the relevant terns af (MU)A,(k) become

where = p1 I p,, and symmetrized multiplication between yj9 and oF is under- s t d . The: v intqration is pflormed:

915 f lerodynarmlcs I Chap, 4

The resulting vacuum amplitude,

can be exhibited as a conventional single spectral i n t e ~ d by c a r ~ n g aut a pwtial integration with respect to M%. Since there are no contributions at the endpoints, m2 and W , this gives the action term (i is onnitteb)

where we proceed to write

Apart from the apparance of 9 rather than IT, this is the electromwetic field dependent tern of Eqs. (4-2.30, 31).

Our final task here is to de~ve , from the double spectral fom, the Emam single spectral form of the spin + charge h factor. The substitutions yq3, -+ - m, ee -=+ -- m, which reject explicit saurce terms, prduce the following coefficient of y"" in ( M @ ) :

As in the derivation of the magnetic moment fom factor in this dreumstance (-- p,%, -- pg2 m2), we also have

Since the vector potential term d e ~ v d for k% = O should be incarprated in the mdified propaption function, we remove it by the effective substitution [Eq. (65*98)1

The resulting coefficient of - k v inferred from (4-6.53) can be amanged as

One recognizes in the first term the complete structure of the spin O form factor, as given in (45.99). This immediateXy supplies the relation

The convergence of this integral at the lower limit means that photon rnass considerations are the same for both spins and need not be repeated, The integrand can be simplified somewhat to become

Then, using the intqrals (v 0)

we get

But, according to Eq, (44.132),

Xt would seem that we h a m failed. That is too somber a conctusjon. But certainly here is a warning that W have

ignored some subtlety in the calculation. Pefhaps it i s titne ta emphasize that the business we are about is more artistic than scientific. I t is the attempt to exploit one double spectral form, without using additional single spctral forms, in order to reproduce the known single spctraf form results. Whether or not this can be accomplished depends upm the organization of the calculation. We have succeeded with the spin mapetic moment effects, doubtless because, as a

dynamically induced phenomenon, there no contact terms are involved. The contrast between the charge form factors of spins Q and 4 possibly stems from the presence of two kinds of propagation functions in the latter circumstance (characterized by opposite intrinsic parity) which are unmixed in the absence: of an electromagnetic field [Eq. (4-6,37)f, but become coupled together by a field in a way that need not have been properly considered. This raises the question whether the difficulty could be bypassed by organizing the spin 4 calculation in a manner resembling that of spin 0.

We are already familiar with such a procedure. Xt is the replacement of the propagation function G$ [Eq. ( P 2 . I ) ] by the propagation function A:, which obeys [Eq. (4-2,4)j

a form that differs from the spin 0 structure only in the acfditional epoF term, In effect, the linear coupling with the electromagnetic field is changed according to

while E+ -.A,, To see the wrkings of this type of calculation, let us return to the considerations of Section 4-4, which examined an extended photon source emitting a pair of charged particles that subsequently interacted, and proceed to in traduce the interaction combination, (4-6,63), by algebraic rearrangement of tbhe known spin 4 stmcture, We refer to Eq. (44 ,35) , which contains the following matrix product :

In view of the prc?jection matrices that flank yL1, one can introduce the substitution

Then, recalling the additional projection matrices that a p p a r in Eq. (44,34), namely m -- and - ,m -- ?p;, we write

and

- K I

.F;: "i-

1C

a" 'N

.

E W

Q

b

r--l

r""r.

t -c !2 9 t &..""I

Form feetors Ill. Spin 4 99

(- m - YPi)yv = yv(- m + YP;) + zpi, -. yvy(P; - p;) + 2piv

(P; + $;)v + &"A(#; - (4-6.67)

The result is to replace (4-6.M) with

which is the statement of the substitution (4-6.63). (Apart from powers of 2m, which would be supplied by external factors.) On omitting the spin terms, we recognize the spin 0 structure contained in Eq. (4-4.6).

In working out the product (4-6.68), it is helpful to note that [P; - p ; = - (P1 - P211

- [- ~ ( $ 1 - P 2 k V - ( P 1 - P 2 ) v l I ~ v ~ ( P 1 - $2) + ( P 1 - P 2 ) V l

m - PI - (44.69)

while

E- Mw(P1 - P ~ ) r l & ~ [ - WVA(PI - p2lA1

E- ?(Pi - P2W - ( P 1 - Pe)vlrJ"[~v~(P1 - Pe) + (P1 - Pdvl

X ($1 - P2)Wo1 (4-6.70)

since

yv#"yv m 0. (4-6.71)

At this stage the integral of (4-4.35) becomes, apart from simple factors,

+ , (PI' + P;).(PI -h). ( (P1 - $2)'

which use the fact that

w e - $ 3 ) = 0 .

1OO El~trodynarmla 1 Chap. 4

We already know the integral [Eq. (@.IS)]

and a similar evaluation eves

Another familiar integral, Eq. (44.44, 451, is encountered in t l ~ e last two terms of (4-6.72) where, since only the antisymmetrical parts of the t e n ~ r s are required,

(M.aa) and

This leaves the integral

where the form of the right-hand side indicates that the integral is symmetrical under an interchange of p, and p;, and vanishes on multiplication with kg or R,. The two scalar combinations produced by the trace and by multiplication with (P1 - P;)~(P~ - p;), supply the information that

and therefore

On combining these evaluatians, one can present (4-8.72) as

Form f%toics ill. Spin 4 101

4- io*@k,Rviav,(pl - p;)" kkYiav,($, -- p;) *iafink,] . (44.81)

To sirnpXify the 1 s t spin combination, we write it as

where the final step is the yroieetian matrix reduction ypl -*- - I

" 8 , yPr -+ P@.

The result,

multiplied by the factors (1/2m) and ( 1 / 4 ~ ) ~ ( 1 -- (4melb-IP))1/2, is in complete agreement with (44.57). It cannot be said that this calculation is appreciably shorter than the earlier one, but certainly the related spin O result i s constantly in evidence.

Now we turn to the causal arrangement that produced the double spectral hrm, and consider [Eq. (4-8.4)]

Here the substitutions are

and

The action of the matrix factors m + yP, and m + yPz converts fields into sources. If the% terms are removed, (M,&Q) is replaced with

1 - [(%Pl - k)' -- iovXR,][(Pl + .Pg - 2k)& - igM@Kq][(2P2 -- k ) r + i g v l k 7 . 2%

(M.87)

1.02, El-rdynamfcsr C Chap. 4

On omitting the spin terms we reeopize the spin O stmcture in Eq. (4-5,fi-9). Since: the momentum pl - p2 in Eqs. (46.69, 70) has been replxed with a

real photon mlnentum (- k2 -1 Q), there is no contribution from these terns and (H .87 ) becomes (apax"E from the factor l/%)

in which

As described in Eqs. (4-6.15) and (46.191, the effect of the K integration is to produce the substitutions

and

(P1 + T"z - 2kf (P,+ P!& - 2k)

where the various coefficients are surnmarimd in Eq. (4-6.22). This converts fM.88) into a f o m that, in a Lorentz gauge, can be presented as

(P1 + fi)@[-- ar(2lr". + M12 -t- M,% + 2m2) +- (a - b)2P1vi~,APBa]

+ kd""K,[2K2 + M1" MZ2 + W $-

Now we observe that

icr~~K,~~~ic~,P~~ =r vyK(yKyPz -+ KPz) =s -- - K P g ~ ~ U v ~ v , PIVie,,Pz%ioYp_M, == ( ~ P J Y X + KP1)yKyIr = - K?PI~' + KP1i6.UVKV,

(44.93)

from which we get

(.in7@@K,, PZV~VRPILA) KB(Pl + P%)'

and

fiSllphf,, P1"iCrVkPg'] = KS(ypX~' - 3fllyPz) + (MEB - MZ2)ifJllpKVS (&G-95)

4-6 Form f;icctors 111. Spin It $03

The folfavving is alfo needed:

A further reduction, in which explicit: source terrns are removed, disposes of the combinations

(?PI 4- m)yB - yU(y.F)z 4- m), (yP1 + .m)yfi;l - yfii"(yP, 4- m). (4-.fie97)

The outcome for (4-6.92) is this expression,

or, introducing the explicit forms of K, a, b [Eq. (4-fi.22), and also (4.--5.79)],

In the earlier spin O and spin 4 discussions we have omitted terrns proportional to 41n2 -+ K2(1 - v%), after introducing the specialisation p12, - mZ, with the argument that the appropriate contact term produces the necessary cancellation. This tactic is somewhat unsatisfactory, however, since the individual terrns are infrared divergent, and may not be handled in the same manner when a finite photon mass is introduced, Let us instead proceed in this way. The re2ation

concentrates all the K 2 dependenw of such contributions to a double spectraf form in the lactor

where the approp"ate Kg-independenl contael tern has already been added. This combination, which is a generalization of that in (4-6,4l), vanishes on placing

and p: equal to - mZ. We have only to divide (4-8.99) by 2m in order to produce the yU and zcUvkv

couplings that are to be compared with the previous calculation. Thus, replacing (44 .56) as the coefficient of - yUk2 is

idcl Eteradynamics I Chap. 4

where the first terrn is again the corresponding spin O result, The relation (4-6.57) is altered to

where [Eqs. (4-6.46, 59)]

yields

In contrast with (4-6,60), we have now realized the desired result. But, on turning to the magnetic fornl factor, we find that the situation has

reversed. The first terrn in the coefficient of a ' ~ ~ ~ k ~ is the expected one. The second contribution involves the x: integral

and the attempt to reproduce the mqnetic form factor seems to have failed. There is, however, a common pattern to these unwanted additional terms tvlzicb suggests that both failures are only apparent. The additions to tlze charge and magnetic form factors in the two calculations are proportional to

and

respectively, where the initial space-like limitation on P,. k2 > 0, is stiU retained. But what are the domains of integration ? They are ordinarily fixed by a square

root factor that indicates the threshold of the multiparticle excbange proc~ws. There are no threshold factors here. As we shall see agstin in the next section, the generalization from the init i d causal situation also implies the removd of initid mass restrictions, subject only to constraints in~gosed by threshold factors. In this situation, space-time extrapolation extends the domain of M2 without limit. The variable M2 ranges from -- a, to co, for such are the values assumed by the momentum structure - P2 in the four-dimensional momentun~ space, The two terms on the right-hand side of Eq. (k6.107) have singufarities at the real values A I 2 = Q and M2 = k2, and similarly there are sinplarities at M% = 4nz2 a2d M2 = - k g in (44,108). But these terms are also related to each other by a finite translation of &I2, A finite but infinitely remote interval @ves no contribution to the individual integrals, Accordingly, the two terms cancel, and the value that is thus assigned to (4-6. IQ?), and to (44.1085, is zero.

Harold interrupts,

H. : I, have a hejpfixl comment and at question, During the spin O discussion that used one extended particle source, the problem arose of establishing equiv- alehce with known results. This was simplified to confirming the identity (&6,55), and that w a done by independent calculation af the two sida. But you expressed the wish to find a suitable transfomation between the differently appearing forms. This is the transfomatian you wanted:

where both variables, z and v, range from - 1 to + f . Since

we find that

or, subtracting 21% from both sides and rearranging,

The choice

then gives the desired result,

106 LSlcetrdynomlcs I Chap, 4

Here is the question. I am a little confused by what you have been doing. The same dynamical modifications of electromagnetic properties have now been derived in a number of different ways. Why is that important! Wouldn" one derivation suffice I

S. : Thank you for the transformation. As to your question, it: is preciwly the agreement of those various derivations that is significant since the consistency of the principles of causality and space-time uniformity was being tested. Different causal arrangements of sources, whicll were sometimes operated as simple sources, sometimes as extended sources, all led to the same conclusions. And, in tke process, we have discussed two extended particle sources, whicll results are particularly useful in various applications. But first, we have more to learn about single and double spectral forms.

4-7 FORM FACTORS !V. THE DEUTERON

The problems to wllich we now direct our attention are, strictly speaking, outside the framework of pure electrodpnamics, They lie in the realm of low energy nuclear physics. Yet no explicit acc~unt of nuclear forces will appear in this discussion, Besides the photon, the particles of interest are the neutron, proton, and deuteron. The latter, in particular, i s described phenon~enologically although we have no doubts about the wrnpsite nature of this particle. For simplicity, all these particles are treated as tflougli they were spinless objects. We shall not distinguish, between the neutron and proton mass, and denote tlte common value by m. The deuteron mass will be tvritten as

where E, the deuteron binding energy, is quite sn1aX1 in comparison with m, The physical relation between the deuteron and the neutron and proton is introduced through the extended source concept, Given enougfl energy relative t s the momentum-a sufficient excess of mass-the source that ordinarily ernits a deuteron can emit a neutron and a proton, This is expressed by the prin~itive interaction

which involves a scalar product in the cfzarge space common to the proton and the deuteron.

We first consider the madification in the deuteron propqation function that is implied by the primitive interaction, which portrays the deuteron field as an effective two-particle source:

4-7 Form fpctorrr IV. The dwtsron IOlr

The resulting coupling between two causally arranged extended deuteron sources is deduced from the vacuum amplitude

In view of the causal arrangement, tlte product of propagation functions is evaluated as

which is an application of the kinernatical integral (4-1.24). Contact terms must be added to this expression in order to satisfy the physical normalization cogdi- tions, They demand that the additional coupling refer to sources and not the deuteron fielib, Otherwise the initial description of the deuteron that is contained in its propagation function would be altered, The needed supplementary terms are indicated by

The additional action term obtained in this way i s

(67 .8 ) When it is added to the initial action expression,

fOB OIlmtrodymrmict 1 Chap, 4

the application af the stationary action p~nciple supplies modifid field qua tkns that are solved By a mdified propagation function,

A1though we hwe used relativistic methds to derive it, the essential domain of application for this result is a nonrelativistic one. This rmtfiction is intrduced by writing

The limiting farm of the propwation function is

I AD($) - - , G(E)t

with

To avoid confusion between E , the deuteron bincling energy, and E -+ + 0, the latter has k e n denoted by -j. + 0. The integral that appars here can be evaluated by contour integration methods, either applied directly, or in simplified form by intrducing a new integration variable, W1fi = X :

This gives

in which we have used the symbol

4-7 form faders LV. The deuteron 1OP

Fot large vaiues of E, E >>. E , this function has the asymptotic behavior

The primitive interaction with the electromagnetic field is

f n order to deternine the dynamical modification s-l the deuteron electmmapetic properties, we consider the following causal arrangement. f l involves two extended deuteron sources, and an extended photon source with space-like mornenta. The virtual deuteron emitted by an extended source decays into a neutron and a proton, The proton is scattered by the photon source and later recombines with the neutron to form a virtual deuteron, which is detected by the other deuteron source, This is the arrangement discussed in previous sections, but with the exchmged photon replaced by a neutron. The corresponding vacuum amplitude is (4-7,5), where the proton propagation function is changed to

Inserting the causal forms of the three propagation functions, we find that the vacuum amplitude for the process of interest becomes

We have designated the neutron momentum by 9 ; other symbEs are chosen as before, The essential change from the earlier discussion is the neutron energy- momentum relation,

replacing that of the photon. Thus, the basic integral to be evaluated now reads

S-- 0. (et)% A'" '

These are the same values as before, with

but the condition for the nonvanishing of the integral is different. I t is deduced, from the requirement

or

rn"Ml2 - Me3' < Kt[(Mla - 2ma)(Mz" 2m8) - m S ( P + 4ma)]. (4-7.26)

A choice of variables that is consistent with this inequality is given by

M1a - 2m4 = m(* + 4ma)1/5 + mK(xa - l)'/%,

Me4 - 2ma = m(Ka + 4ma)lfax - mK(x8 - 1 ) l 9 1 , (4-7.27)

where

v a < l , % > l . (4-7.28)

Some other useful quantities appear in these variables as

A =c K*[K" +ma + 4m(Ka + 4rn')lfax + 4ma(x4 - l)v'],

dMltdMea = 2msK(K' + 4m3*(~4 - l)"% dv. (4-7.29)

The algebraic property

K(P1 + Pz - 2P) S ($1 - P,)(Pl+ p,) P. 0 (4-7.30)

again fixes the form of the vector expectation value

where

AparZ: from contact terms, and e v e s & in a Lomtt gauge, the spce-tim extrapolation of the vacuum amplitude is then given by the double spectral in tegmal

f l $01 (- PX)~($I + Pz)RgA (k)4m(Pe)

+ -- ie peg +

(4-7 33)

In the field-free situation expressed by h" 0, we have

Using the integral

we find that (67.33) becomes, correctly,

X * I .,... l

-- it3 p,' + M' - iE (MS - mDg) (67.37)

in which we have inserted the purely local contact tern that is implied by the gauge covariant generaGzation of Eqs, (4-7.7, 8).

The double spectral integral (4-7.33) will now be applied to deuteron fields that are associated with simple sources. The action term inferred through the replacement P:, 9; 4 - m: is

where

(MIa - mDa)(MJ - m,') = (m(ka + 4m8)*x + 2ma - m,?" m2ka(xa-l)d.

(4-7.30)

At this point we confine ourselves to the nonrelativistic situation, expressed by such restrictions and simplifications as

and

Thus,

which indicates that the important values of x - 1 are small, of the order of k'lm8 and y2lm8. We therefore introduce the simplification r' - 1 2 2(x - l), and also write

The spectral integral of (4-7.38) then becomes (including a factor of

(4-7.44)

The variable y ranges from 0 to m. But an equivalent version of (4-7.44), apart from the multiplier 2*JP/ma, is

and the translation

now gives

Farm factors IV, The deuteron t l f

The k dependence of this factor appears in spectral forrn on noting that

The vector potential coupling deduced for k2 = 0, which is to be associated with the mdified propagation function, must still be removed. (This is automatically produced by the contact term.) Accordingly, the spectral integral actually occurs in the forrn

Adding the resulting act~on term to the primitive electromagnetic interaction of the deuteron, we obtain the deuteron forrn factor:

For large values of k , K >>By, the forrn factor approaches a constant limit,

NTe have met this eo~nbination before, in the asymptotic behavior of the deuteron propagation function, Eq, (4-7.17). Evidently there is a particular significance to a zero value for the combination, fixing the coupling constant f relative to the deuteron binding energy :

The effective electromagnetic interaction of ttte cIeuteron with high frequency photons would be zero, and the characteristic particle behavior of the deuteron propagation function would disappear. Both remarks make ckar that this is a composite deuteron, which is dissociated wit h eert ain t y if probed with a sufficiently high energy disturbance. Under these circumstances, the two terms of the form factor (4-7.50) can be united to give

For spatial values of k one can provide this farrnuia with a, conventional interpretation in terms sf a charge distribution. On recaliing that

I t 4 Elarodynamles t Chap. 4

which uses &r as a psition vector, we get

Exhibit& here is the deuteron wave function as a function of the neutron-proton distance; the vector 6r locates the proton relative to the center of mass of the deuteron. This wave function is familiar as the zero effective range limit. ft is a solution of the Schrijdinger equation for internal energy ---- E , with no interaction energy at: any finite die;tance between the particles.

We shdl do k t t e r if we recognize that the limiting behavior (4-7.61) for k Xwge compared to y must refer to values of R that itre small still in camparison with the mornenta of the virtual particles that art; exchanged between neutron m d protcm. The deuteron is not yet completely dissociated and the limiting value ( 47 .5 f ) need not be Era, Let us desimate it as - yr, /( l - ?re), so that

NOW the form factor &comes

where

We still describt: the charge distribution in terms of the motion of proton and neutron, with the exchanged particles implicit in the neutron-proton farce, This mwns that the charge density is deduced from a wave function #(r) :

(dr) expf - ilr &r)

(dr) exp(- 'dk " fr) (df @W(-- z'h * -@)

(4-7,59)

The comparison of the two expressions, (4-7.57) and (4-7.59), has this interpretation. The wave functions #(r) and $o(r) coincide except for such values of r that RP is small in the restricted range of K being considered, That provides the identification

Farm factors W. The deuteron 1 1 S

This relation is more recognizable if we introduce radial wave functions u(r) and % @ ( F ) by wri"ring

Now (4-7.W) reads

which is the conventional definition of the effective range r , , The usual boundary condition, zcf0) = Q, suggests, correctly, that r , is a positive length,

It is interesting to use our present standpoint in order to derive two familiar sets of results involving the effective range. One of these refers to cross sections for the photodisintegration of the deuteron . The primitive interaction is enlarged by including the effect of an electromagnetic field on the charged particles. I t is not the influence of the phobn on the motion of the deuteron that is important, but the disturbance cif the internal state through the action on the proton. Accord- ingly the relevant interaction term, derived from

The T matrix element, the coefficient of the source products i K b p i ~ G ~ ~ ~ ~ i J ~ ~ , is stated as (g =: + 1)

with momentum conservation expressed by

P = Pp + p n = PD + '4'

We use the gauge in which the plafization vector has no time component in the

3

E N

l li )-5c

e I 0 r

.CS,

S

2 Ill C

-SCZ

E i

5

iA,

m I

116 EIetrodyimmia I Chap. 4

rest frame of P. It validates the omission of the electromagnetic interaction with the deuteron. The invariant flux factor is [Eq. (3-12.70)]

and the final state integration in the center of mass frame for prescribed solid angle dS2 is given by [Eq. (3-12.75)]

We deduce the differential cross section

This formula will be applied in the nonrelativistic regime where

M g 2 m + E = 2 m - - & + k O (4-7.71)

and thus

Y = E + e . (4-7.72)

The kinetic energy of neutron and proton,

is computed from the relative momentum of the particles in the center of mass frame,

P p = - Pn = P.

We also have

leading to the nonrelativistic expression

in which (4-7.56) has been used to eliminate P. Averaging over the initial polarization and integrating over the final angular distribution gives us the total cross section

4-7 Farm faetam IV, The deutarsn ZI7

fts maximum value, at p y , is

These are well-known results of effective range theory. The ~ e o n d example is neutron-proton scattering. This lakes place through

the exchange of a deuteron as described by the modified propagation function, The implied interaction term is

and the comesp~nding T matrix element is deduced as

where

and

(4-7.80)

p = Ppl + P n l = fip2 -F Pn? (4-7.8 1)

p,, - pp, lc= .AI W p,. 14-7-82]

Since this is an elastic collision, similar kinematieal factors involved in the cross section definition cancel, giving

The second version is the nonrelativistic one. In this circumstance the d,(pn2 - - p,,) term, necessary to satisfy crossing symmetry, is negligible. We have also used the coupling constant relation 667.66). When the Iatter is employed in (4-7.16) we find that

7 2 1 I - - ---- 1 sin 6 exp(z"8), 1 - yr, m G'E)=-y++r,(pe+ye)-i$ p c o t 6 - i P p

where the real angle 6 is determined by

The differential cross section

118 EIiectrsdynamics I Chap, 4

is the standard one, expressed in terms of the S-wave pllase shift S, and the pllase shift fomula (4-7.85) is the well-known effective range result. W7e also recognize, in (4-7.84), the usual complex form of the scattering amplitude in terms of the phase shift, This means that the unitarity requirement has been satisfied automatically.

Mrearing a serious look, Harold speaks up.

H.: X think that scientists, like generals and statesmen, should write their memoirs, However influenced by a specific viewpoint and the natural human tendency to self-aggrandizement, the report of an individual who took part in events is irreplaceable. Surely, an appreciable fraction of the scientific history that is mechanically repeated in papers and books written by people who were not there, so to speak, is fictional or, in the phrase of Josephine Tey (The Daughter of Time), is Tonypand?;. I say ali this in connection with the effective range phase shift formula, Altllough there doubtless were qualitative precursors, I believe that you were the first to appreciate its significance and give a derivation in which the effective range, as you called it, had a precise meaning that could be transferred to other problems, That derivation used a variational technique. Later, other people produced mare elementary derivations and one of these can be found described in textbooks as though its author had originated the formula. Since it is hard to believe that you were unaware of the passibility of an elementary derivation, why did you prefer to use the unfamiliar variational approach ?

S,: It is an interesting question in motivations. Perhaps Z should point out that both the variational method far scattering and the effective range formula had their origin in the electromagnetic problems of wave guide propagation. Some of that history is described in Biscantz'rc~a'ties iut Waveg.t&z"des# Gordon and Breach, 1968, One will find there formulas for the frequency derivatives of certain electromagnetic quantities which also have stationary properties. These frequency derivatives are given as the difference between total and asymptotic expressions for energy. X was quite familiar with the Schriidinger equaticm analogue, in which a triganometric function of the phase shift replaces the electromagnetic quantities, and probability appears in the place of electromagnetic energy to determine an energy derivative, What was needed, however, w~ not an exact formula with uncertain variability in energy of a parameter, but an approximation valid over a limited energy region. For that reason it seemed preferable to use the stationary property of the phase shift, XncidentaXly, the effective range formula was an early embodiment of the nonspeculative viewpoint that later found its full expression in saurce theory. It is rather pleasing, therefore, to rferive the effective range results once again by using saurce theory.

Xn previous sections we llave deduced electromagnetic form factors by considering extended photon sources that emit time-like momenta. Let us apply that approach to the deuteron form factor, A virtual photon decays into a proton and an antiproton. These particles later interact by exchanging a virtual neutron to form a deuteron-antideuteron pair. The scattering reaction is described by the interaction term

where the effective two-particle proton source is represented by

and

= p, t p;. The resulting probability amplitude can ;be prexnted as

(47 ,W) with

dw,, do,; ( 2 ~ ) ~ 6 ( $ ~ f P{ - 1 k ,

($1 - P,Ie + m2 - P;)"

The scalar function is evaluated in terms of an expectation value:

do,, do,; ( 2 ~ ) ~ 6 ( p ~ + p; - 1 (PI -@;)(P, - P;) k ,

(P1 - P,)p + wZ ('$1 - p; 1%

All this is quite similar t s what was done in Section 4 4 , But here it is necessary to distine~uish between the initial particles of the collision, which are protons, and the final particles, deuterons. That distinction appears in the magnitudes assigned to the spatial momenta of the parlicles in the center of mass frame,

Accordingly,

ttL6 Electrodynamics I Chap. 4

where x is the cosine of the scattering angle. Before discussing this integral in detail, let us complete the formal space-time

extrapoilation, We write the scalar function S(M7 as

and identify the deuteron current vector to obtain the vacuum amplitude

do, exp[ziZ(lr: - %")l ( - av~ f lv fx" ) ) , (67.96)

where the gauge invariant substitution (34.19) has also been introduced. The space-time extrapolation is performed, as*always, by replacing the causal form of the pmpagation function with d& - xt, M2). After adding the primitive interaction of the deuteron, the Lorentz gauge form of the deuteron form factor is obtained as

The explicit expression for @ ( M ) is

where

Since a pair of deuterons are created, the threshold would seem to be at M = gm,, where'p I=; 0, But the function of p that appears in (G7.98) i s even and, for small p,

giving

4-7 Form factota; IV. Ths deuteron 121

There is no threshold here, As M becomes less than 2m,, p turns irna@nary,

but @(M) remains real. For that reason there is no distinction between z'r and - 2.u. If we wnite

where

A pair of protons is also created, and the next threshold approached with decreas- ing M is at M == 2m. If it had happened that

which represents a very large binding energy, greater than (2 - 2u2)m, the denominator of v. MP - 2mk, would remain positive throughout and the angle q5 would return to zero at M =2 2m. More preeixfy, we have

which is normal threshold behavior. Bu"rhe inequality (&7,1M) is not: obeyed. At a value of. M intermediate

between 2mD and 2m, namely M = 2l/;"inz,, r is infinite and 46 = E. Then, when M 4 2m + Q, v approaches zero through negative values and + -+ 2z. The limiting value is

Can we proceed below M -- 2nz 3 Yes, f n this regon,

122 Electrodynamia I

while

Chap, 4

Leaving the ambiguous terms explicit, we have

Since there is no physical basis for choosing either of the signs, we use the average of the two forms as in the computation of principal values. That gives the real expression

which joins continuously wit h (4-7.108). Where does this extrapolation procedure stop 3 There is another singularity,

a t p = 1, which is explicit in (4-7.132). If we notice that the value of p for M = 0, namely 2m/mD, is quite close ta unity, it is evident that p = X will occur for such small values of i%I that we can use the expansion

Thus the singularity occurs for M &%fO, where

This time, however, as we proceed below the singularity, where p > X , the ambiguous imaginary part of the logarithm implies ambiguous real terms in crfM). The undefined discontinuity at M == .Mo tells us to call a halt.

For nonrelativistic values of k , the farm factor (47 .97 ) is dominated by small values of M, where

Thus, the deuteron form factor is

identical with (4-7.60). It is reassuring to have this independent evidence for the correctness af the mass ex trapofation procedure.

4-8 Saaerlng of light by light l, Low frquenc9m $23

4-4 SCATTERtNC OF LIGHT ELY LIGHT I. LOW FREQUENCIES

We have been discussing multiparticle exchange effects that modify skeletal in teractians, Now ufe turn to examine how multiparticle exchange introduces new classes af interactions, The simplest example, which has @eat conceptual importance despite its lack of immediate experimental contact, is the scattering of light by light,

Processes involving two spin 4 particle sources and various numkrs of photon sources are contained in the coupling term

This is illustrated by

which describes a, combination of two photon fields as an effective. electron- p~sitron source :

The corresponding physical process is the collision of two photons to produce an electron-positron pair, or its inverse, which have been discussed in Section 3-13. Now let us consider a causal arrangement in which an electron-psitran pair is created by photon fields, labeled A%, and the subsequent annihilation into two photons is detected by sources, which have associated fields that are desimated as A,, The vacuum amplitude describing the two-particle exchange coupling between effective sources is conveniently written, as in Eq. (4-3.71, by using ac trace notation :

The insertion of (19-8.3) gives

(dx) * * (dx""" tr[eqyAlfxf6+(x - xf)trgyA1(x")G,(x" X"'")

or, using the more compact notation in which the space-time coordinates join spin and charge indices as matrix labels,

Chap. 4

(4-8.6)

This vacuum amplitude can be presented as a unified action term,

in which the additional factor of 4 records tile four equivalent places where one can begin the particular sequence A 1A IA dAC", all giving equal contributions in virtue of the cyclic symmetry of the trace,

Here is a process ttlat has no counterpart in the interaction skeleton of Section 3-12, since no charged particles are in evidence. I t is stated in a generalf y applicable space-time form. The validity of that unqualified assertion must, howeverr be confirmed by applying pbsical tests, There are two of these, gauge invariance and existence, Granting the existence af the structure, the property of gauge invariance can be verified by a. formal matrix calculation, in which it suffices to use an infinitesimal gauge transformation,

This induces the change

and the rearrangement

implies that

The existence question refers to the situation of complete overlap of the four fields, which is not contemplated in the initial causal arrangement. As the mast exkeme possibility, we consider such a small space-time region that the vector potentials are sensibly constant over it, Then the multiple space-time. integrals of ( 6 8 . 7 ) become, apart from numerical factors,

(dx) * * (dz"? trr[qyA (x)G,(x - x7)e~lyA. (x)6,(xr - z"")

4-43 Satteriing of Ifght. by light l. Low frquencies t2S

where the last form conveys the complementary restriction to very large momenta. Certainly this expression should vanish since it is not gauge invariant. The momentum integral does not seem to exist, however. Yet, the correct value to assign it is zero, provided the extension to unfimited rnornenta is performed last, in accordance with the picture of extrapolation from initially nonoverlapping fields. This is a consequence of the Lorentz invariance of the integration process, as expressed by the covariant form

a straiglrtfarward reduction sfrows that

Wfhen the derivatives of the vector potential needed to produce field strengths are included, corresponding inverse poMrers of mornenta appear and the integrals are abmlutely convergent at high momenta.

One can elihibit a generalization of (4-8.7) to any number of field products, ;v 2 4, Consider the exctlange of a particle pair between the two effective sources associated with a very weak field 6 A , and with an arbitrary field distribution A . The first effective source is

while the other, obtained from the cornparison of iW,,,. with

is represented by the efkctive field product

The coupling between the two sources, expressed by the vacuum amplitude

Also included here is pair exchange between the sources symbolized by dA, and A acting once, which we do not want to consider again, It is simplest, however, merely to strike out this te rn in the final result, As the notation QW(A) indicates, when we add the infinitesimal field BA to a preexisting field A we generate a

differential expression for an action term W(A), which contains all WO,, v 2 4. A formal integration is produced by using the integral equation [Eq. (3-12.21)]

CA, = G+ + G + C ~ ~ A G $

and its formal solution [Eq. (3-12.22)]

G: = (1 - G+eqyA)-'G+ = G,(1 - WAG+)-'. (4-8.21)

This gives

6 W(A) = H i Tr[(l - WAG+)-leqybAG,]

= - +id Tr log(1 - WAG,)

= - fit3 log det(1 - WAG+), (4-8.22)

where. the last version refers to the differential property of determinants,

b log det X = Tr(X-lBX). (4-8.23)

The integrated statement is

W(A) = - ti log det(1 - WAG,) = - +i Tr log(1 - WAG+). (4-8.24)

or, in expanded form with v = 2 omitted,

where odd powers are missing since q has a vanishing trace. The Ww term has been reproduced, which is a reminder that different causal arrangements of a given number of sources can be used to infer the same general space-time coupling of the sources,

This last discussion is easily repeated for spin 0 particles. The particle coupling is

which describes the photon sources in terms of an effective two-particle field,

i+(x)+(xl) l&. = AA,(%, X'). (4-8-27)

The effective two-particle source description of a weak electromagnetic field is [cf. Eq. (4-3.31)]

iK(n)K(xl) Id. = eq(p6A + BAp)(n)6(x - X'). (4-8.28)

We can then express the causal coupling between the two photon sources, symbolized by 6A and A, by means of the vacuum amplitude

Scatrtcring of light by fight I. Low frequencies 127

(dx) (dx'f tr: iK(x')Iij.r) jeff,i4(~)4(x8)

= 1 Tr[cq($GA + GAp)d:j. (4-8.20)

There is one subtlety here, hawever, tvXtich will be brouglrt out by the formal solution of the equation for A: [Eqs. (3-12.27, 2811 ;

If the expression for eZfl/(A) i s to be integrable, in the nlallner illustrated for spin &, one must cllange (4-8.28) into

There i s no objection to tllis sirlce bi;t and A are disjol~~t and tlteir product vanishes in the causal arrangement for wl~lclt (4--8.29) was derived, ?Ve can now state that

== ?ji Tr logjl - (eq(pA + A p) - e2A 2)d ,]. (4-8 3%)

In particular,

W,, = - i& Tr[cqfPA + Apjd.,.j4 -+ i$ Tr' ( eq jpA + A ~ ) d , ) ~ e ~ A ~ d , ]

wllich can also be derived by considering particle exclrange between two pairs of photon sources. The existence and gauge invariance of this elipression can be verified, rnuclz as in the spin Q discussion.

There are other prese~~tatians of 11'(A) that are particufarly useful under special circumstances, Let us begin wit11 the spin Q situation and write (17 .=

p - 6 9 4

where E -+ + O is implicit in the integral as a convergence factor, exp(- CS).

Now the differential expression (&8,3t) reads

and

128 eharodpamicr l Chap. 4

although only terms containing at least four field factors are to be retained. A simple proof of gauge invariance becomes available since A - + A + aA implies

n + exp(iqA) n exp(- ieqA) (4-8.37)

and

which leaves the trace unaltered. The spin 4 analogue of (4-8.36) is based on the construction Pqs. (4-2.1,3,4)]

where we now write

1 P- q F + m2-ie

= i [ tis exp(- - -F + m?}. (H.@)

Since the trace of a product containing an odd number of y matrices is zero (yY and - y" = y;lyYyS are equivalent matrices), and

- wdAyl7 - yReqydA = 8[(yn)q

= - 8[n2 - ~ u F J , (4-8.41)

the differential form (4-8.19) becomes

which gives

The utility of these expressions is confined to slowly varying fields, which are effectively constant over appropriate space-time regions. In such situations, the formal similarity of + ma or - equF + me to a particle Hamiltonian, and of S to a time variable, can be exploited. While this is always possible, it is the constancy of the commutator

[17,.17,1= icqF,"

that produces simple results. We note that

Sclrthwing of fitM; by Ilght I. Low fmquclacim (29

[D,, = 2icqFCav, (-.W and therefore

I!&) = exp(isDe) D, exp(- islI3 (H.w obeys the equation of motion

The solution is, in matrix notation,

D(s) = exp(2qFs)lZ = D exp(- 2eqFs), ( 4 - 8 4

which recalls the antisymmetry of F,,. Now consider the following tensor, which is defined by a trace that does not refer to charge space,

T,, = T T ' [ ~ ~ ~ I I ~ exp(- i sP ) ] = TrP[& exp(- isR7 &(S)]

= Tr'[&(s)n, expf - isfl)]. (44.49)

An equivalent form is

Tu, = Tr'[llPnv(s) exp(- i sP ) ] - Trt[[&, K(s)] exp(- islIe)]. (4-8.88)

The commutator that appears here is evaluated as

14, nVwl = rn,, nArexp(- ~ ~ ~ I ; S ) I , ~ I ~ M F exp(- ~ F ~ I I , , . ~4-8.51)

and a return to matrix notation gives

T = T exp(- 2eqFs) - iepF exp(- 2eqFsf Tr8[exp(- i sm] , (443.62)

or

We vse this result to evaluate

d i - Tf[exp(- is17131 - Tr'[P expf - i sP ) ]

ds

- - icq tr'

W - W The sdution of the ensuing differential equation is

IM Elsctrodynamics I Chap, 4

TrTexpj- if12)] = C exp

where the latter form emplays the dimensionality of space-time, and notes that the s i p of q is immaterial,

The constant C can be determined by car~sidering the limit of small S . This situation is dominated by large If values and the noncornrnutatlivity of different 17* components ceases to be significant. fising four-dimensional forms of conventional quantum relations, we get

The four-dimensional integral over the (3 + X)-dimensional momentum space is computed as

and therefore

To complete the evaluation of WCA) for spin Q we have only to supply the additional factor of 2, associated with the charge space in the trace, thus obtaining

The reality of the Lagrange function is made apparent by deforming the integration path of is to the positive real axis (but, see a later remark) :

4-43 Scaaaring of light by lfght I, Low trquanclss 13s

in which we have now explicitly removed the unwanted terms, The notation used here is

S = - gFr""F,, ==: i(E2 - H%-), (4-8.61)

to which we add

9 5. --- ) *F""Ffl, ;=- E: B (4-8.62)

and X, --- 2 ( F -& M ) = (E rJ1: i%f)%, (k8.63)

The general evaluation of the determinant can be given by finding the eigenvalues of the tensor F , Xt is convenient: to use the self-dual tensors

Considered as matrices, tlre two tensors commute, and the square of each is a multiple of the unit matrix, That can be checked by explicit use of the small number of independent components. The squares are

where. the coefficients X, are found by forming the trace, Equivalent statements are

from which we deduce

(F4),, - 2F(F2),, - g%u, = 0, (4-8.67)

the minimum equation af the tensor F. The eligenvalues appear in oppositely signed pairs, -& F" -& F', where

Accordingly,

"\F" - s F f s m -- 2(@$)%@

sin @F's sin eF"s cos(,~~2) - cos(essyP)

where we have .finally written just .%? in place of X-. Here, then, is the spin 0 result:

132 Ilmrotfynamficlr C Chap. 4

afthough all that we really want is the term qua&ic in the fielids. For that it is possibfy simpler to return to (k8 .W) and use the determinantal expansion

This gives

and

or2 1 spin O : = - - ( 7 F 2 +- 9%)

80 M"

The corresponding spin 4 result is produced by inseding in the integrand of 9 the following trace over the 2 X 4 dimensionaf charge-"pin space:

where the substitution is -+ s has already been made. The algebraic properties of the spin matrices are such that

‘ H @ K A ~ @fiJ = ~ K @ ~ A V - gxvgifi - ~ X A B V Y ~ * (4-8.75)

Therefore,

and the eigenvaltxes of crF are

(oF)' = & i ~ y , + iXy. (&8,77)

That gives

- g trtctl eoshfes~F) = - 2 Re cos(esSEfi") (68.78)

and

(4-8.79)

We use the expansion

Re cos(esZXlz) = 1 - (&S)%* + i(es)4(S2 -- g2) + (&&.80)

to derive the term of principal interest:

Several applications can be made of these low-energy La~grange functions. Inasmuch as comparisons with experiment are not at immediate issue we shall be content to omit numerical factors (but not R) and infer general orders of magnitude. The T matrix element for photon-photon scattering is derived from iWSq as the coefficient of iJ*ktlliJ*kt~n,~iJk212iJk22i2~. In the center of mass frame where all photon energies equal +M, the presence of four field strengths introduces the factor (+M)& and

Since this is an elastic collision, the ratio of the kinernatical factors involved in the differential cross section definition is .IJ (X/st2)fl/M2) and the total cross section emerges as

As a variant of photon-photon scattering, we note that a region containing a macroscopic electromagnetic field is an anisotropic medium for photon propagation. In the example of a. magnetic field of strength H, the deviation from unity of the propagation parameters is of the order of or2H2/m4. The qua&ie coupling of gtioton fields also states that an extended photon source can emit or absorb three real photons. This is of interest because the process exists, if weakly, betow the mass threshold for particle pair exchange at M I=. 2m, The weigllt factor a(M2j in the structure sf the modified photon propagation function [Eq, (4-3.82)] is inferred as

The existence of this effect, with a threshold at M = 0, implies that the initial long range deviation from the Coulomb interaction of static charges has an algebraic rather than, an exponential depndenee on distance:

134 EtacCrodynmmlcs 1 Chap, 4

Finally, let us note one use of the general Lagrange function, say (4-8.79) refening to spin 4 particles, which concerns a region occupied by a strong electric field E. In the limit of vanishing magnetic field finvariantjy characterized by

= Q, F =, O f , we find that

mpf--- 112%) [&S cat (egs) - l + &(eEs)%], (4-8.86)

One remark is needed, I-rawever. The variable now called s originally ranged along the positive imaginary axis, and then i t s path was deformed to the positive real axis. We must recall that sense of approach to the real axis, from above, since the integrand has slngularities on the real axis, They occur at

The necessary deformation of the contour into the upper half-plane near the% singulaxities pravides 9 with an imaginary part,

Since the vacuum persistence probability is

exp(iW)12 = exp(-- 2 Im W), (&8.W)

we recognize in 2 Xm 3 a measure af the probability, per unit time ;m& unit spatial volume, that an electron-positron pair has been created* This process is interesting conceptually, for no finite number of encounters with the static electric field, in the sense of a scattering description, can produce the enerm needed to create the particles,

4-9 SCATTERING OF LtGMT BY LIGHT 11. FORWARD SCATTERING

Xn the preceding section we exhibited the space-time form of couplinf3s that involve only the electrornqnetic field, and we also used these forms directly far calculations, in the special circumstance of slowly varying fields.. With more general

4-9 ScaeesrOng of light By tight #I, IFo~wwd rcaetering $33

situations, however, it is usually preferable to consider an appropriate causal arrangement and t ben perform the space- time extrrapola,lion. We are recognizing now that source theory is flexible; it is not committed to any special calculationai metllod and is free to choose the most convenient one, Indeed, it is the interplay and synthesis of various calculatianal devices, each adapted to specific circumstances, that constitutes the general source theory computational method.

Let us consider the arrangement in wltich two photons collide to create a charged particle pair, and then the two photons imitted in the subsequent annihilation of the palrticles are detected. For spin O particles, we can use the coupling (4-8.33), where

A szzz A I - + A.%, (4-@,l)

and retain just the terms, symbolized by A 1A lA 2A2r that describe particle creation or annihilation through the combined efforts of two sin~ple photon sources rather than by individual extended photon sources. The corresponding vacuum amplitucfe

where we have simplified the writing by replacing PA + A$ with %PA, as is appropriate to a Lorentz gauge. We can see here the individual factors that describe the effective two-particle sources which are associated with the twofold action of the electromagnetic field. This i s the causal slmcture with which one could have begun in order to derive Eq. (k8.33) . The causal form of the fields is given by

and the introductian of the causal forms of the propvation fanellion d, produces the vacuum amplitude

where, in a simplified notation with red polarization vectors,

2 ezpez'9" I

(P' - kg)" +g (p' - kg)% + ma -

t3CC t f ~ r d y n a m 8 c s I Chap, 4

The kinematieaf proprties of the rnomenta reduce this to

This structure will be used only to prduce the, coupling that describes fomard (and brtekward) scattering of the photons. For that situation, considered in the rest frame of the total momentum, there is one preferred spatial direction with photons moving in either sense along this direction, as expressed by the momentum relations

Xn the gauge where polarization vectors are purely spatial, a11 are perpendicular to this common direction of motion. The integrations of (4-9.6) involve the variable n, the cosine of the angle between p = - p k a d the preferred direction, and an angle in the plane perpendicular to that direction. Averages for the latter integration process are given by

and

The use of these averages prdaces

+ e, * e,e; * ed + e, * e;e; e,)],

where

[l -- (4n%jMg)] (1 -- 9) a ---vs a(M2) = f - dz

M%)]z% ' v B (4-9.11) - - - = - + vlog-

-1 l - v

SaWerlmg cif light by f Oght 81, Foward

The results of the integrations have been stated in terns of the v&able

The causal picture is emphasized by d t i n g

which makes explicit the individual coordinate dependences of the field stren@fts, and the propagation function that causally connects the two reeons. In order to give a covariant space-time fom to the vacuum amplitude, we must replam the polarization vector strue ture of (4-9.10) by equivalent field strenelt combinations, Consider first the situation in which an palarization. vectors are parauel, thereby reducing (4-9. f 0) to

We obxrve that, generally,

and that the p~lanization vector factor reduce, under the eircumstaxrc~ now being considered, to

- fk,k; = -- f (kI + h;)% = i M a . (k9.17)

A similar remark applie to @%(X". Accordingly, for the spciiaX situation of p a d e l polarizations, the vacuum amplitude (4-9,4) c m be presented as

1% Ilwtrodynamlcs 1 Chap, 4

The corresponding action expression is

8dc2 (dx) ( d x i ) S ( x ) d ,(X -- X' , Ilr12)S(z".

If we consider the limit of slowly varying fields, where 1/(2m) sets the scale of length, one can replace F(%" by S ( x ) in (4-9.19), which introduces

1 (dz")d,(x --- x', M%) = -.

M2

Then the action term can be represented by a Lagrange function, which is

The integrals that appear here are

and the coefficient of (a"lm4)S4 becomes 7/90, in agreement with that part of the Lagrange function in (4-8.73). Note that the initial limitation to forward scattering which is actually a restriction on momentum transfer in tbe collision, ceases to be a constraint on the scattering angle in this limit of small momenta.

The choice of parallel polarizations is one example in which the polarization vectors of the photons do not change in scattering. The other example refers to perpendicular polarizations of the initial photons and of the final photons:

e, * c; = F, e; = e, = P; = c, e; = O, I I

e l 0 e 2 = e I e o , =22 1,

when (&f).lO) becomes

A related field structure is

Scrttrkrrlng oC l l g k by light 11. Forward mmrfng 139

the general polarization vector factor here reduces to

where the & sign depends upon the particular orientation of the perpendicular '

vectors cl and e,. But the same sign appears in g2(x7 since the palarizatior~ vectors are unaltered in tire scattering act. Tflerefore the vacuum amplitude (4-9.4) becomes, in this situation,

(dx)(dx"%,(x)&f+(x - x', h1Z)$2(x'),

and the corresponding action expression i s

In the limit of slttwly varying fields there is a Lagrange function term:

which also agrees with tile corresponding term of (4-8.73). Now that we have established contact wit11 the low frequency results by

considering special polarization assignments, what can we say about the general polarization vector combination in (4-9.10) ? The term with the coefficient a is easily given a gauge invariant representation since the polarization vector factor in Fl (x)Sz(n." is

But the more elaborate structure with coefficient b cannot, in general, be expressed by products of the two scalars, F 1 ( x ) F 2 ( x 8 ) and $ , ( x ) 9 , ( x ' ) , Xt is necessary to use the tensor

F2 = Fg ( X ) F$ ( X ) . (k9.31)

Now,

where, omitting the compensating & i factors, we have

Evaluated in the center of mass frame, and restricted to forward scattering condi"rons, the polarization factor in (4-9.32) is, indeed,

As we have seen, the tensor combination can be replaced by scalars in the two situations where no change of polarization occurs:

[parallel : 6 F l ( z )Sz (z f ) F : ( ~ ) Y ~ F ~ ( ~ ' ) , , -.

perpendicular : 2g1(x) ).

Thus the additional term that is required is given by the difference,

~ : j x ) ~ ~ ~ g ( n ' ) , , -- 6 S 1 ( r ) 9 , ( x f ) - 2g1 ( x )$~(x ' ) . (&g .36)

I t would seem that we should add the space-time extrapalation of this coupling to the ones already known, as comprised in Eqs, (4-9. f9) and (k9.28). But, if we do so, the static interaction will be changed, That follows from the nonzero value of (69.36) in its unified form, when the distinction between x and x' is removed,

t r F4 - G f F Z - 2g2 = 2 ( F 2 + P) + 0. (4-9.37)

The proper procedure should be clear. The static interaction pro\rides a normalization condition for the more general calculation. To avoid altering the already correct result, the space-t irne extrapolation of the contribution containing (4-9.36) must be performed with an additional contact term, designed to remove this contribution a t low frequencies:

The complete action expression obtained in this way, presented in momentum space, is

or, alternatively,

One can also rewrite tlze tensor combination in the farm

so that the relevant term of (4-9.39) reads

Perl~aps it should be emphasized that the additional factor of - K2jMz, wllich appears in j&9.39), is not needed to bring about tile existence of the spectral integd. Only the normalization requirement imposed at low frequencies demands its presence.

When spin 4 charged particles are involved, one can study these photon- photon scattering plienornena with eitt~er of two ealculational, methds, The first one starts from the vacuum amplitude expression (68.6) while the other expfoits the similarity between spin Q and spin Q couplings. On comparing (4-8.36) and (4-8.431, we see that tire substitution Pr2 --. IT2 - e q ~ F and an additional, factor of - 4 converts IY (A),,i,, into W (A),,i, t, provided one extends the trace operation to include the spinor indices. The second method is more economical since much of the calculation is common to both spin values. The vacuum amplitude derived in this way from the spin O structure (4-9.2) is

One facilitates the spin part of this calculation by writing

wIiich uses the field strength construction of the radiation gauge, far from sources, Xf we retain the vacuum amplitude representation (4-8.4), the appropriate

$12 Elw$rdynamica I Chap, 4

structure of I,,,, is derived from (69.5) by introducing the spinor operation (- 8) tr, white perfoming the modifications indicated by

where n is the unit vector associated with the preferred direction 05 (4-9.7). Note also that

and

(+a f CS . nI2 == 2(1 r f r +$a n). (4-9.47)

The values of the spin traces encountered here are given by

and

the latter can be derived by reducing the o products or by systematically corn- muting one factor from left to right in the trace. The outcome of (be calculation i s expressed by

where .

We a j s ~ observe that

4 9 Sattarittg of tight by Hght 11, Forward mttertng $43

This is the only integral needed to discuss the low frequency limits, for parallel and perpendicular polarizations that do not alter in scattering. The changes in the factors contained in (69.21) and (4-9.29) are indicated by

and the comesponding numerical values are

These are indeed the coeffieien ts exfribited in (4-8.81). To produce a space-time extrapolation for arbitrary polarization vector

assignments, we observe that the two vectar combinations of (4-9.50) can be wdtterx as follows,

i' r I I

e, e;e, * P, -- e, eZeI * ep + cl e 2 ~ 1 ep = el a * ei + (el X Q;) (aI X e j. C I I I

2(e, e,e, e, -- c, e;n, P,) = (c, c;@, * e; + e, * o,e, g e,

The first of thee will be recognized as the polarization vector stmcture of P 1 ( ~ ) P 2 ( ~ 3 + +l(x)%z(x"), while the second refers to the combination of (69.36). The latter does not contribute to eitlier of tlre elastic polarization vector assignments, and its space-time extrapolation requires a contact te rn that awids a low frequenc y cant ribution from this coupling. The result is

144 EIerodynamlu t Chap. 4

which has the equivalent form

w01, spin f ~ W O P , spin 0 i- (p(- k ) s ( k ) + g(- k)@(k))

4-10 SCAmERING OF LIGHT BY" LtCHT Ill, DOUBLE SPECTRAL, FORMS

We come at last to the general situation of photon-photon scattering. To produce the required coupling tbe following causal situation is considered, An extended photon source, Jz, emits a paik of charged particles, Each of these particles is individuauy deflected "oy extended photon sources oprating in a, space-like E n s , J,, J,, and finally the two particles are detected by an extended photon source, JI.

The fields amciated with the four sources do not overlap, and A, + Ab is causaUy intemediate between A I and A,. Accordingly, the spin O vacuum amplitude deduced from (68.33) has the form

where account has been taken of the four equivdent positions in which AI , for example, can be placed, The requird sequence is such that A , and A, occur between A I and A%, corresponaing to the causd order. The specid feature of this causal arrangement is that all four propagation functions describe real padieles. We shalt write the fields sf the four extendd sources as

e t h the associated vacuum ampfitude taking the fom

The structure of the sedar intqral I is indicated by

since there are four invariant momentum space measures for that number of real padicles, faar vector ptentials representing the actian of the rehted murces,

and three delta function factors that establish momentum consrv&ion at the comesponctjing interactians. The fourth such delta function is &ready exhibited in (410.3). I t is mare convenient to replace the three-dimensional momentum measures by restricted four-dimensional ones,

and then use the delta functions to eliminate all but one of the particle momenta, The choice of the latter is arbitrary, and the pssibilities are illustrated by

Here the variable p is not a particle momentum; the actual particle rnomenta are displayed in the various delta functions, The momentum factors that accompany the vector potentials are produced by adding the two reEevant paflicle momenta, each multiplied by the appropriate charge sign factor, This procedure removes the gauge restriction that is impEeit in the combination 21)A.

The four delta functions in (4-10.6) can also be written as

They supply four conditions on the camponents of 9, which determine this vector (almost) completely. In the rest frame of the time-like vector A,, where h: = M,, we have p0 = 0, according to the first factor in (610.7) . The second one fixes the m w i t u d e of the momentum p,

The other two &lit& functions determine components of p along the directions of k, and k,, which vectors define a plane, Then the mqnitude, but not the s ip, of the P-component perpendicular to this plane is supplied by (410.8). If a real vdue does not appear, the integral vanishes. Since p is uniquely betemined, apart from the s i p ambiguity W have noted, the eswntial integration p rmes in (610.6) refers entirely to the delta functions. One can easily state the value of this i n v a ~ a n t integral in the spcial coordinate systern where k,, Ir, occupy the xy-plme and k, coincides with the x-axis,

$41) ECmf@dynanrfeo t Chap. 4

Tbe reference to a special cooranate system is remove-d in

where

The symbol (-- anticipates the possibility of giving the determinant that appears in (610.11) an explicitly invariant form by squaring and using the deteminantal multiplication property (Gram determinant). There is, however, one pitfall associated with the indefinite Minkowski metric which we can avoid by expressing all vectors in Euclidean form (V, = iV@). Since this introduces an explicit factor of i in the determinant of (4-10.1 l), we see that

This fornufa does not necessarily provide the easiest way to construct A, however. A general expression for the vector p, which satisfies explicitly the constraint

k2p = 0, is stated in

The last term gives a covariant form to the p, component. Multiplication by the vectors K, and & suppfies the equations

which determine the coefficients a and b. To find c, we square the combination (k10.13). In doing so, one encounters

and also

where D is the determinant of the cwfficient: array in ( 4 1 0 . 1 4 ) :

1 = kzkf - (k,R,)a - - [kf (kpk,)g + ki(kgk.)f -- 2kakokpkekaRb]. (P10.17)

k:

The condition for the nonvanishing of the integral IQ(#: > 0) reads

it has been left in implicit form for compactness, Another useful result is produced by multiplying (P-10.13) wit11 p,, This reproduces the structure realized by squtsring (4-10.13), with the exception that the c% term is replaced by me linear in c. The cornparisan of the two supplies the relation [vvhich is also contained in (&X0.9)]

- d = (k;Dc)% = ( - k i ~ ) (-- kiDcP) . (4-10.19)

The latter form combines (4-10.16) and (410.18) to give an alternative evaluation of d.

The causal arrangement under consideration refers entirely to extended photon sources, with k,, kz tirne-like, and h,, k, space-like vec to~ . After the corresponding space-time coupling has been estabfished, an extrapolation will be made to the situation of interest, where k: = R: = k: = k i = 0. We shall illustrate the algebraic relations just discussed by utilizing these kinernatical simgtifications appropriate to real photons. The two variables needed to discuss the photon- photon collision are convenient f y chown as

where the second version refers to the real photon situation, Some quantities derived for this circ~xmstanee are

where we have used

k; = - (k, + k, + -. Q

to evaluate 2Rak, -+ M: + M!.

The positiveness criterion (4-10.18) now becomes

and, with the aid of (4-10.19), one obtains

Of course, the same result is derived from the determinant of (4-10.12),

0 - #M: - #M:

- +M: 0 - A = det

f (M: + M:) , (4-10.26)

- M + M:) 0

0 ,M: - W: - nra

but the work seems more ponderous. The introduction of the variables

which range between 0 and 1 , converts the inequality (P10.24) into

wa $ f6 > l r

and the expression for A becomes

The momentum coinbinations appearing in (4-10.20),

are associated with two different ways of considering the causal coupling among the sources in terms of two-particle exchanges ( M , , > 2 4 . Sources Jz + J, exchange a pair of real particles with sources J1 + Jb, and Je + Jb exchange a pair of particles with J1 + J,. We shall make these momenta K, and Kb explicit by introducing the factor

l (dKa)(dKb)d(Ka - kz - ka)d(Kb - kz - kb)

4-1 0 h & A n g of tlglrt by tight ill, hubre s

I t is combined with the Pot& momentum delta funetion,

The causal significance of this farm becomes apparent on cansidering the structure

=!= - d ~ : d ~ ; P- (dx) (dt,) (deb)A l (x)A ,(x - 5,)A,(x - F~)Az (x - t a - [a) 2n 2z

According to the causal arrangement of the fields, the vectors 5, and 6, are time- like, with positive time components, We recognize: the appmpriate causal t oms of the propagation functions

Although other details must be added, involving questions of gawe invafianm and contact terms, this is the essence of the space-time extrapolation process. As a convenient way of expressing the result, we return t a four-dirnensiond momentum space and state the space-time extrapolation of (4-10.34) in the doubie spctral f o m

1 1. . (6110.36) (k, + k.)" Ma2 -X (kp + k,)' + M,' - ie

Xt is instmetive to appty what we have learned to a simplified x d a r fii?ld problem, in which we compare a noncausal calculation of the coupling, analog~tls

$!#l E f ~ r a d m m r i c s 1 Chap. 4

to (4-1O,l), with the double spectral farm evaluation. This wiU be done only in the limit as all photon momenta approach zero, The phrase "noncausal caileula- tion" mfers to the direct use of the propagation function in its four-dimensionaf form, as distinguished from the causal calculation leading to the double spectral form. The alternative computations are displayed on oppsite sides of the equation

The left side can be evaluated in several ways, One can transform to a Euctidean metric (po = if4), and then perfom the single radiaE momentum integrd, utilizing the surface area of a unit sphere in four dimensions, 2z2,

Or, we can exploit the technique intraduced in (48.341, which is usr;d here in the f a m

ds sS expf -- is(@% +- PE?], (kXO.39)

and then apply the integral (4-8.67) to get

When we turn to the other side of Eq. (P.10.37), an impartant aspect of the extrapola lion procedure is brought to our attention. In the causal amangement that produced the integral (4-10.9, 10), the quantity d is necessarily negative. But after the extrapolation to real photons has been performed, d has become positive, as exhibited in (4-10.25). This poses the question of which square root of - l to adopt, in evaluating (--- A)"%. The comparison of the two sides in (610.37) shows that

We complete this test by verifying the nurnerieal equality of the two sides, fn the notation of (4-I0.27), this is stated a

where u, and H, range over the interval betwen O and 1, subject to the positiveness restriction that is evident in the denominator of the i n t ~ r d , Performbg the inkegrations successively, we get

Scattering of fight by light 111. Double spectral forms 151

as required. iVe must now turn tt2 the vector potential factors in (4-10.6) and the problem

of makil~g explicit the gauge inr~ariance that does fiold in the causal arrangement, in order to preserve it after space-time extrapolation. Since our interest in photon- pllotoll sratteri~lg is primarily didactic, we sf~aXt avoid tlie relativeIy unrewarding complexities of ar1;itrary polarizations by considering only the simplest polarization assignment. It is tlte one in wl~icii a11 po1arim"tion vectors are parallel to each other, and perpendicular to t l ~ e plane of scattering. In that circumstance the vector potential factors a i (4--10.6) beconlc

wllere the various fui~ctions are the si~tgle nonvaltislring vector components, those perpendicular to the scattering plane. To Itttroduce ficXd strengths, let us consider tlxe product

wl~ere kfAA(k) and kA(k7 have been set equal to zero, as is appropriate to the situation where the spatial palarization vectors are perpendicular to the plane of scattcrir~g. One way ta use this relation, in order to give gauge invariant expression to the product of four vector potentials, is indicated by

(F(k)F(kr))(I;(K"")f;(k"'")) = (kk")(kt'"k""") A(k)A ( K ' ) ) A (h"") ((K"?), (4-10.46)

which employs the notation

The same product of four single-component potentialis can be provided with differexlt gauge invariant interpretations depending upon the pairing of fieid strengths. Mere are examples:

All agree in the initial causaE situation, but differ after the space-time extrapolation

1s Elaetrdparnlcs I Chap. 4

h= been perfamed. The three field stmetures in. (&f0,48) can. be expressed gexreraBy as the prduct AI * * A%, multiplied by momentum factors which arc., respective1 y,

When used in the context of the double spectral form (&10.36), the various psibilities differ by single spectral foms, as illustrated in

(k2/M32 (k2/M2) (kfg/Me2) (k' + MS)(ktl + M'? - (kg + M2)(k8" MM.%)

This element of xbitrariness in enforcing gauge invariance indicates that added single spctral forms can be present, thus requiring additional pfiysieal information. That information is forthcoming in the single spectral form appropriate to foward scattering, which w a derived in the preceding sectiion.

The results obta-ined thus far are united in the following action expression:

( 2 ~ 1 4 Sfk + * + KU')")A (K) * * A (K""")

dM@% $M,% (k" kf'""")2 (K" " kk 'y M?) (K' + km')% + iE (k" + k"')8 + Mba ie

where

We have elected to use the most symmelricd of the gauge invariant repreenta- tians det;aLiled in. (4-f0,48,49), f t has dso been =cognized that, owing to the symmetry in k', K", and those arising from the equalities

(K' + kf"") = ((K + h")', (km -+- kK""") 2 ((k + k'f2, ( k10 .58 )

the initial causal process is contained in eight equivalent terms of ( 6 1 0 . 5 1 ) . The single spctral intepaj, with its as yet unknown weight factor +1(M8), is the

4-l Q Scslctsrtng of light by tight 111, Double rpsrctral farms; IS3

momentum version of the fornn seen in (4-8.19), with field strengths replaced by vector potentialls in accordance with

(F"(k)f;(k") = - *(k + k"%A(k)A (k". (kl0.M)

The four photon rnornenta obey the relation

f n the situation of forward scattering, one of the three combinations vanishes, and the other two are equaX in magnitude but opposite in sign. This leads to the foltew- ing effective substitution,

(R' + km""% (k" " k@#')% (k" kk""")2 M@% - kc: (k" " kK""")2 +&a - it.

which is verified by comparing the three terms produced on each side by sym- metrizzrtion in K, k" kK'" wwfliEe utilizing the symmetry of 4% in Nag, M,%, The identification of the resulting single spectral farm with (4-9.19) then gives

f . + v = . i v + # a ( l - v ? + f i i - v 2 ) ( # f l - v % ) - 1 ) 1 o g - . (P-10.67) l - v

Performing the necessary integration, we deduce that (zt = v%)

which completes the specification of the spectral forms in (610.51). To produce the spin Q counterpart of these considerations, we have only to make

the replacement

where the mulltiptication order [Eq. (610. X ) ] reflects the causal amangernent , The right side of (4-10.69) is calculated as

"c)

g

S

2 2 I! El

'"d P4

'CC E

&

+ a

S

+ Sr,

41" -l- CJ

25 -I- -%

.

2

-t

m

S

U

I II

c

G'a

'CC

8 ccl

m-

- CI gm

S

" &

S Q

)+

Tij W

#".

"c)

n

i;?" u.? Q

ls l

-4' v

8 4

v

U

E: 0

- ri

.Clr

2 23 B 8 4 1

C,

.M

"-4

C4

5 h0 G

.M

&

l

Q;,

J-,

*.' ctJ 8

"c)

! 0 cc1 8 P

cl*

c@ F: (d

Q,

9

T3 9

- 2[(2pJ4Al -44% + (2p,)Tl + Tell (4-10.60)

in which

T i = 2 AA4 tr UFUF m A iAa(F2Fb) + A2Ab(F,Fa) + A 1Ab(F2Fa)

+ A2“4a(F1Fb) + A 1AdFaFb) + AaAb(F1F2) (4-10.61)

and

The latter assertion, which is analogous to (4-9.49), refers specifically to the situation under consideration where all electric fields are perpendicular to the magnetic fields. In the causal arrangement the value of T1 is

T1 - [ ( K Z + kJ2 + (k2 + + (k, + kb)yA1 A2 = 0, (4-10.63)

while

Accordingly, the weight factor of the double spectral form is derived from that of spin 0 by the substitution

and

(1 - ua)(l - ub)(ua + Ub - l)"2 spin 4: $2(M,e, Mbs) = +(U, + ub - l)-1l2 - -.

(2 - U , - u,)~

The discussion is completed by using forward scattering information in the manner of Eq. (4-10.67), but applied to the convenient combinations

(6'1(M2) = +I(W)*~ + 241(W) ,~op

(6'2(Ma2* MbB) , +2(Ma8, MbE)*, + 2$2(MaSt MbS)spinO a 4 ( ~ e + ub - l)-ll'e

(4-10.67)

and the analogous forward scattering amplitude combination [cf. Eq. (4-9.53)]

1 l + v L- 2(a + 46) + cl + 2(a + qb) = c(W) = V log - . (4-10.68) l - v

4-5 0 haeriing of tfghr by light l l@. Double spectral forms t55

Performing the intevation in

we get

and therefore

spin : (Mg) = v + (3 --- 2 ~ 2 ) (4-10.71)

For the actual application to photon-photon scattering in the center of mass frame, with tataI energy M and scattering angfe 8, the three momentum combinations that appear in (4-10.55) are, individually,

The 2" m a t ~ x element is

and the differential cross section is computed as

Concerning the details of the angular distribution we shall only remark that at law energies, M < m, the scattering amplitude t i s proportional to

1 + sin4#@ + cos4+@ =;; +(3 +- cos2@), (4-li0.76)

while at high energies, M m, there! is a logarithmic variation with scattering angb.

l36 Rlrsctrd~anla I Chap. 4

4-11 H-PARTICLE ENERGY IDCSPLACEMENTS. WON RELATIVISTIC DISCUSSIQ N

Qecasiond reference has been made to the energy level displacements of bound systems, There is even an explicit formula stated [Eq. (M.84)11j in which, however, the precise meaning of an atomic excitation energy is left open. Unlikc: the sattering discussed in Section 4 4 , which is of interest psimrily at the high enerees that are attainable experimentally, the problem of energy displacements is dminated by the characteristic low energies of bound systems. Accordingly, we shall initiate the discussion of H-particle energy displaeements with an in- stmctive nonrelativistic treatment ,

There has been frequent consideration of nonrelativistic limits to relativistic dynamical derivations. This time we shall work directly with nonrelativistic dynamics, although, in doing so, we utilize without extended explanation the procedures that have already received their general space-time description. In the nonrelativistic circumstances conveyed by the energy expression

the prapagatian function d,(x - x" has the causal forms fxO = t )

The nonre1at;ivistic origin af energy is introduced by mul"cip1ying A,.(% -- X ' ) with exp[im(xD --. A?')]. (The alternative of multiplication by exp[- kn%(x0 - F)] is equivalent. I t interchanges the roles of particle and antiparticle, which are given separate descriptions nonrelativtstically .) In the limit that m is regarded as an wb i t r a~ ly large energy, we reach the nonrelativistic form of the propagation function :

G(r -- r', t -- 8') = Lim(-- gm) exp[im(x0 - #')l A,(% - X ' )

- 0 exp{i[p (r -- r') - T(p)(t - 8'));). ( 2 ~ ) ~

This retarded function is a Green" function of the inhomogeneous Schradinger equation,

The n~ontentum space version of this equation is

a i - - at

C@, t -- t ') = 46(t - t'), (4- l l .G)

where

The general time transform function,

dt expjz'E(t" - $'))l G( , I - t'),

obeys equations appropriate to the choice of spatial variables:

Since the Green" function is retarded, in time, the transforn~ function (611 .7 ) exists for complex values of E that are confind to the upper half-plane, Irn E 2 0. Accordingly, the appropriate solution of (4-11.8) for real E is produced by approaching the real axis from the half-p2ane of regularity:

Qne verifies directly that the implied time behavior is that of a retarded function,

An action expression that incorporates the inhomogeneous ScfirGdinger equation is given by

The implied field equations are

Chrp. 4

(4-11.12)

As the notation indicates, these equations are in complex conjugate relationship. The field equation for $ is solved by

$(r, l) = 1 (drl) dtl G(r - r t , t - tl)q(rl, t r ) . (P1 1.14)

One uses this solution to find the explicit source dependence of W,

W q * , 1 ) = - (dr) dt q*(r, t)$(r, t ) S = - 1 (dr) dt (drl) dtl q*(r, t)G(r - rl , t - t')q(r6, t l ) . (411 .16)

The minus sign that appears here, in contrast to the relativistic forms, reflects the sign factor contained in the relation of (4-1 1.3) ; it has been introduced in order to conform with nonrelativistic conventions concerning Green's functions. The corresponding action principle definition of fields is

I t implies the additional field expression

(dr) dt v*(r, t)G(r - r', t - t'). (4-11.17)

Notice that this is not the complex conjugate of $(rl, t'). The field JI is related to its source q at earlier times, while $* is linked to values of q* at later times. Alternative choices of variables enable the explicit expression of W, for example, to be given such forms as

c=j I)

&--.

X).

(4-11.19) and

W(?*. ?) = - j -?*(P. E)C(p. E)?(p, E).

in which

'I( 1 E) 5 exP(i~t) r)( .t). v*( . E) = 5 dt exp(- iEt) v*( , t). ( 6 1 1.21)

Electromagnetic interactions are introduced by the radiation gauge substitutions

which are written for a particle of charge - e. In the radiation gauge attention focuses on the vector photon source J. After removing the instantaneous Coulomb interaction of the charge density JO [cf. Eq. (3-15.51)], we can eliminate this time component with the aid of the conservation condition

Thus, the photon contribution to W can be exhibited in the dyadic form [cf. Eqs. (3-15.52,63)]

(dr) d4 (dr') dt' J(r, t) D(r - r', t - 1') J(r', t'), (4-11.24)

where

t > t' : D(r - r', t - t') = i dm, exp(i[lr (r - r') 5 The nonrelativistic situations we shall consider are such that the momentum carried by the photon is relatively negligible, which is to say that the typical photon wavelength is large in comparison with the spatial dimensions of the

(60 Elutrodynamio I Chap. 4

system, [k. (r - r')l l. Then the photon propagation function can be simplified to (a factor of c8 is also supplied)

t > t' : 8D(t - t') i - dRo R0 expt- ikO(t - tf)Jl. (4-1 1.26) 332 "I

In arriving a t this expression we have introduced spherical coordinates for the k space,

and performed the angular integrations,

after which the photon momentum property,

lkl = K",

was used. First let us derive some known nonrelativistic results concerning modified

propagation functions and form factors. The action term

c - I (dr) dt #*p, l) - p A(r, t)$(r, t), p = - V , (4-11.30) m

characterizes an extended particle source q2(r, t) as the effective two-particle emission source

and similarly for an extended absorption source q*l(r, t),

C t) Jl(rf. tl)ldf. - #*,(K t) G p d(r - r') d(t - t3. (4-11.32)

It is actually only the transverse parts of these vectors that are effective, but these are selected automatically by the photon propagation function (4-11.26). The vacuum amplitude term that represents the exchange of a photon and a free particle is

8% =--,.p (dr) dl (dr" ddt" qb*,fr, t)p G D(r - r f , t - b")E(r - r" 6 - 8') * pJIZ(r; if)).

m%

The use of the simplified photon propagation function (4-11.26), and of the particle propagation function (4-1 1.31, converts this vacuum amplitude into

. 2% l -$..-...--

l 3n m"

dt dl' $*,(p, 1 ) exp{- i[T(p) + ko](t - 1')) pe$,(p, t ') .

(4-1 I .34)

We recognize in (l/i) exp(- z[l'(pf -+- kol(k - I")) the t thexpression for the propqation function of a particle that has the energy T(p) + k@. Its general form will be given by (4-11 .lX), modified appropriately by the substitution T --+

T + k@, to which can be added time contact terms, that is, B(t - t') )and a finite number of its derivatives. Accordingly, the lime extrapolation of (4-11.34) is

where the contact terms now appear as a polynomial in the energy parameter E. A sufficient form fur them is fixed by the requirement that this additional coupling should refer to sources, rather than fields, to avoid altering the initial description af the free particle, The needed factors, those displayed in

are prduced by the combination

The modified propagation function is then obtained as

One will recomize here the structure of the soft photon result that is stated in Eq. (41.83) far spin 0, and in Eq, (4-1.95) for spin $.

The motion of the particle in a static potential V(r) is represent&, initially, by altering the Green's function differential equation to

An equivalent integral equation, presented in an abstract notation, is

= G + G V G ~ , (4-1 1 .M)

which has the formal solution

GV 5 (1 - CV)-lG = G + GVG + . . (4-1 1.41)

A more explicit statement of these first terms is given by

GV(p, 1; p', t*) = 6(p - pl)G(pf t - P ) + dtl G(p, t - t t ) v (p - p f ) j X G(p', t1 - 1') + ,

where

The modified description of motion in the potential V is produced by using the propagation function GV in Eq, (4-11.33).

Let us consider first the linear V term in the expansion of (4-11.41). I t represents the effect of single scattering by the potential in a causal arrangement involving two extended particle sources that exchange a particle and a photon. The observation that

exp[- iP(t - l*)] dtl up[- iT(p)(t - tl)] V(p - p') exp[- iT(p1)(t1 - t')]

- Id1 up[ - ~ ( T ( P ) + WoV - 4)1 V$ - P') up[-- i (T0 ' ) + P)(t1 - 1'11

(4-1 1.44)

leads directly to the extrapolated form of the vacuum amplitude:

A contact term, which is independent of E, has also been added. Its existence is implied by the requirement that a constant potential V produce only a displacement of the energy origin, E -, E - V. When this substitution is introduced in (4-11.35,37), and the terms linear in the constant V are selected, we recognize the counterparts of the two contributions in (4-1 1 .M) where a symmetrization has been used to give general meaning to the product p8V.

4-1 1 H-particle energy ditpla~mants. Nonmlativistic discunion 163

In the application of the vacuum amplitude (4-11.45) to single scattering by the potential, the fields obey

$*(P, E)(E - UP)) = 0, (E - T(p'))$(p', E) = 0 (411.46)

and the vacuum amplitude reduces to

$*(P, ~ ) ( p - p')'V(p - p')$(p8)'. E). (4-11.47)

The potential effective in scattering the particle from momentum p' to momentum p is thus given by

We recognize the essential structure of the charge form factor (44.81). The comparison is sharpened by introducing into the spectral integral J' dkO/kO an upper limit K, which represents the,boundary of applicability of the nonrelativistic treatment, and a lower limit associated with a finite photon mass, p. The effect of the latter, as contained in the altered momentum

Ikl = ((KO)' - p*)"', (4-1 1.49)

appears in Eqs. (4-1 1.25.27). and brings about the modification

the integral is performed conveniently by using the transformation

KO = p cosh 0. (4-11.51)

Now the comparison with the spin 4 charge form factor (4-4.81) produces a precise limit to the nonrelativistic discussion,

2K 5 3 11 spin*: log- = -- - -

m 6 8 - 2 4

For spin 0 particles, according to the relation (44.132) and its consequence

444 Elsctrodynamla t

this is replaced by

The situation in which an unlimited number of interactions with the potential V can occur is represented by the use of the complete Green's function G' in (4-1f -33). Alternative integral equations obeyed by the function are given by

according to the algebraic equivalence stated in

The 'two equations are combined in

C? =: G +- GVG + G V G ~ S I G , (4-1 l .B?)

which gives an exact form to the remainder after two terms of the expansion in powers of V. The formal treatment of this remainder term is straightforward, Multiplication of G" by the photon propagation function effectively adds the photon energy to the kinetic or total energy of the particle, and there are no further physical normalizatian conditions La require the presence of contact terms. We present the complete result as an additional action term,

where, written in a m a t ~ x notation,

1 l E - T E + & - - T - k @ + k ~ ' ( k o ) a

and

This additional interaction will be appiied to bound states which obey, initialiiy, the eigenvalue equations

4-1 1 W-particle energy displacemcrnts~ Nonrelatlvistie dlscutsion 465

The use of these equations in (4-1 I .58,59) produces the eanwflation of the p2(B - Tj and, - +(;p2V" + t"p2) terms. A furfifter simplification results from the algebraic identity [essentially (4-f 1,55)3,

which, applied, to the first term of (4-1 1.591, makes explicit the linear dependence u p n the potential V , We write this version as

&V = Cflb~l"- SF(=), where

and

Note that, in the approximation of single scattering, where the field equations (4-1 t ,611 are replaced by (4-1 JL ,461, the additional scattering potential of (4-1 l .48) is regained from 6Vt1'. The stmcture of &V(@ is well-defined nonrelativistically, avving to the three powers of kO in the denominator the spectral intel;yraf converges at high energies, And, there is no infrared difficulty since ?" - E never vanishes for bound states, where E .L= 0. Not even the existence of a zero eigenvalue far H - E, yielding the singular factor - Ilka, is significant since the multiplicative matrix elements vanish in the limit k@ -+ 0:

All this suggests, correctly, that the 6 V 2 ) contribution is a relatively minor one. It is 6V1) that produces the major part of the energy level displacement, quite apart from the logarithmic dependence upon the large energy value K ,

flCd Er~ctrodynamicr I Chap, 4

To facilitate the calculation of this dominant contribution we rewrite S'lrru as (is is without effect and is omitted)

The Coulomb potential

V(F) =;; - Zgjr

is such that

For a given state of interest, one with wave function $(r), the delta function selects the value at the origin of an auxiliary wave function which is defined by

The energy displacement is given by

It i s convenient to present it in tems of dimensionless variables, which are constmcted from the Bohr radius

and the Bohr energy values

(We have also introduced the so-called Rydberg energy unit.) The variables are

H-particle energy displacemenb. Nonrelotivirtic discussion 167

and the wavefunctions are cox~espondingly redefined by

$(X) = [n(-o)T1'v(r)*

%(X* 4 = [Jdnao)TuS~(r, P ) , (4-1 1.76) which gives

(4-1 1.77)

The simplest example of this calculation is provided by the n = 1, I = 0 ground state where

$1o(x) = ~ X P ( - X).

(1 - s)%exp(- X) - exp{- [(l + s)/(l - s)]x) xlo(x. S) = exp(- X) - - 2s X

(4-1 1.78)

and

x10(0, S) = S.

The successive integrations are exhibited in

where the result involves the number

17 -- 4

2 log 2 = 2.8637. (4-11.81)

Concerning the 131/ (~) contribution we note that, on introducing the complete set of wavefunctions &,(r), we have generally

For the ground state, in particular, all the matrix elements that appear here are

such that E - E, > 0, and this additional energy displacement is negative, Its numerical valuer is indicated by the folfowing Ateration in the ad&tive constant (4-11.81),

Thus, quite apart from the dominant logarithm, Xag .m/ 10.6, the additive constant is fixed ta within four percent by this elementary calculation.

It is advantageous, for a similar discussion of other s stares, to b @ n with the following generating function (essentidly that of the Laguexre polpomiab)

whi& has the same expnential f o m as the wave functian $lofx), and reduces to it on placing t = 0. Accordingly, the generating function for the zBa(~, S ) ,

also resembles xlo(x, S) . Produced by solving the differential equation

this generating function is

S* t ) = (l. - $)%(l - t)%

(4-1 f .S7)

On evaluating it at tke oridn we get

"The necessaxy caleu'lstions have been performed mafit effmtivefy with the aid, of a,

momentum space constraction of the H-particle Green" function, which is described in Quantum Xt'rrternatics and By~amics , U". A. Benjamin, Inc., Mento Park, 1970. The details can be found in the Harvard, 1887, thesis of Michael Liehr. See also Pkys. RIV. 174, 2037 (1968).

and therefore

~ n o ( O 8 S ) == Sn*

Although it would be possible to construct generating functions for the integrals that appear in (&11.77), we shall be content to consider only the 1 . ~ =; 2 s-state. The required wave functions are

and

The value of the numerical constant that a-ppears here, and of the additional. effect associated with bV@~, which again involves only negative terns, is in;&cated

'&Y 2 . W 3 + 0,2665 = 2,8118. (4-1 1-92]

While the contribution sf cliV(%) h a increased, relative to the lz = f situation, it is still. only a few percent of the complete effect.

The Is and 2s states we have been discussing are rather special ones since they are stable. (Of course, the 2s Ievel is not completely stable but tbis refinement does not appear in our physically limited treatment.) The instability of other levels is manifested in the stmctrxre of 6 V 2 ) [Eq, (4-IX.8233 by the appearance of an imqinary part,

where the inteual that occurs in (4-11.82) has been simplified with the aid af the energy restriction imposd by the &It& function,

We reeolifnize the structure of the spontaneous emission probabilities per unit time, Eq, f%li6.69), which are added to give the total Beeay rate [cf. Eq. f%16.41)],

This modification of the energy, apparing in the time propagation phase factor

c ~ n e c t ly represents the instability of the nl state through photon emission transitions to H-padicles of lesser energy, (Perhaps we should recall that the earfier discussion of H-particle instability was completefy phenomenofogcal concerning energy values. Now we are in the process of deepening our under- standing of the energy level structure.)

For states of nonzero angular mczmeatum both wave functions, $(P) and g(', kO), vanish at the origin. That removes the major term of 6t""I, Eq. (4-1 1.77). The residual contribution and that of SV"t2) combine to give a quite small displacement. I t is illustrated for the 2p level by

Relativistic. effects, in the experimentally interesting example of spin Q, are partjy included by the relation (4-4 1,52), which converts (4-lX.f)f, 92) into

There is also the magnetic moment effect displayed in (H.124), for example. I t produces the following energy displacement,

Tbe wave funetions that appear here obey the Dirae equation, which can be written a s

(m t- E -- V)$ =. fy5e' v + my@)tf(rt (&lf*lOO)

and

$*M 4- E - 'C") - $*(yJcre +snyof, +*v' G - v+*. fkltr,ror)

These equations are combined in

and, using a, nonrefativistie approximation that neglects E --- V relative to m, we express the energy displacement (4-11.99) as

Comparison with (4-1 1.67,69) and its multipIicative factor of (a/3z) ( l /ma) shows that the first term of (4-11.1032, which is efkctive only in s - s t a t ~ , adds the constant 8 to "re dominant logarithm,

In states of nonzero orbital a n p l a r momentum and a @venEtotaI angular momentum quantum number j = k -& &, (4-1 1.103) becomes

To review the derivation af the known expectation value that appears here, one notes that the average radial force vanishes in a stationary state,

whife the familiar linear depndenee of the principal quantum nurnber n upan the orbital quantum number I, .n = n, + I -+ 1 , shows that

This Efives

and the enerw displacement induced by the additional xnapetic moment in states with I S O appears as

Xneidentally, the p v i o u d y stated result for the S,,, levels is also produced by this fomula.

The combination of a11 the effects we have discussed, including that of vacuum polarization, @ves the following expression for the relative displacement of the initially degenerate 2s1,, and 2pltl, hydrogc3nic levels:

l f. 5 Ez,,,, - E,,,, = -- Z4aS Ry - - %.SIX8 + --

3% pa2 43

Xn writing the last line we have alw introduced some of the more obvious mass corrections for a realistic W-particle. These recognize that it is the reducd mass of the electron (m) and nucleus (M) that enters the Bohr radius

and the Bohr enerw values

(the Rydberg unit for infinite mass is retained), Using the value

and the energy unit (conventionally stated as a frequency)

we deduce from (4-11.109) this fxequency shift for hydrogen,

A recent meaurement of the levd splitting gave IQ57.90 -& 6-10 NHz. The ameement to better than one percent is impressiw, partievlarly since one can expect an improved relativistic treatment to produce additional effects of relative magnitude - a = 7.3 X 10-s,

Stal preoccupied with mcient history, Harold asks a question.

4-1 1 W-paelcle ensay dlrpt~ements. Nonrsla&Ovistic discussion f 73

H, : Since you, were first to state the additional mapetic moment, and the radiative comection to scattering of the electron, it would be surprising if you had not been the first to a k v e at the energy displacement formula you have just rederived. Were you ?

S, : I believe that I was, although a t the time (1947) I was not a t all convinced of the correctnes of the result. Let me review some of that history which can also be traced thraagh, the various papers colfeeted in Selected Pgeers on Quantum Electrody1-zanzics, Dover Publications, IRC,, New York, 1958. Discussions between V. Weisskopf and myself, prior to and dudng the famous Shelter Island Conference of June, 1947, produced agreement that relativistieatly calculated electrodynamic effects should @ve a finite splitting of hydrogenic energy levels, Shortly after, H. Bethe performed his nonrelativistic calculation, which left unsettled the value of the constant that accompanies the dominant logarithm. That month of June, I947, held two other significant events far me, f stopged smoking, and X got m a ~ e d i , Following an extended honeymoon tour of the country, I returned ta the relativistic problem. Using the noncovariant operator field theory then in voee , I intrduced a canonical transformation tbat iwlated the physical effects associated with an externally applied electromagnetic field.. WXlen this was chosen to be a homogeneous m a ~ e t i c field the additional mqnetic moment of or/f?n: magnetons was obtained. An inhomogeneous electric field gave just the results that are stated in (4-1 1.98), but the accompanying spin-orbit coupling had the wrong factor to be identified with the additional maeetic moment - rejativistic invariance was violated, Pnseding the right factor gives (as we now know) the correct answer, but there was no conviction then in such a procedure, Attention therefore shifted to the development of manifestly covariant calculational methods. Although they greatly reduced the labor of the camputations, there was a geriod of confusion concerning the proper joining of high and low frequency contdbu- tions. By this time (1948) other groups, variously using noncovariant and co- v a ~ a n t methads, had attacked the problem and arrived at a variety of answers. Xt was Weisskopf, by the way, who insisted on the particular number with which all eventualfy agreed; this was the almost forgatten result of my earher calculation, (I still recall the shock X experienced when X happened to compare them and discovered their identity. The classification of procesws in the first method was not rt particularly physical one and only the final answer for the additive constant was stated, as a, certain fractional number. The eovariant technique, on the other hand, gave a clear physical separation of various effects which were therefore left explicit in the fom of corresponding fractions* Somebow, with the memory of the noneova~ant cdculation resolutely suppressed, it haid never sccuned to me to combine &XI, thwe fractions into one.) As in the history of the

scattering formula, there is a moral here for us. The artificial separation of high and low frequencies, which are handled in different ways, must be avoided. We shall see that the relativistic treatment of energy &placements has this desirable feature.

Before leaving the nonrelativistic arena, however, it is interesting to follow the workings of a method which unites elastic and inelastic processes in scattering, thereby avoiding that artificial separation. The vacuum amplitude representing single particle exchange in a causal arrangement is, for a free particle,

- i (dr) dt' q*(r, t)G(r - r', t - t')rl(r', t'), I (4-11.116)

where

t > t': G(r - r', t - t') = - iz b ( r , t)$#, t')*, P

Using the source definitions

p = ( ) $ * 1 , q: = I (d.) dt v*@, t)&,(r, t). (611.117)

we find that the vacuum amplitude (4-11.115) is

(- iqi)(- 'qP)D P

which represents the fact that a particle, emitted with momentum p, is detected with certainty in the same circumstance. Now let ns replace G with G' and again extract the coefficient of (- iq;)(- iqp). The absolute square of this probability amplitude states the probability that the particle persists in its initial state despite the action of the potential V. We retain only the first two terms of the expansion (4-1 1.41), which gives the persistence probability amplitude

1 - i (dr) . dt' $,(r, t)*[V(r) d(r - r') d(t - t') J + V(r)G(r - r', t - tt)V(r')]$,(r', t'). (4-11.119)

The time integrations reduce to one extended over the entire interval, T, between the emission and absorption acts. The probability amplitude then reads

where E -. T(p) and V(O) is V(p -- p). From this the persistence probability is derived as

2n d(E - T(pP)) /V(p - p') IP,

wtnich also states the total probability of a scattering process. That probability, per unit time, divided by the incident particle flux [(dg)l(2~)3] f pllm], gives the total scattering cross section

Spherical coordinates in momentum space are introduced by

where dR is the element of solid angle. The evident implication concerning the differential cross section for elastic scattering is

which is indeed the well-known first Born approximation, AI1 this leads up to the analogous consideration of the scattering potentiai

V + &V. The two parts of SV are stated in Eqs. (Pll.@4), or (4-11.67) with ie reinstated, and (4-11.65) where H is to be replaced by T for our limited purpose. The probability amplitude (4-11,120) csntinues to apply, with the substitutions

and

where the last factor (not its complex conjugate) also applies to V @ ' - pp), There is complete cancellation, in the analogue of (4-1 f ,1520), between the terms that conbin p2 explicitly. The resulting expression for the total cross section is

which form emphasizes that no singularity at RO = 0 appears. Then, to facilitate the extraction of the imaginary part, we write

1 1 E + i e - T(p')E + ie- T(p') --P

and this gives the modification of the cross section in the form

The two terms clearly exhibit the decrease in the cross section for purely elastic processes, and the increase associated with inelastic processes. The practical impossibility of distinguishing the two classes of events, as kO -+ 0, is offset by the precise cancellation of the two kinds of contributions in that limit. We can use (4-11.123) to infer the modification of the differential cross section in angle,

- b(E - T -P)], (4-1 1.130)

which stiIl leaves free the selection of the range of kinetic energy for the scattered particle.

Let us consider the situation of essentially elastic scattering, as characterized by

where R:,, represents the accuracy with which scattered particle energies can be measured. Under the assumption that k&, << E we have Ip'l 2 Ipl, and the integrals of (4-1 1.130) reduce to

4-1 2 A rufativtstfc aaaerlrrlp calculation l77

The resulting modification in the differential cross section for essentially elastic scatterir~g is

In the example of spin O particles, relativistic effects are introduced by the substitution of (4-lX.54),

K un I log ,- = log ----- + --

kmin 2kynin 12

(we are still considering $ow particles, of course), and by adding the contribution of vacuum pulatization. The latter, for spin 0 particles, is of the spin 4 csntribu- tion wllich alters the added constant in (4-11.134) to

\Ye recognize the spin O resujt stated in Eq, (4-4.99).

&I2 A RELATIVISTIC SCATTERING CALCULATION

It has been remarked that, in processes where photons can be emitted, a decomposition into elastic and inelastic cross sections is artificial owing to the experimental impossibility of making that distinction for sufficiently soft photons. More realistic t l~an tl-te elastic cross section is the cross section for essentially elastic scattering, as defined by the limited eqerirnental ability to perform the necessary energetic carltrul. We have just described, in a nonrelativistic context, a technique for the direct calculation of the essentially elastic crass section. Now let us apply that idea to the relativistic scattering of cl~arged particles in a Coulomb field. Since we wisll to illustrate a method, rather than obtain new results, it will suffice to consider spinless particles (the spin $ results are then easily inferred). ft is convenient, however, to modify slightly the objective of the calculation relative to that of Section 4-4. There, processes were selected in which the emitted photon l~ad an energy less tllan k:i, << p@ - rtr. \lie propose to replace this criterion for almost elasticity by a restriction on the mass of the final tw-particle state:

Accordingly, using the method of Section 4-4, we shall first construct the essentially elastic cross section that embodies this criterion.

The probbility amplitude (3-14.61) for soft photon emission yields an emision probability with the kinematical factors dw,, dwk. We transfer attention to the total momentum

(d*l) (dP) 8( (P - h)% + m2) do,, = - d($l" m2) = - ( 2 ~ ) ~ (2~1%

Then, using the total momentum for the comparison, .rather than the particle momentum, the refative probabilit y for photon emission, replacing Eq. (44.93) , becomes

The lower limit of the M2 integral recalls that the photon is temporarity assigned the mass p. The basic photon mamentum inteeal, evaluated in the rest frame of P m d then simplified in accordance with MZ - m2 < 2m 6M .m2, is

X. l -- E (2n)"M

( ( M -- m)2 - p2)1/24

We exhibit it by rewriting the integral (4-12'4) as

where the averee indicated by the expectation value notation refers ta an angle, Tt is illustrated by

The last evduation has been performed in the rest frame af P g PI, where, in the notation of Section 4-4,

4-1 2

we have

To complete the computation of (4-1 2-41, uTe note that

while

leads to

according to the relations

The spectral integrals that appear here are of the form fcf. Eq, (4-4-97)]

and the resulting evaluation af (4-12.4) can be presented as

180 Efarodymmlu; C Chap, 4

Here, expressed in the variable 6,

cosh C) == 1 + q & / 2 d ,

we have

where

E* = 3- v ) 4- -W - 4 expfrir: of. (612.19)

This is not the most felicitous presentation of the function 8, howver. We shall do better by observing that

&* = exp(& a@) [cosh &B F v sink +@l, (P-XZ.2o)

which leads to the expression,

Now, if we explicitly enforce a spmetrization between v and - v as integration variables, and perform some partial integrations, there emerges as the equivalent of the right side of (4-1 2.21) :

log(l + (1 -- ve) sinhaf 8) -- log(1 - v') dv 1 -+ (1. --- v2) sinfig+@

2e log 2. (P12.22) -+- sinh @

The recognition that

combined with the identity (W. 1091, then produces the result

4-1 1 A rslaelvtolfe saetering calculation 484

The relative photon emission probability stated in (4-12.l6) thus becomes

X t combines with the factor that gives the modified elastic cross section, Eq. (44.86), to produce the essentially elastic cross section, as expressed by the modification factor

This is what we must reproduce by direct calculation of the essentially elastic cross section, Under nonrelativistic circumstances, incidentally, wklere no distinction need be made between 611.1 and h:,,, one expects (4-12.26) to yield (4-4.99), which it does.

Let us review the direct calculation af scattering cross sections, using the example of a Eocaf vector potential A. This is the relativistic generalization of the disclxssian given in Section 4-11. The vacuum amplitude representing single particle exchange in the presence of the potential is

where, only exhibiting terms linear and quadratic in A ,

On extracting the coefficient of (iKG,)(iKZ,,), we get the persistence probability amplitude

The charge eigenveetor p, can be omitted if one replaces the charge matrix

182 Elsctrdynmmfcs I Chap. 4

with its eig-envalue. For the situation of a time-independent potentid that can act upon the particle for the time interval if", this hcornes

where three-dimensional Faurier transforms are used, and P@" $@. From the implied persistence probability,

we deduce the total scattering crass section,

and, by remarking that

infer the differen tial cross section

The last version in (4-f 2.35) refers to a purdy scalar potential, Using the example of the Coulomb potential, we indeed recover the familiar result stated in Eq. (3-14,8), for example.

This brings us to the consideration of the modified, nonlocal potential that is implied by the coupling with photons, We are, of course, acquainted with one aspect ol it, as expressed by the forrn factor of Section 4-45. It is essentiaf. to note here that what is required is not the f o m factor applicable to free padicles, where both momenta of the double spectral forrn are on the nnss shell, as the expression goes, but the farm factor ufith only one momentum on the mass shell. That is evident in (kX2.31) where the momentum # ' i s not restricted to free particle values. Thus, the form factor that multiplies both A (p - p') and A (p' - p) is, primarily,

I t has k e n derived from (4-5.86) (mark the notational change k -* q) by using the relation (&fj,99), but it does not include the contribution involving the combination (&6.200), which will be considered separately, On placing p'%

- m%, we recover the form factor expression of (4-5.98,99). In computing the total cross wetion one takes the i m ~ i n a r y part of the wuare of this function multipGed by (p'% + m2 - ief-x, which, with a factor of n removed, is

As in the nonrelativistic discussion, we see combined here the elastic process, indicated by 8(pf2 + m%), and the inelastic processes setected by S(pa + MM,%)" The cancellation as M 2 --. m means that no infrared singularity oeeurs in (G12.37). However, since it is convenient ta utitize the form factor results already obtained, we shall inerduce a photon mass and consider the two classes of processes separately.

The integrations in the inelastic term are canstrained by the photon rnass at one end, as expressed by the restriction ( 6 5 . f 02) and, at the other end, by the requirement that f i f z 2 - nz2 < 2.m 8M. Throughout this interval, x <e I, and one can simplify the inelastic coefficient in (4-1 2.37) to

where

(612.39) This giws the folfowing value to (&12.38),

where symmetrkation betvveen v and --- v induces the refiacernent

Then, the use of the identity [it is equivalent to (M,IQ9)]

Chap. 4 "1 EEI&rdynamlcr 1

produces

As far as the ckllnninant logarithmic term is concerned, this aeees completely with (&112.25). But, as we have stated, the above form factor consideration i s incomplete. I t behoova us now to examine the whole picture,

The vacuum amplitude far the causal exchange of a photon, and a particle that moves in a potential A , can be expressed as

where the appearance of the gauge csvariant combination

in&cates that we me now cansirIering emission and abwrption acts occurring in the reeon occupied by the potential, We shall again use the expansion (4-1 2.28) for A,". The initial term, A,, appears under causal conditions its

exp[iK(x - x"] A+(% --. X ' ) = i dw, exp[i(p +- k) (x - x')]

The corresponding vacuum amplitude contributions are

where one can introduce the effective replacement

and the inteeal (P.12.5). The ensuing space-time extraplation, with appropriate contact terns, eves the action expression

We recwnize the additional action for a free particle with the contact terns needed to supply an effective ($9 + m2j2 factor [essentiaUy Eq, (4-6,40), modified for a nonzero photon mass]. There are also terms linear and quadratic in the vector potential. They have been provided with the simpler contact term that suffices to avoid modifying the primitive interaction when applied to a slowly varying potential and the fields of essentially real particles.

Since the field 4 in (4-12.49) describes particles moving in the potential A, one cak make the substitution

referring to a Lorentz gauge, with the A2 term omitted, which is sufficiently accurate for our limited purposes. Tbe resulting action term k

I t makes a contribution to the probability amplitude which, since its imaginary part contains 6(p2 + M2), represents inelastic processes. In view of the restriction to almost elasticity, it is only the infrared singular term of (4-12.51) that is significant. This supplement to the modification factor (412 .43) is

f d(M -- m) ( (M - m)z - p2)1/2 --- =: -

(M --. m)%

The next tern in the expansion of containing q ( p A + Ap) --. e2A2, makes three distinct contributions in (412.M), ail of which are expressed as double spectral foms. Xn the appGcation of the A2 term, b t h momenta are placed on the mass she11 and no imaginary contribution ensues, Now consider the tern that combines the linear A dewndence of the propagation function with the A depndence of the effective ernision or absorption source. Here, one momen- tarn is on the mass shell. The comesponding ima@nary part, which desc~bes Jnehtic processes, has this appeamnce:

where ( * * a . ) indicates factors that have finite Iimits as x -+ Q.. Thus, in contrast; vvith the andogous terrn in (4-12-37), there is only one spectral denominator, which does not yield an infrared sinelar structure. What remains is the double spectral fom of Section 4-45, which has d r e d y been considered in its major impfications. Now let us eomgEete that discussion.

According to the complete stmcture of Eq. (&6.86), and the relation (4-6,100), the f o l l o ~ n g terrn has been omitted in the h r m factor (&1%,3@),

Its contribution to the imaginary part of the squared form factor multiplied by (p* 2 wm2 - ie15.)-', and divided by n, is

which represents inelastic pmcews. Since the x integration is limited &s in (&12,39), the coefficient of interest is

(412.60)

or, after padial in teeation,

It is worth pointing out that this contribution would not occur bad we adlopted the provisional procedure of Eq. f&5,90) where mare elaborate contact terns are used,

We come now to the last term of the expansion (4-12.28). When inserted in the vacuum amplitude (4-12.4) it describes a process wherein two scatterings by the vector potential A occur between the photon emission and absorption acts. The causal arrangement that is mast convenient for our purposes is this. A free particle emits a photon, and the resulting virtual particle is scattered by the potential to produce the real particle which, in company with the photon, is detected by an analogous arrangement. This will lead to a single spectral form associated with the mass ?21 of the two-particle system. Its irna@aary part correspondingly represents inelastic processes. We are interested only in soft photons for which the process under consideration can be factored into the elatic doubk scattering of the particle and the photon exchange process, Accord- ingly, this contribution to the modification factor for essentially elastic scattering is just the part of (612.4) that refers entirely to the initial momentum cornbin- ling the photon emission by the initial particle with the photon detection by the final particle. i t is

according to the various evaluations illustrated by (4-12.8) and (4-12.23). When one adds the'. three additional effects given in Eqs, +12,52,67,58),

alf referrz-nce to photon ma5s and the criterion sf inelasticity disappears, Ieavjng the fdlowing tern :

i;B8 Elmtredynamict C Chap, 4

Its addition to (4-12.43) yields precisely Ey. (4-12.25), which combines with the elastic factor to give (4-1 2.26). Note again that, while we have retained the adificial distinctions of elasticity and inelasticity for a certain convenience, the mettro-cl directly supplies the modification factor for essentially elastic scat te~ng, deriving it from the double spectral version of the form factor arid requiring only relatively simple considerations to obtain a small supplementary term.

For a given 6%%f, the modification factor (4-12.26) depends only upon the momentum transfer measure q2. This i s in contrast with (M.114), where an explicit energy dependence also occurs. For values of q2 <CC m2, the nonrelativistic evaluation applies,

At the other extreme firnit, we find that the modification factor for almost elasticity is

In view of the simple relation that is known between the spin O and the spin problems [cf. Eq, (44.235) and relajted discussion], we can immediately supply the analogous spin 4 results,

-+ 3 log 2(1 - log 2) -

The preceding and following sections are concerned with different physical aspects of charged particle couplings that involve the repeated action of a Coulomb field, This section has a similar preoccupation, but with the fields of photons. The physical process is the scattering of photons by charged particles-Campton scatte~ng-as it is modified through the mechanism of two-padicle exchange. One systematic approach to that problem is provided by the additional action terms that are inferred from the vacuum amplitude for the causal exchange of a

4-1 3 Phot~n-charged particle scatecrlng l83

photon, Eq, (&12..14), The terms independent of A describe the modified propagation function of the charged particle, terms linear in A represent the modification in the single photon emission and absorption mechanisms, and terms quadratic in A do a, similar service for two-photon processes. The ensuing effective coupling that contains two free pstr-ticle fidds and two photon fietds represents the process of interest, We: propow to attack this problem in another, and possibly simpler, way by considering the process as a unit instead of building it up from various individual acts, Since there are no experimental results of great accuracy for electron-photon scattering, the question is primarily of methodological interest and we shall restrict the discussion to spin 0 particles.

To illustrate the possibility of a more direct calculation, let us consider the following causal anangernent. The charged particle and the photon collide to produce a new configuration of these particles. After a Iapse of time the padicles again collide and the end products are subsequently detected, This is a two- particle exchange process, with the scattering mechanism used both as effective prdrxctim and detection sources. The action term of (3-12.92) implies the following effective emission source,

where

ineav~ra tes the simplifications of a Lorentz gauge, and a reduced notation has been used for the sources of the particles that enter the collision. The analogous effective absorption source is

with

The vacuum amplitude that represents the two-particle exchange is bczrived from the coupling expression

It can be expresxd as the following transition matrix element (the subsc~p t

190 Elutrody~mlcr t Chap. 4

c is appended as a reminder that this refers to a causal arrangement) :

where

P = kl + p1 = ke + P2

is the total momentum. We shall use the rest frame of this vector to select a convenient gauge, as expressed by the conditions

Since each polarization vector is also orthogonal to its own momentum vector, we can introduce the following substitutions:

Plc1 = Peez = 0, Pc2 = - RC2, pc1 = - kC1. (4-13.9)

Accordingly,

where parentheses are used to distinguish scalar products from dyadics. One can also exercise the option of writing

The relevant terms vanish on integration, however, since the vector

for example, can only be a linear combination of the vectors P and pl, both of which are annulled by multiplication with e:. A similar remark applies to (k/kp2) and the polarization vector ez. Thus the only integrals required are comprised in the tensor

RukV l - v

) (4-1s.a)

The last form makes evident that the calculation is not appreciably more complicated for PI # p%, compared to p1 E; pz. Nevertheless, since we shalI use the result of this causal arrangement only forpl = pe we now adopt that specializa-

tion. In the interests of the following calculational device, we shall also introduce the symbol q to denote an arbitrary vector that is finally identified with P1 = P,:

The necessary integration thus reduces to

(log(@)) - [l f d. [ogMI&ma + log [g + z (p + pgr)"]} (613.15) where, in the rest frame of P, the single integration variable is the cosine of the angle between q and k. A first differentiation gives

But, the vector P that appears here will multiply one of the polarization vectors and can be omitted. Accordingly, the effective value is

At the next and final stage of differentiation, both the vectors P and q + 9% can be discarded. Thus the only contribution arises from the differentiation of the vector q in the numerator. This gives

where we have made the replacements

in which M is the mass associated with P. Our results are expressed in terms of the following function,

which has simple limiting values,

W = m" = jZ,

W > > m s : %(M%)=l . (4-13.22)

The c a d transition matrix element is

Its space-time extrapolation proceeds by writing

( 2 ~ ) ~ 8 @ 1 + p 1 - k2 -p2)

.rup[iP(x - X')]] e q [ i ( k ~ + pJx7.

(4-13.24)

where

One also removes the causal distinction between the fields of the initial and final particles; this introduces the crossing symmetry ,

c: t ,es , k l w - k z , (4-13.26)

or

Pi* - Pe,

But, in order to maintain gauge invariance after these extrapolations have been

pdormed, the polarization vectors in (4-1 3.28) must first be replaced by equivdent field strength combinations, as indicakd by

Then, evaluated in the gauge where the polarization vectors are orthogonal to the particle (or total) momentum, the resulting transition matrix element describing forward scattering is

where the spectral integral can also be written as

Note that we have now included the direct implication of the p~mitive interaction, as &ven in Eq, (3-12.98) and speciagzed to foward scatkring,

There is a test of this statement, in which it is applied to predict the total cross section for photon-particle scattexing, Such ideas have appeared in Sections 4-1 l and 4-12. They involve computing the probability that the initial confipra- tion of particles has persisted, despite the effect of the interaction. The &sussion of Section 3-12, where the transition matrix was introduced, in&cates that the vacuum persistence amplitude for the given two-pa~icle state is (in simpEfied notation)

where V measures the four-dimensional interaction volume. The implied persistence probability,

exhibits the complementary total proba-bility of a scattering process, That probability, per unit volume, and per unit initial invapiant flux F, @ves the total cross section :

$94 Elactrodynamim I Chap, 4

In the situation of immediate concern, where one padicle is massless, Eqs. (3-12.68,69) supply the flux; expression

F = do,, dw,, g(-- 2k2Pz), - 2k2pz = M' - m2, (4-13.34)

and we deduce that

The limiting forms suppiied by (4-1 3.22) are indeed those stated in Eq. (3-1 2.1 M), and the total cross section produced by integrating Eq. (3-12.1 17) agees with the general expression for %(M2), Eq. (G13.21).

The real part sf the spectral integral (4-13.30) is conveniently evaluated by inserting the integral expression for ~(hilz) given in Ep, (k13.21) and carrying out the M2 integration first, The result is presented (without subscripts) in

At Eow photon energies the right-hand side reduces to

while tfre otber limit is

Expremd in terns of the photon energy in the pad--ticEe rest frame,

the two limiting foms of the differential cross section for foward scattering are

Note that there is no infrared problem here, since the charged particle is un- deflected, and that the cross section is increasied relative to its skeletal interaction value. The summation aver final polarizations replaces independently of the initial polarization,

En the discussion of photon-photon scattering, the single spectral farm supplied by foward scattering considerations, taken in conjunction with a double spectral form, compfetely specified the transition matrix. The present situation, involving two particle and two photon fields, is more complicated since there are two possible foward scattering arrangements. The second one can be chosen as the collision sf two photons to form a particle pair, which subsequently undergoes a scattering interaction. This is the two-photon analogue of the arrangement used in Section 4-4. Mere, however, one considers real photons instead of the virtual photon of that discussion, The forward scattering restriction is realized, in the center of mass frame, by requiring that the oppositely moving charged pariicles have the same directions as the oppositely moving photons. The effective two-particfe source, which is again infemed from (3-12.92), is

with (dyadic notation)

This is the source that is to be inserted in the Coulnmh scattering part of the vacuum amplitude (44.4). The annihilation mechanism term is also present, of course, but it selects effective sources that are antisymmetrical in 9% and p'%, whereas (k13.42) is symmetrical. The implied transition matdx element for the causal amangement is (superfluous causal labels are omitted)

The inteeation probiern conveyed by the tensor expectation value

1% Eloctrodynrmler I Chap. 4

is simplified on specializing to the forward scattering circumstance that is expressed covariantly by

4mt p - ( l ) (k -k ' ) .

where

M2 = - (k + K')% = - ( p + p1)2. (4-13.47)

The tensor A must then be constructed from the vectors k and K', in the various combinations kk, k'k', kk', k'k, and the unit tensor. But there are also restrictions implied by the gauge properties

kV = k', Vk' = k. (4-13.48)

They are incorporated in the form

However, only the coefficient a(M2) appears in the final result :

The information needed to construct a(M4) is obtained by forming the trace of the tensor A:

(4-13.61)

and by computing k'Ak:

(4-13.62)

The reIation

enables (4-13.62) to be simplified to

and adding this to (4--13.51) gives

The equivalent integration over the scattering angle in the center of mass frame reads (inserting the photon mass p),

M2 - 2m2 M2 - $m2 tr= 2 -- log -- -

M2 - 4m2

The space-time extrapolation of the probability amplitudt? dessribed by (4-13.50) is produced on writing

where additional functions of M 2 are placed under the sign of integration. The corresponding contribution to the physical transition mat fix elemnt is

198 Elscgrdynamlrrs 1 Chap, 4

which has been infemed directly from (4t3.W) since the Iatter is already in gauge invarimt fam.

Now we consider a carnal anangernent that leads to a doable spectral form, (The reader is encouraged to draw the diamond shaped causal diagram for the process to be described.) An extended photon saurce, emitting the time-like momentum Kg, creates a particle-antipartieie pair. One of these particles travels to the ecinity of anather extended photon source, vvbere the space-like momentum K, is transfened to it, The other reaches the neighborhood of an extended padicle source where, by combining this particfe with a virtual (anti) particle of momentum P,, a photon is produced.. The photon and particle that have appeared at these intermediate stages subsequently join, and are detected by an extended par"ticle source that absorbs momentum PI, where

This arrangement contains a real photon of momentum. k , and three real pa~ic les af rnornenta p', p'". The identification of these mornenta is indicated by the conservation statements appropriate to each interaction :

Thus, the momentum of the particle that is detected with the photon is

that of the particle which contributes to the production of the photon is

and the remaining momentum is

p" .K2 + P, - k = PI - k - K,. (k13.63)

The prwess in question is one of single photon exchange, with the particle interacting successively with the fields 14, m d A ., in the sense of the symbolically w ~ t t e n vacuum amplitude

This is explicitly disp;layed a

P h o t a c ~ b m particle sattarin# 199

where

I., - I (M) d(k' + p? &(P1 - h)' + m2) d((P1- k - KO)' + m? d((PO - k)' + m%)

X k)(2Pa - k)I?(P,- h) - KJfi[2(Pa - R) + K2],. (4-13.66)

Note, in the last expression, that the momentum factor in the coupling to the field A," is (p' + ~5")~ , since this refers to the deflection of a particle with given charge, while the field AeY, which creates a pair of oppositely charged particles, is multiplied by (p" - p"'),. The associated conservation, or gauge invariance statements are

The two independent ways of viewing excitations as proceeding through this system are indicated by the spectral masses

and

M" = - (Kq + K,)' =: - (P1 - P,)' > h'. (4-13.69)

They refer to the propagation of a particle and photon, and a particle-antiparticle pair, respectively. [Of course, the first inequality reads M2 > (m + p)'; we have stated the physical lower limit in (4-13.68).] We also introduce the masses

they will eventually be extrapolated, from values appropriate for the causal arrangement, to the value of interest in the actual scattering process, namely, me. Thescalars K,', KZ', which later will be extrapolated to zero, complete the list of six kinematical scalar quantities (the twelve components of three independent vectors with the six parameters of the Lorentz group removed). The delta functions in (4-13.66) supply such evaluations as (the photon mass is omitted here)

- 2kP1 = Mla - m*, - 2kPa = MS4 - m',

- 2kKa = MMf - W , - 2kK2 = W - M,'. (4-13.71)

Other usefuI combinations are

2P1Pa m M'" - M1' - Me2, - 2KeK0 M* + Ke4 + Ka4 (4-13.72)

2W Efactrodynamia I Chap. 4

- 2K2P, = hi2 - Me2 + Kz2, 2K,P1 = M2 - M12 + K,',

- 2K2P, = M" WM'2 - M,' + I<," 2K,P, = M2 + iW2 - M12 + Ke2.

(4-13.73)

As an application, we note that

(2P1 - k)(2P4 - k) = 2M12 - M12 - MO2 - 2m2. (4-13.74)

The basic integral in (4-13.66) is

in which

This integral is essentially similar to that of (4-10.7), and, analogously,

I = 1/[8(- A)'/'], (4-13.77)

where

(- A)''' = ~ E ~ ~ ~ ~ K ~ ~ P , ~ P ~ " K ~ ~ . (4-13.78)

The general form of the vector K' is given by [cf. (4-10.13)]

Multiplication by the vectors P, and Pl supplies the information to determine a and b:

The determinant of this array of coefficients is

1 = M,2Mle - (PaP1)' + [Ma2(K2P1)' + Mls(Kd'a)2 + 2PaP1PaK2PlKzl.

K2 (4-13.81)

The square of the combination (4-13.79) is

where

C'(- KzzD) > 0

is necessary for the nonvanishing of the integral (4-13.75). The alternative form produced by multiplying (4-13.79) with K,,

shows, on comparison with (4-13.82), that

To illustrate these relations, we consider directly the situation of interest, where

M12 M,2 r me, Kaa = Kza 0, (613.87)

although K2Z = 0 must be realized by a limiting process. For convenience, we list the values of various scalars in this limit:

2P1Pa -, M f 2 - 2m2; - 2K2Ko + M";

%R Electrodynrrnics I Chap. 4

2K,P1, - 2KzP, + M2 - m2; 21<,P,, - IKaP1 -r -112 + .lI'Z - rtlf. (4-1 3.88)

Then,

KZ2D 4 fM'2[(.1.12 - nt2j2 + 9IZll2'2j (4- 13.89)

while the solutions of the equations in (4-13.80) with M,2 = i ~ l , , ~ = ~ 2 ,

lead to

The condition for the existence of the integral, evaluated for M12 = Mug = 7122, is

The linear equations of (4-13.80) can be used to give two equivalent evaluations for the combination of (4-13.92),

The resulting limiting form of (4-13.92) is

When it is permissible to ignore the fictitious photon mass, this inequality simply combines the independent spectral requirements

But, if the finiteness of the photon mass is significant, neither of the lower limits in (4-13.95) can be realized. A sufficiently accurate expression of this spectral domain is

4-1 3 Photon-charged particle scattaring %l3

The construction of (4-13.86) supplies the limiting forrn of A as

Corresponding to the simplification used in (4-13.86), the coefficient of p2 is well enough approximated by replacing , I f2 with 8t2,

As in the photon-photon scattering discussion, the extrapolated value of A is positive, requiring a selection to be made between the values

Again we make this decision bj- comparing the causai and noncausal solutions of a related, simplified problem. \Ye ask to evaluate the integral (here D, includes a photon mass)

first in the noncausal nlanner that introduces the four-dimensional constructions of these propagation functions,

J == (2n)"d(P - Pi)Jos

and then with the double spectral forrn that gives the space-time extrapolation of a causal arrangement, The kinerrlaticak integral is (4-13.75), where WC: are interested only in the extrapolated situation with K g = K, =;; 0, and PI = P, = 9. The introduction of double spectral forms, in the manner illustrated by the discussion leading to (4-10.361, then supplies the alternative evaluation

The first computation a l Jo proceeds by combining the garan~etric representations of the two propagation functions in (4-13.101),

Chap. 4

On changing the variables according to

this reads

The momentum integration is performed by completing the square,

R8 - 2k@ = (k - + m%*, (4-13.106)

and then applying the standard integral (4-8.67). This gives

Since the integral that appears here is positive, a comparison with (4-13.102) settles the question of f i in favor of + i. But let us also verify that the two numerical factors are the same, at least for p/m << 1.

Inasmuch as the pt tenn in the denominator is only significant when u is quite small, the 1 - u factor can be replaced by unity. Then, introducing the variable

we get, effectively,

The alternative computation is

or, in terms of the variable y defined by

4-1 3 Plrrotan-charged partlct rs scattering 11;OS

The explicit construction of the pl-toton n~omenturn vector is obtained from (4-13.76) and (4-13.78) as

where

according to (4--13.93). Note that the various Xixtliting values are related by

Let us also recall that only the magnitude of c is fixed and tlrerefore an average of both signs is to be used (the basic integral 1 already contains the factor of two). The extrapolated value of the square of c is derived fiam (4-13.86) as

The vector combinations that occur in (4-13.661 can be presented in this way:

which make use of such reanangements as

There i s a relation htween a and b that is demanded by the gauge invariance conditions,

0 F=;: K,ljqCPl - k ) - K,] = K2[2(P, - k) + KZ] = ( M s - m2)(l - a - b) - Mqb.

(Q-13.31 9) I t is indeed satisfied since

The space-time extrapolations that we are about to prform will not maintain this gauge invaniance, apad from the terms having the coefficient c, unless we make it expgcit at the causal s t q e , That can be done in various ways, as in the pboton-photon scattefing &cussion. One possibility is described by the following g a q e transf o m a tion :

The new structurm vanish on. multiplication with K'% + Ka. I f we then adopt a Lorentz gaage, scalar multiplication by either photon momentum annds these combinations, This also enables us to replace PI + P,, when multipl9ng the vwtor potentids, by either 2Px or 2P,. As a result, we have

where terms linear in G have been omitted, Note that the two terms introlve field stren@hs and dual field strengths, respectively.

As in the discussion that leads from (4-10.W) to (&10136), the space-time extrapolation is performed by introducing the double spectral form structure

dW dM'8 " m r " Z P v -

1 f 232 29t (P, + K%)% + MS - ig (K, -f- KZ)" Mp2 - ie (613.123)

In order to pregnt our resdts vvith a minimum reptition of complicated structures, we shalE state directly the double spectrd part of the physical transition matrix, which is produced by the extrapolation

supplemented by the crossing transfczmation

or the equivalent one,

dM2 dM'2 (dw,, * - dw,,)vg4a2 ---- ( M a - 2m2)

1 l/% ($2 + + - i~

f - ~ ' ~ ~ ~ ~ , ~ ~ k ~ , ~ ~ ~ ~ ~ ~ ~ ' k ~ ~ e ~ ~ k ~ ~ p ~ ~

where

The procedure we have just followed to make gauge invariance explicit is straightfomartX, but has the disadvantage that the resulting numerator momentum dependence is not irreducible. In this situation there are two baiiic cambinations, involving the pzllarisactian vectors, that are gauge invariant and crossing symmetric, namely

which gives the structure of the skeletal scattering process, and

Written in terms of these, we have

and

These statements can. be verified by algebraic rearrangement or by comparing the two sides in some coordinate system such as the rest frame. Since it is only the demand of gauge invariance that motivates the apparance of momentum factors in the numerator, sugerfluous ones should be removed by transforming them into added single spctral farms, Xn doing this k,kz is replaced by *Mt2, while - K,$2 and ktPz, multiplying the corresponding denominator, are replaced by i ( M 2 - m2). The outcame is this double spectral. form:

2 Electrodynamics I Chap. 4

X l 1 f

-I- ( k , -- k2I2 +M'B (Pz + k2I2 + &f2 - i~ ($2 - k1)2- dM2 $Mi2

- (dw *)1/24aPm2G2 11%

which uses the relation

The added single spectral forms are determined by fsr~rard scattering infsrma- tisn. We first construct the single spectral form that gives the generalization of (413.29) to arbitrary scattering angles. As one can recognize from the gauge invariant structures (4-1 3.281, the space- time version of this coupling involves fields in the combination

(dz)(dx" ~ g q b ( x ) l ; C V ( x ) d , ( x - x f , ~ l f 2 ) F v n ( ~ ~ a F A 4 ( ~ ' ) , (4-13.138)

which is to be integrated over L W 2 with the appropriate weight factor. The implied single spectral form contains tlae reducible gauge invariant combinations

The reduced version is

wbich has no infrared singularity in the forward scattering situation of actual interest, where p2k2 --- $"L1. This is to be compared with the tonvard scattering limit of the double spectral form, which is selected by imposing the kinernatied restriction (kI - I=, 0.

One of the resulting spectral integrals is fp = 0)

4-1 3 Photon-charged particle battering ?M

(M2 + .m2) 2 -J- u2(M2 - 8 %(PM2) d ~ ( 1 + V $ ) ---- - -

[(Me + %%)g - v 2 ( ~ 3 T 2 E e - M% M2 .....- pp$ '

The function X ( M 2 ) that appears here is

which definition is equivalent to that of (4-13.26). We also need the integral

The ensuing single spectral forrn is

where the brackets indicate the propagation function combination that is displayed in (613.137). The first two terms that appear above reproduce (4-13.13"j". Should we conclude that an additional single spectral form is needed to cancel the last term of (4-13.141) 3

The answer to this question is negative; no single spectral forrn involving the momentum combinations ( p , + k2)2 and (P2 ---- ks)2 is required. We have allowed this apparently paradoxical situation to develop in order to emphasize a subtlety in the study of special kinematical circumstances, Does 'Yoward scattering" nlean k1 ==: kz , or ( k , - K Z I 2 = O ? The calculation from wllich (4-23.137) was inferred used the k , == k , characterization; the reduction of the double spectral form to (623.141) exploited ( K s - = 0. The distinction between the two approaches appears on noting that 6% vanishes for k1 = k 2 ; only the'single spectral form with G , as a factor can really be inferred in this way. What we must do is return to the calculatictn of (4-13.13) that culminated in (4-13.X8) and retain the additional numerator structure tllat introduces G%, while continuing to use the denominator simplifkations that express ( k l -- k2f2 = 0. Then, with the notation

we have

2fQ If~@rodynamtcr I Chap, 4

f l h.-------- -p------

[ ( (Me + me) /$M)% - zg((Me - maj[2zB2 ' (413,243)

where, in the context of multiplication by c; and el,.

The last step gives the result of the v integration that is required in (4-13.13). After performing a partial integration in the additional term of (&13,X43), we find that (4-13.10) is now evaluated as

or, with the gauge invariance substitution (4-.13.28),

We recognise, in the relative coefficients of these three terms, precisely the combination of (d-13.X41),

In order to apply the double spctral form in the pair creation regime we make the crossing substitutions

and also write h" d, $P instead of kg, ez, 15,. As a result,

In the forward scattering circumstance expressed by

we find that

while

2 - X -_*_

M2 - m2 - kk" - M% (k - kt)2 + [(M% -- m 2 ) ~ m q '

The combination that occurs in the Boultre spectral forn~ is

The function [ ( k - + (M2 - ,mZ)2/M2]-X describes the transfer of a spacelike excitation, which is distinct from the pair-exchange mechanism represented by [ (k + K')% + M'% --E]-l, The contribution of the latter is

The M2 intqral that oceurs here can be evaluated by decomposing it: at a value of M% - m2 == X such that

This enables one t o write it as

where

with the result

The comparison with (4-13.56, 58j indicates that the following single spectraf form, vvhich is stated for the +oton scattering situation, is needed to supplement the double spectral form,

fdu,, * * *)1/24a%Gz dMrB 2ma 1 + [l - ( 4 m ~ I M 1 ~ ) ] ~ / P -- l

(2,p M'% M'% log 1 - [ l -- (4m%/~'%)I'" (k , -- kg)% + '

One question remains, however, The forward scattering c o d t i o n (4-13.149) leaves undetermined any stmcture involving Gx + +&Gg, Does the double spectral form specify the latter correctly, or is an additional single spetral form required? As one might anticipate from the earlier discussion concerning G%, the double spectral form does contain the necessary information. In order to v e ~ f y this, we return to the calculation in (k13.44) and consider, in the center of mass f r m e , an arrangement in which the particle momenta p = - p b a k e an angle a with the photon rnomenta k = - k f . We also subtract the results for the two arrangements where the polarization vectors e = @Vie in, the pfane of p and Er, and are perpendicular to it, respectively, This isolates the 6, dependence, since G2 assumes the same value for both polarization choices, whereas

The integral that now appears is

x [(- sin cx cos 8 + cos a sin 8 cos 4)" ((sin 8 sin t$)%], (G13,lCIO)

where

1 -- 1. 8 +-72=---

1

-- $pk - Pzk M2 1 -- (1 -- (4m4/M1)) (COS O! COS @ + sin o! sin 6 cos +)a '

which employs a spherical coordinate system that selects P as the z-axis while k lies in the xz-plane. It should be noted that the - 1 term of the first bracket makes no contribution since, lacking reference to "Ee p vector, the two golarization choices are equivalent, and cancel.

I t suffices to consider the limit cx -+ Q, after differentiating twice. The result of this operation on (4-23.160), in which the azimuth& integration has b e n performed, is

(4-13.162)

which employs the abbreviation

Completing the integration gives

where

The implied spectral form is

Comparison must now be made with the single speetral lom that: is extracted from the first term of (4-13.133) by imposing a scalar version af the forward scattering condition. As one can recognize from the coefficients of ee3n (.1-13,148), this statement is

it suffices to bring about the reduction given in Eq. (4-13.151). The spectral farm can be written in the notation of (4-13. I66), with

The outcome of the integration is just (613.165). We conclude that the modifica-

tian in particle-photon scattering is described completely by the douMe spe~tfal form (623,133) and the single spetral form (413.158).

Harold intejects a question.,

H, : Before you go on, please tell me this, Why has there been no mention of single particle exchange I After all, the skeletal Compton scattering process can be infexred from a causal arrangement in which a particle is sattered by an electromagnetic field and the resulting real particle detected by another such xattering process, We would be concerned with the dynamical modification of each scattering act.

S. : In ~ n e r a l , you are right; single particle exchange must also be considered, Spin O is an exception. If one considers the scattering of a real particle by an appropriate field and then extrapolates to the field of a photon, the charge fom factor F(&) is finally evaluated for k2 = 0, and there it remains unity. Single particle exchange continues to be described by the skeletal coupling, for spin 0.

Only the double spectral form involves the photon mass, There is a simple test of this dependence, since it must balance the addition& scattering that is accompanied by a. soft photon, the so-called double Cornpton scattering. Our eancem now is only with the coefficients af Iog f fp and not with additive constants, The relevant contrjibution of the double spectral tom, which comes from the neighborhood M2 - m2, is

according to the relation

This is superimposed on the amplitude of the skeletal scattering process [Eq. (3-1 2.98)] to give

Thf. implied fractional modification of the elastic scattering cross section, in its dependence on the photon m m ,

dws indeed combine ~ t h the soft photon emission probability s t a t d in (4-1 2, M), where-

ta produce the total dimination of the fictitious photon mass, The low e n e r e li&t of the scattefl"ng process is cchmacterized, in part, by the

conditlisn

and therefore resembles the bxvvard scatte~ng discussion that produced (4-1 3.141), except that attention must be paid to the infrared phenomena near MS .= m2. The latter consideration would repeat the photon mass discussion just FS;iven, subject to (6113.1 741, except that we are now interested in the constant added to lag(1lp). Let the first inteeail of (4-13.141) be terminated at the lower limit W == mZ + X, where X obeys the restrictions of (4-f3.ft54). Then, added to it is

where 1?" is defined by f613.1156). The double integral here i s evaluated ais

The soft photon emission that removes thr: p dependence is represented by (4-1 2-26] where, for the nonrelativistic situation that occurs in the rest frame of the initial padicle, &M can be identified as k:ia, the minimum detectable frequency of an additional soft gXloton, The &ependence on these photon parameters i s cont&ecX in the inteval

Thus, the effective value of - the inirared sensitive tern is

To find the low energy Emit of the cantribation stated in (4-15. X41), we agply an &teed, which i s related to ("&13,36),

216 Electrodynamics I Chap. 4

As stated, it refers to x > 0. With x C Q, the corresponding proper value integral is obtained on writing in the arpment of the first logarithm. When one extracts the part of (4-13.179) that i s odd in x, the formula (4-13.36) is recovered, The leading terms of an expansion far small x are displayed in

The type of integral that appears in the Gl term of (4-23.141) is

The first of these integrals on the right-hand side is obtained from (4-13.180) by considering x < X < m2:

(bX3.182) namely

This i s the term that combines with (4-13.178) to eliminate the arbitrary parameter X.

In the low energy Xirnit, the East of the three terms of (4-1 3.141) supplies the integral

the exhtence of which is confirmed by rewriting (613.139) as

On introducing the variable

we get

The last of the needed integrals is supplied by the single spctral form (4-13.158). It is

The various contributions are put together to gjive the low energy farm of the transition matrix :

omitting the imaginary term since it has negligible effect in the computation of the differential cross section, which is our present concern. Considered in the rest frame of the initial particle, and in the gauge where polarization vectors are orthoganal to this momentum (pz& = 0), our result reads

(1 / T / 2 ) = - (do,, e)'/28nael* * e, + (dw,, * 0)"'

Here, n, are the unit photon propagation vectors,

and h@ represents the photon energy, which is essentially unaltered in this low energy collision,

218 Elatrodynamtcs I Chap, 4

The mo&fied differential: cross section for polarized photons is obtained irnme- diately from (4-13,190), and g-eneralizes the result of Eq. (4-13.40)- We shall only state the modification in the afferential cross section for unpolarized photons, The required summations over final plarizations and averages over initial plarizations are given by [cf. Eq. (3-14.101)]

and

+ ~ e 2 * g e 1 n z ~ e l * n l e e 2 = - ~ ~ o s 8 ( 1 - c o s ~ 8 ) . (4-13.193)

The conclusion is that

In the particular situation of back scattering this becomes

while the mdificatian in, the total cross section is

m 1 1 log - $3 - 3 (4-13.196)

which can be an increase or a decreae, depending upon the quantitative relation between ko[kt i , and (nz/2k@)g. We shall not trouble to give details about the high energy behavior, excepl to note that the k t i o n a l modification in the differential cross section is of order a, multiplied by logarithms that vary in form with the pmticular anwlar region [an example is Eq. (4-13.4131.

Historical note : The energy- and angle-dependent factor of Eq. (P-13,194), in an equivalent: form, was stated in 1948 by E, Corinaldesi and R. Jost, Ncrlv. Phys. Ada 21, 183. These authors used the unitary transformation method that also gave the first results concerning the electron magnetic moment, energy displacements, and Coulomb scattering modifications.

Occasional use has been made of noncausal calculational techniques, most notably in the treatment of low frequency scattering of light by light and related questions.

The cajlculation of the probabiEty that an electron-positron pair be cre;lled by a strong homogeneous electric fidd w;ls particularly strikinf: since no finite number of singlle scattegng encounters can produce this act. We now waart to recapize the special ability of noncawal rmethd to hanae the problem of bound state enerw dispfacements, vvhich similarly involve an unlimited number of interactim. This is welcome since the c a m 1 meth* have not suggested any very elegant solution to the problem of finding a unified treatment of high and law energy phenomena. f ndeed, much t ine and energy were expnded on one such calculation of the 2% r r r~f ica t ion to the energy displacement in a Coulomb field before the unnecessary complexity of the procedure vvas adnzitted and the attempt discarded. I t is w & h remarking here that the freedom to chaow between, or to combine, causal and noncausaf calctxlational methds emphasizes the uniiieation that source theory has brought about between the causally oriented analytic S - m a t ~ x theory (by removing the hypothesis of analytieity) and the noncaus~ll operator field theory (by removing the aprator fields),

To indicate the kind of approach now to be studied, let us return to the spin 0 vacuum amplitude of Eq, (4-12.U). There the particle propwation function retains its e n e r d space-time fom, but that of the photon has been specialized to its causal version. f"Ve shdl remove this rest~ction through the replacement

(-- ze is omitted) and no Xonger insist that and 4% be in causat relationship. They are, however, still required not to overlap. Thus, the vacuum amplitude now reads, in a symbolic notation,

PI useful rearrangement is stated by

where the dot indicates symmetrized multiplication,

'4.B = h(A, B).

We have, furthermore,

220 Eleetrodynmnial Chap. 4

in which the first term on the right can be omitted since the corresponding local structure in (4-14.3) vanishes under the nonoverlapping circumstances being considered. Accordingly, our actual starting point is the vacuum amplitude expression

We shall use the propagation function representations [cf. Eq. (4-8.34)]

and the implied product representation

In the latter, the parameter transformation

S , = su, s2 = s(1 - U ) , dsl ds2 = S d~ d~ (4-1 4.9)

produces the form

where

X(%) = u [ ( n - k)% + m21 + ( 1 - u)k2

= (k - ~ f f ) ~ + u(l - u)P+ m%. (4-14.1 1)

Note that if we want to consider a "photon" of mass p, the term $(l - U ) should be appended to ~ ( u ) .

Let us illustrate the use of the product representation in the simple situation without an electromagnetic field where, according to the commutation relation [Eq. (4-8.44)]

[ f f , II] = ieqF, (4-14.12)

the components of L' are commutative. Then, the last term of (4-14.6) vanishes. Also, the permissible redefinition of the integration variable in the first term.

k - 11 -+ k , exhibits that cantribution as a local one, which can be omitted. The analogous transformation in X(%) , k ---. wP7 -+ k , yields [a photon mass t e rn i s included],

according to the momentum integral (4-8.57). I t is convenient to carry out a partial integration on "U:

The first term on the right contributes, a t u = l, the value exp(-- ismy which, being locaf, vanishes in Eq. (4-X4.6). The s integration is now performed, and (kI4.6) becomes

The continued reference to nonoverlapping conditions enables one to reduce 11" in the numerator according to the substituGion

This yields [for simplicity, p2 is set equal to zero in the numerator]

I 4%. (614.17) n2 + fm2/1 - "U) 4- ( ~ ~ 1 2 6 )

I t is a t this stage that the full. space-time extrapolation is performed by adding contact terms, in the known manner,

When p is set equal to zero, and thus

ECaietrsdynamfcr 1 Chap. 4

the spectral weight factor that appears in (4-1 4.17) is

in complete agreement with (4-6.40). For finite photon mass, the spectral representation that is provided by (4-f4.1vp in which

uses the parametrization already encountered in Eq, (4-1.32). In this connection, it should be noted that

We are going to make extensive use of expansions of exp[-- i s ~ ( u ) ] that are patterned after the quantum mechanical perturbation expansion

[for a quantum action primiple derivation, see Quafzt.tlm Kinemtics and Dynamks, W. A. Benjamin, Inc., Menlo Park, 1670, Section 7-6, although the time dependence is there feft implicit]. Xt is eonlrenient to use fractional elapsed times as the integration variables, and to give them appropriately syrnmetfical toms. Thus, for the situation of interest, with

X =" XO -k XIS we write

If is an infinitesimrtl quantity, the expansion terminates with the linear X, term. That is the situation when X(%) is subjected to an infinitesimal vaiation, as illustrated by

In this examplie we have

[fl, X(%)] = %[IT, (IT - k)" = 2zli~qF",(17 - k) . (614.27)

Let us also discuss here the evaluation of the dioubIe commutator [IT, m, exp(- & X ) ] ] , in which the scalar product of the il! vectors is understood, Consider, for that purpse, the transfomation

where A is an arbitrary constant vector, We shall compare the expansions of both sides, based upan the general commutator expansion

1 edBe-A = B + [ A , B] + [ A , [A, B]] + *, (614.29)

which e m be ve~fied through successive differentiations of a scale parameter in A . Thus, up to terns quadratic in A, we have

@V(-- isg) 3- [m, exp(- kg)] + *[A@, expf - is^)]]

= =p(- --. &[m, %] - isifm, [;l.ZTI %]]) . (4-1 4.30)

The expansion of Eq. (614.26) presents the ~ght-hand side as

Etccrrodynamiest Chap, 4

Comparison with the left side then restates (4--14.26) and gives the desired expression :

X exp(- ist (1 -f- v j / 2 1 ~ ~ ) W, XI exp[-- - ~1x1 In x expi--- isf(1 - v ) / 2 J w ~ ) , (614.32)

where

[H, X(%)]] = - 221eq(fl- k). J - 2.ue8Ffi"Z"",, (4-1 4.33)

and we have introduced.

Note that the symrnetrlization in the associated term is unnecessary, since

The technical problem before us is to camy out the k integration when the substitution K - -+ k cannot be made owing to the nsncommutativity of the components of 17 in the presence of an eleelronna~etic field. To that; end we propose a device suggested by the following quantum mechanical considerations, In a system of n q- and p-variables, the expectation value (4" evaluation

which is independent of 9'. Accordingly, let 6 be the (four-vector) caordinate complementary to k, and write

The advanlt~e offered by this reformulation is the possibility of introducing canonical transformations that do not affect the expectation value, but alter its f o m in a useful manner. Thus, we shall bring about the nearest correct version of the invalid substitution k - %IT 4 k through an oprator transfomatisn. Indeed, when the field vanishes and the eompanents of .l2" coxnnnuk, we have

exp(- zwfDJ / (h - uf1) exp( ig6f l ) = f ( K ) (614.38)

and

(P = OV(k - ~n>lP = 0) = (g = Olf(k)lP = 0). (4-14.39)

The simplest procedure is to use the same transformation in the presence of the field. We therefore seek to evaluate the transformed quantities

One can exploit the presence of the variable U to produce differential equations:

where

E = F(&), 3 = exp(- iu(l7) x exp(iutl7). (4-14.42)

The chain of transformations stops with &, since

in virtue of the commutativity of the E components among themselves. Hence,

& = x - u E (4-14.44)

and

d -fi = - qF(x - U€)€, du

(4-14.46)

which is integrated to give

Integration of the differential equation for & then produces

k = k + ul7 - eq du" F(x - U"€)€

We also note the combinations

Elsctrocrfpamics I

and

The transfomed version of ~ ( w ) is

As a first application, we shall extract just the terns that are explicitly linear in the electrornametic field, When specialized to particle fields that obey (nB + mB)lrf) = 0, the r e d t should imply the very well-known fom factor for real particles. In this situation we must include a photon mass. Let us begin with the last te rn of (4-14.6) and use the analysis of [TI, [If, exp(- z ' s ~ ) ] ] given in Eq. (k14.32). To avoid higher powers of F than the first, we retain only the first tern on the right side of (614.33) and simplify (4-14.32) to

When the transformation signaled by is performed, we have, to the desired accurac y

X(%) -+ k X -j- ~ ( f -- @DZ + 112211, 11 - k -+ ( X - gl)JiT --- k, J(x) -+ J(x - (k14.53)

Now consider a typical harmonic component of J l x f , exp(i$x), and examine the et k operator structure

expf - &[(l + v) /2 ]k2) e ~ ( - - v)/2lP6) exp(-- i@[(l f ~)/qPt) x exp(- isi(I - v)/2lkZ3

This rearrangernenwes only the momentum translation proprty of the 6 exponential factors, When the F'= 0 diagonal matrix element is extracted, these factors, in their final psitions, are replaced by unity, leaving just the factor in brackets. There also occurs in (4-14.52), with the substitutions of (4-14.631, a linear k term which, as written, would become k - %[(l - v)/21@ afker the transla- tions indicated in (4-14.55) are performed. But pJ = 0, ancl the residual odd function of k vanishes on integration. Accordingly, to terms linear in F ,

exp(- isue[(l -- VP)[4]pP) ~qnj exp(-- is[(l -- v)/21#)* (4-14.56)

where

and the Gaussian function of' $P is defined through its multiplicative action on the Fourier components of J ( x j . When the specialization of particle fields indicated by 17% -+ - mVis introduced, we get for the last, tern of (4-14.8),

Etmrdyrramfcs L Chap. 4

Note that, with p = 0, an infrared sinpliaity appars in the limit u -+ 0, Next, consider the middle term of (414.6) for which we need the expansion

which is limited to the first power of F, The corabination

exhibits the angular momentum stmcture EUKv - %,k,, This commutes with any function of KZ, md annuls the ratationally invariant states ( E ' = q, That restricts attention to the last te rn of (4-14.59), where we encounter

expf - &[(l + v)/2)k2) expf--- izc'pt) f: exp(- i s[( l - v) /qkZ)

1 - v2 --P t2szct - p expf - iskzf exp(- isd2[jl - ~a)/4]fl~)~ (&l4.61>

4

After the real particle specialization flr;12 --+ - mz), and with only the term hnear in l; exhibited, we have

(614,622

The latter double integral can be performed in either order: for example,

4-2 4 Noncausal methods 229

Concerning the first term of (4-I4.8), we observe that its exponential form contains

and correspondingly the form of the f transformation with .u == X is to be used, As one recognizes from the factor 1 - u in the last term of (4-14.591, no finear term in F results from this contribution.

In this way we get, as the operator standing between ifbr and +2, the following :

in which, we have introduced the permissible simplification (p <g m)

The a62 term in uff ---- N) gives a well-delined integral as p --+ 0; here, the photon mass can be set equal t a zero. The u term supplies these integrals:

With the aid of the identity (k12.421, and of the relation

we arrive directly a t this result for (4-l4,@),

When set between particle fields 4, and multiplied by 1, i t constitutes an addition t o the action. Since J = p2A, in a Lorentz gauge, we recognize here, as the coefficient of 2eqnA, the anticipated expression for F($) -- 1.

E l ~ r d y m ~ i n i u r 1 Chap. 4

For a relatively simple example of effects that are quadratic in F, consider a field far from its source so that J .=I O. This describs the situation of photon scattefing where we shd restrict attention to foward scattering. That enables us to discard the field combination FMvFfi, [it prdacm the poldzation vector combination 6% of Eq. (Q-13.130)]. In this situation both terns of [II; [.L!, g]] disappear and

[D, [n, exp(- .is%)]] 8sh2ea dv rrzt expf -- &[(I. $- v)/2]wg) F~~(l"li! - h),

Since the required powers of F are already in evidence, the calcdation is padieularly easy ; the 6 transfomation is simgy

The restriction ts forward scattering means that no net momentam is supplied by the two fields. Consider, then, the combination

-. expf - is[(l + v ) m w ( k + M$)" eexp[-- &(l - w)Kg]

which, with the intemation vlzdable transformation

In the application to photon scattering, where p2 - 0, the last faetor reduces to unity. The transfomation (4-114.73) has no effect on the linear k factors of (414,70), since = 0. We also encounter the momentum i n t ~ a l

but, since this produces the, field combination FMVF,,, it does not contribute to our limited objective, In this way, we arrive at

x exp[-- &(I - w)u(Z ---. u)fna + m33 Fy,nR. (614.76)

I t is convenient to perfom a partial integration on ~ r l l according to

where

X = %(X - %f(lZ2 + m2). (4-14.78)

Then, when the s intfsgration i s perfomed, we get

Now consider the expansion of exp(- is i) , where only the t e rn containing F f l quadratically is of interest to us. When simplified by the restriction (H2 -+ m2)+ = 0, It is

x exp(- is(I - W ) [kg + %(l -- %)(H' + m2)]) ell expi-- &[(X - v)/2]wk2).

(4-1 4.80) where

The k, E operator structure apparing here is reduced by liberal use of the prop- e&ies J == 0, p2 = 0;

expf - + v)/2]wk2) EY ~?xp("ta6@$E) expl- is(X - w)kZJ P,

+- s2(1. - v2)w2k,kA expf - ;sky), (4-14.82)

where the last step employs the integration variable transformation

The disappearance of the parameters up, u" makes F effectively equal to F. Accordingly, with only the qrradratiie F item stated,

A partial inteeation on w is again advisable:

The outcome of the S integration is given by

Concerning the first term of f&14,6), we note that placing zl = I in {&f4,84) removes the contrr"bution of interest .

The result of combining (4-14.79) and (4-14.86) is the following expression for the owrator that stands between z'+, and 4j2:

where

On eliminating the variable zl, and writing

the function called XfMy becomes

This will be recognized as the causally derived function of Eq. (4-13-21), which is as it should be.

We now face the major problem before us: how to handle neatly the repeated interactions that characterize the low momentum excitations of a bound system. An indication of the proper course comes by comparing the evaluation of the double commutator given in (4-14.56) far weak fields with the farm obtained by refraining from exercising the second commutator. For the latter we t-eturn to (4-24.26) and (4-t4.2?),

and, retaining only terms linear in the field, perform the transformations indicated in (4-14.53). This gives

(k14.92)

where

+(H) == %(L - %)(n2 + m%) + m8212. (4-14.93)

The reduction of Eq. (4-34.55) is now employed, tagether ~ t f a

which prduces

At this point, we write out the t w terms of the second commutator with f7! and, where If2 acts directly on the particle fields, replace it with - m2. That has the f o l o ~ n g consequence,

a sa -- i-- 1 -- dv exp(-- is@[( l -- v"j141p2)

2n s 2

x { n ~ exp{- &[(l + v)/2]u(l -- u) (17% + mP)) [(l -- u)eqF,..ny + f i@vcq],l

- [(I -- ~ ) L ~ F , ~ . I I * + iilcvegj,] exp( -- is[( l -- v)/2]u(l -- N) (17% + me)) nu)B (4-1 4.96)

and then

If IT% + m2 i s set equal to zero in these denominators, we regain (4-l4.58). But retaining it provides a natural "cut-off" for the infrared sinelarity at u- = 0, which has athemist: h e n produced artificially by invoking a, photon mass.

Let us abo rmord a version of (4-14.97) that is applicable to stourIy varying fields (Zak 1). To t bat end we omit P2 in the denominators, and drop the J term, Fadhemore, we decornpse the u integral at a value of u ;= uo such that

where the - ie that has been left implicit in the denominator tells us that

This simplification of (4-14.97) is expressed by

which uses the integral

According to the commutator

[r"b,, I?IB f- m7 = ZiqF,,,ffY, (414,103)

the result can also be prexnted as

Now we must praduce a similar improvement to the ealeulations of Eqs. (4-14.62,63). This will first be directed to-cvard the situation af slowty varying fields. Then the important field-dependent term in 2 (B) is the one containing LT, together with its quadrati~ paftner, the last t e r n of Eq, ($-14.51). The others either vanish or lead to quadratic field terms without infrared singu'tarities. These s@nificarmt cantrihtions 4 0 i ( u ) are:

Etsetrdynamics f Chap. 4

where only the first two tenns in a 6 power series expansion of Fjx - ~'6) have been retained, With the aid of the reduction,

- expi- + v) /2]kz ) (-- ~ ( 1 + v)k,)(s(l - v)k,) e x p i - &[(l -- v ) /2]k7 ,

(4-14.1106)

and the integral (&I4.75), one gets the following :inear and quadratic field terms,

The commutator of Eq. (4-14.103) can be used to give alternalive forms to the last factor:

We shall average the two forms, after partial integration, Then the commutator that appears a t W = X,

gives a contribution that cancels the first term on the right side of (4-14.107). What remains is [recall the definition of ,H in Eq. (4-14.78jj

Ot - - ~ 4 ~ ( 1 - g) expf - i ~ r n ~ 2 ~ ~ ) cl.w w2[17" exp(- &(P - W ) # ) eql";,y.Bv Pn

-- e@,,.IT%exp(-- is(L .-. W)#) j7@],

where we shall in traduce another partial integration :

The s integration then gives, for the field-dependent terms,

The la integral that appears here is the same as (4-14-99), with 1 - W replacing i ( 1 -+ v). But the remaining parametric intetgrals are different,

(4-14.f 13)

This si~npliiied version of (4-14. t 12) can be written as

or I m2 X. 2" --- - f l u + m%) log -- - - 3% m2 n2 + fj

Again we note that, since (4-14.1102, for example, vanishes a t -- l , there is no such contribution from the first term of (4-14.6). When (4-14.1M) and (4-14.114) are added, and the factor af i removed, we get the following operator to stand between 4, and +z,

2or L - -- m2 I log 4- 12 (4-14.115)

3% m2

The implied addition to the action is produced by forming a scalar product with the particle field (B and multiplying by 4,

The energy displacement predicted by this addition to the action can be inferred by standard perturbation theory frarn the modified -field equation, or, by considering a causal anrangement, Emission and detection sources separated by the time interval T , exchange a particle with the associated field

The vacuum amplitude convey& by (414.1 15;) then bcomes

iK:[- iTbE]iK, (414.111 7)

where, uskg thre-&mensional d a r produet notation,

is the enerw displacement of the state. We haw only to show that, vvith the neglect; of terns of relative order Za md higher, this is the result found in %ction 4, X 1 by combining nonrelativistic and relatiktk alcuXlirtions, Ta that accuracy, we have

where

log .K == log *m -f- h. The ornimian of the fkst tern on the hght side of (614.119) combines the nq1ect of the piece that is quadratic in V with the recognition that 4 is an eigenvector of H d t h the eilgenvatlue E, Now, if we wnite

(&l.rl;;l%f)

and note that the fador 2na converts t# into #, the nonrelatiktic wave betion,

W have just the stnxcture of Eq. (P.Xl.59) [taking into accoant the candlation of the contact terns exhibited there], with the uppr limit cif in te~at ian fked according to (414.12Qj at just the value @yen in Eq. (4-1 l,&$), Here, indad, is the unified de~vation we have b e n seeking, The related one for spin & h @yen later. Incidentally, the effect of vwuum polarization, which has not been considered explidtlfy, is to alter the adcfitive constant h. AGindieated in Eq, (4-ILli .l 35), this is #ven by

Qur objective here is to find the modification of these enerw &splacement calculations that is of relative order Zcc. Perhaps one should first show that there are such relativistic eorreetions af order Za, rather than the getzerdy smder effects suggested by the ehaxacteristk fine-stnxeture fador &)g. The shplmt place to look is the vacuum plarization calculation of Section 4-43, where the approx;irnatian of replacing was intrduced, As an improvement, one includes the variation < a@ == ( m Z ~ ) - l . For OMP restdcted accuracy, this is adequately described by the ScbrWnger equation,

The kbavior near the o ~ @ n is insensitive to the enerw of the S-state, l@ the mlution

Accor&ngly8 the energy 8splacement; fomula (&$.B) is Atered into

where the ad&tianail tern is indeed of relative oder Z'a. The new integd a p p here is, for spin 0,

The effect is conveyed, in the ad&tive constants of (&14,122), by the substitution

As a first step W shaU present another treatment of the intqral

which will be advantageous for our present pttrpox. It is based. on the related in tegrd

where

We have wfitten I(P) since the integral is clearly an even function of R. The first point is this, The quantity

is the result that is obtained for 1 = I ( l ) in the absence of an electromagnetic field, where the trlmsfomaticzn k - -+ K can be used, Accordingly, the rearrangement

xgarates, in the last term, the expllicitly field-depndent part. Second, the additional rearrangement

isoIaLes, in the A = Q component, just the infrared sensitive part that is exhibited in (4-14,110), thus p m i t ting a simpler handling of the residual tern.

Let us been with the derivative expression

which, we proceed to rewfite by using the fact that

Differentiation of the individual exponentials gives

In the latter version the three parameters of unit sum have h e n given the more symmetrical, form used in (614.28). The transformation of the first te rn in (4-15.14) induces

and the second one is andogous, giving an identical contribution. This identity converts (4-15.12) into

To verify the remark about the A - 0 value of this derivative, W have only to simplify it for use with fields obeying (piFa + =; 0:

whieh will be recognizd as (d-I4.f 10), after inserting the identity (4-14.111) and using the cammutator (4-14.103).

The situation is even simpler for the double commutator tern, where

(dk) - CrJ; [LI, expf ---. i s ~ ~ ( 2 ~ ) ) ] ] ~ ~ ~ ( 2 ~ ) "

produces just the bading approximation to (4-14.87), in which J and are cEjiscarded, Be@nning with

we get for (4-16.1 9) :

X qF.JT =p(-- G[(X - ~)/2]fl)h

which reduces [fn" -)-g)# to

This is indeed the ~ s u l t extracted from (4-14.87) in the manner mentioned. Our study of: the Zcx modifications starts with the errors intraduced through

the allproximations used in Eqs, (4-l4.99) antf. (4-14.1 19). Let us compare the (incompXete) energy shift expression [combining the approximate version of (4-1 4,971, and (4-1 4. f X2)j

with the fudher sirnplrified one of Eq, (4-14.118) :

To that end we evduate the intqral

where the differentiation dws not extend to the 7ng in 11% + m" The simplified version is obtain& by neglectiw 11% -4- m2 in the denominator, relative to #. Note, in this connection, that

The difference of the two,

can again be represented as

Th? appearance of turo factors of RZ + m V n the numerator of the resulting expression means that it is explicitly quadratic in the fields,

As a high energy effect, only the behavior of the field + near the spatial origin is sieifieant. Accdingly, we replace +(X) with

w&ch is equivaltelit to nwlecting the spatid momenl.um dependence of this function, Now, one can w ~ t e (416.29) ant as

2 EIecfrodyn~imilca I Chap. 4

in which has been approximated by m. The quantities occurring here are

while 11% +- %g, in the cfenoHlinator of (4-15.28), is approximated by p2, since n o m.

The energy shift tltus obtained is

But, it is essential to realize that the comparison of interest is not with the simplified expression (4-15,24), but with its nonrdativistic reduction (4-14, .l 1 9). The difference resides in the time component of the scalar product, where 2 m + V gives the additional contribution

N O \ ~ observe that

4-1 5 H-partfele energy displ~emenb, Spfn O rcelatfviotlc theery 24s

where it has been convenien"tto return, temporarily, to the explicit function of p2. The cantribution to (4-15.33) of the last t e r n in (4-16.35) precisely cancels (4-15.34), and the remainder gives the partial energy shift

(4blEi.36) The momentum integral is

(&P) 4 --....-p

1

(2nI3 $ + v m2u -+ - - (l --- %)p2

2

and the subsequent g integration gives

as a simple example of the r-function relations

;rt r"(a)P"(1 - a) == -.

sin za (4-15.39)

The result is

In what follows, the complete neglect of spatial momentum in the particle field, as given in (4-15.30), is not always permissible. One needs, for some purposes,

246 Ilwerolrtynarnics I Chap. 4

the short distance behavior stated in (4-X5.2). We now want to show that the Iatter can be regarded as the initial result of an iterative solution of the bomoge- neous field equation

one that be@ns with the field

With only the term linear in A retained, the first iteration eves

or, inse&ing the Coulomb field and the charge relationship necessary for a bound state,

To be quite precise, what is claimed involves a specific interpretation of the three- dimnsional Green's function symbolically represented by Xt is stated in

where the Cauchy principal value is employed. We v e ~ f y this directly [Lim P is understwd] ,

in which. the last integral. can be presented as

The evaluation uses the complex computation of a principal value integral, averwing the results supptied by the two cantours d r a m above and belaw the sinpfarity. Of the four contours that interlace the p i n t s E and - c, the one traced entirely above the real axis vanishes, while the analogous one drawn below the real axis duplicates the sum of the other two contributions, each of which effectively encircles axle singular point, We have shown that

w ~ c h dif fern f ram. the ef ementary inveme statement

through the special, interpretation of the Green's function, preventing the &pp=- a c e of an added constant. This confims that the construction, of Eqs. (616.43, 44,46) prduces

Next, we consider the enors introduced in replacing f (l) by I(O), or by I(@) + ,&. Since the= residads refer exclusively to high energy phenomena we

shall reso& to an etementwy method : expansion in powrs of the vector potential, Ac~30r&ngiy, we now M t e X,(#) [Eq* (4-1. 6.8)j as

(k15,fil) where

CA(%) = (k - A%$)2 + %(l - %)p2 + m%. (416.82)

After p d o m i n g the k inteption, the fea&ng tern of i ( A g ) ,

has h o m e independent of if. ancl, therefore d w not contribute to either of the: differeaces of interest. The tern Iinertr in A is

For a, tygicd matrix element, r e f e ~ n g to momenta p b n d $", one encounters the momentum integals

2 Efrsctrodynamics I Chap. 4

f dk) p" l - v k - it%- (f13.tI4 2;

( k - AN$')' -+ -F ( k - A u P " ) ~

f l v = -- g -- S 2% l (p' - p") exp (4n) zs

which evaluations use the transformation

f f a Lorentz gauge is adopted, so that

(p" Pp"")A == 0, (4-15.58)

the latter integral will not contribute in (4-.15.54), and one gets, for the matrix element :

where

Should we evaluate (6-15.59) in the null momentum state 9, the quantity D, tvauld become independent: of il and no contribution to the A difference appears* kire must use the once iterated field (4-15.63). Retaining only the cross terms between the two parts of 4 produces

where D, has beqorne

There are ttvo applications for this structure, or something similar. The first one refers to the middle term of (4-I4.6) where, according to (4-16.10) and (4-15.11), we must form A differences that amount ta rejecting the linear c22 dependence

of I ( l ) . Thus we encounter

with

(kxs.64)

This contribution to the energy shift is

The principal value interpretation of (p2 - E ~ ) - - ~ S not involved in the elementary p integration, which yields

1 f-zs! '$@ - [(L - u;m)"2 - (1 - %)3/2

0 %W2

- &(l - %)li2(l - W ) ] , (4-11 6.66)

where

After performing two partial integrations with respect to W , this simplifies to

The general integral of this type is (a, b > 0, c > - I)

Using it, we get

(616.70)

The second application of (4-15.fif), in a related versim, occurs in the double commutator term of (&14.6). When one ignores the vector potentid in II; the double commutator introduces the allditional factor of (p' --- p"fZ into a matrix

element. In the context of the expectation value computed from Q, where one spatial momentum is zero and the other f p, this supplies an extra factor of p2 in the integrand of (4-16.61). The residual effect in the double commutator term is just the difference between A = 1 and A = 0. That produces the combination

and the energy shift from this source is

l - U = - 1 6 z h a l p , 1 8 ~ l dw +n[ du [ ( l - uw)'" - ( l - u ) Y . (4-15.72)

A single partial integration produces the form

and

Concerning the first term of (4-14.6), we note that I(A2), vanishes for U = l ; no analogous contribution is forthcoming here.

We must finally examine the terms that are explicitly quadratic in A. They have the simplifying feature that Q = p, suffices. As a consequence, no contributions of this type are obtained from the double commutator expression [Lf, [n, I)]. Such terms are

where IAz states the part, in an expansion of I (P) , that is quadratic in A. Since we are now taking a diagonal matrix element for a state of definite momentum. the first two commutators vanish. Concerning the third commutator, the identity of Jacobi [Eq. (1-1.22)] tells us that

&P Id] = [p, I d ] - [W, @l, IAI- (4-15.76)

In the Lorentz gauge that is being systematically employed, the last term above also vanishes.

The explicit expression for I(R),= is

x [zlAu(k - A~p).eqd - 2 4 1 - zl)$,eqA]

x expf- is(l -- w)CA] [2Rzl(k - Agp),egA - 2 4 1 - u)p.eqA]

x exp(--- &[(l - v)j2]w<,)

X (As2 + %(X ---. %)feu2 expf - is[(l - v)/2]<,). (4-18.77)

We need only the diagonal matrix element in the state gi of rnomenturn p' (pq = m, p'== OQ Thus the momentum integral in the last terrn of (4-15.77) becomes

(dk) - expf - is(k --- A@$')g] .= ( 2 ~ ) ~

exp(-- &k2) = - -. (4-15.78) (4n)2 i s2 '

As a result, the last term is a linear function of A2 and does not contribute to the A difference of interest. The basic momentum integral in the first terrn sf (4-15.77) has the fom [it is (4-l5.85), with $(l -f- v) -+m]

There also occurs an integral with the additional factor of K - ;l@" which, as in (4-.15.56), results in a multiple of p". This gives a. vanishing contribution in the Lorentz gauge. And, there is an integral with two K - Asp' factors, The use of the Lorentz gauge again produces a simplification, effectively reducing the integral to (4-14.75). In consequence,

Rl-rodynrmlcs I Chap, 4

where W have elected. ta write

*.- 4(1 - so that L), h= the structure even in (G15.62). When the s integration is performed, we get

The 2 differences that are needed here are those of Eqs. (4-16.63) and (4-15.7 I). The expresfian thus obtained for the energy shift contribution is

or

with

(4-1 6.86)

and

where we have seen fit to return, by means of Eq. (&15,67), to a variable called W.

PL singb partial intqration converts &&E into

H-particle enarm disp#wemonts. Spin O relativistic dhsory 115.3

according to (4-Xfi.69). As for S""4, two partial integrations give

The sum of these two parts is

Remaining to be considered is the first term of (4-14,(i), which has not done anything for us, sa far, We set %I; = 1 in Eq. (4-15.81):

e2+*I(A2, u = lIA2$ = i ~ x Z ~ E ~ / ~ expf - z"sD,), (4-15.91)

where

is an even function of v ; accordingly, the term linear in u has been discarded and the v integral suitably rewritten. Let us carry out a partial integration with respect to v,

The first term on the right side produces, in (4-15.91), a linear function af Aqthat does not contribute to the jl difference we need. In effect, then,

and

W Iterdynmmlcr I Chap, 4

The fwction (A2)q with its first Awderivative, v a ~ s h e s act A = 0. Hence, the enerw displacement contribution dedved here is

The sum of thew five Ipi.eces [Eqs, (4-16.40, 70, 74, 90, 9711 is

When the vacuum polarization effect [Eqs, (&115.3,4)] is included, this rea&

So witten, tbe factor in brackets represents the Za modification to the ad&tive constants 1/12 - 1/40. Concerning the magnitude of this effect, we shali only remark here that the large numerical factor 322 roughly matches the dominant logarithm of the principal contribution so that the fractional measure of this additional upward displacement of s levels is, for Z == I, fairly well @ven by E =: 7.3 x 10-9, P~ecise numbers will be reserved for the andogous discussion of the more experimen tally accessible spin system.

4-16 H-PARTICLE ENERGY" RISPLAGEMENTSe SPIN 4 RELATIVISTIC THEORY 1

The spin 4 counterpart of the vacuum amplitude (614.2) is

We employ the analape of (4-1 4.102,

with

4-26 Wmparttclc. anew dlrptacements. Spin 4 rclatlvittic theory I =!!l

A useful reanangement is given by

P(% -- - 4) exp(-- y,

= yg(nz - (1 - f y(K --- ~ n ) ) exp(--. is^) y,

= ylblm - I1 - @)yfl) f i =p(- ~ S X ) + yBfm --. f f --- ~)yD)[exp(- is%), y,]

+ P y f a - ~ f i ) expf- i sx f ?U, (4-1 6.4)

where we shall vvfite

v(% - (1 -- u)ynf =; (m -4- (1 - @)yn)yU + 2(X - %)P, (416.5j

Then, syrnmetrization between left and right produces

We shall also discard all t e r n involving y R + m appearing on the left or right, which are not involved ixl appGeations to energy dispiacements. This results in

- ~ ( 1 7 -.-. 4) ew(- i sx ) y,

-+ - gm(1 4- U) exp(--. & X ) 3- (l - @)[I;I\ [expl- isg), y J ]

4- &m@l~C, [exp(-- is^), y,]] 4- yfiy(k - %n),exp(- isx) y,, ( M 6 . 7 )

where it is helpful to note that

In the ;Jbsnee of an electromagnetic field only the first term on the right side of (416.7) sumives the rformation of commutators and the prformance of the K intwation. Under these conditions,

Chap. 4

Then, we have

- j ds du esj g4 y"(m - y(h' - R)) u p ( - isx) y.

or, after a partial integration with respect to U that discards local terms (which vanish for nonoverlapping #l and a) :

(4-16.12)

The last version includes the contact terms necessary to satisfy the normalization condition; they serve to introduce the factor (yh' + m)a. In any application to particle fields obeying (y17 + m)# = 0, this structure vanishes. We are interested only in explicitly fielddependent terms.

As a first example, consider a weak, homogeneous electromagnetic field. In this situation there is no effect of the noncommutativity of l7 in the k-integration of exp(- isx), since it can only involve [a, 17,]FYV, which is quadratic in the field. I t is merely necessary to include the spin term in modifying (4-16.9),

where just the first two terms in the expansion of that exponential have been retained. The vector obtained through k-integration of (k - un)Y.exp(- isx) can only be a multiple of F&*& (these elementary observations are also confirmed by the results of explicit integration, of course). But

which does not contribute to energy displacements. Also, the terms in (4-16.7,8) involving commutators with h' disappear in the homogeneous field situation. When we note that

which is to be applied to the full spin term of

4-1 6 W-particle energy displacements, SpZn 4 relat4vistlc theory 1 257

we obtain the following two field-dependent contributions to the momentum integral of (4-16.7), as simplified by 8 --. 0:

cr nz a m -+- - - H" (f .t zl) exp( - i sn t22~2) & ~ G P +- - - exp f - ispngg2) q a F

2n s ?E S

The parametric integrals indicated in (4- 16.1 1) then give

and the additional action term obtained by space-time extrapolation,

states the familiar aj2z supplement to the magnetic moment. This derivation is surely as short as any.

The discussion af energy displacemeats in a Coulomb field will follow the pattern faid down for spin 0. Thus, we now consider

and the integral

fdk) --- expf - isx, (24) l. (f?zl4

The latter is such that

reproduces the n u l electromagnetic field structure of Eq, {4f-16.9). When applied to the first term on the right side of (4-16.7), with the contact additions stated in (&ICi,12), no contribution to energy displacements emerges. The computation of the first 2% derivative differs from the spin 0 discussion only in the appearance of the spin term with coefficient A2. Thus, based on Eq. (4-15.17), we have

Chap. 4

Wen fieI& obeying ( y u + m)# = O directly multiply this slnrcture, it reduces to

and we get:

in which we have replac4 the parameter W :

Next, we t u n to the commutator of exp(-- isg) ~ t h y,

4-$4 H-prrQicE8 e n t m displacements, Spin + rstaftivlstlc thwry t

where

(Fy)& = FUVyv.

A related quantity is produced by inserting X , in place of g on the right side of (616.28). The evaluation at A = Q 8ves

and

From thk we defive the reduced [S# = Q3 fom of the double commutator expressian involving 17:

Let us note here that

which Xets us replace (klff.flS2) with the equivalent fom

The denofinator terns appeadng in (kX6.26) and (616.34) &ifer though the wcsmenee, in the numerator, of the second. order and fkst order foms of the Dkac &fferen.lial oprator, rmpetively, The connection btween them, w&ch foUows from

is given by

For the first two terms ad (4-16.7) we have now indricated stmetures that, according to the spin O experience, should correctly display the lour energy characteristics of the bound system. The remaining terms are not sensitive to low energy phenomena. We show this by an evaluation for weak, slowly v v i n g fields, That has already been done for the third term of (4-16,7), containing the double commutator with y, and we restate the result:

Notice, by the way, that the two contributions to the magnetic moment are a g ~ n in. evidence [Egs. (6116.28) and (&l6,37)], We now consider the last term of (416,7), and its decomposition as given by (k16.8). Beginning with the second part of (&16.8), we use (&li6,31), with its reduced weak field consequence

(&16,39)

In order to ded with (k - wirPj.exp(- k q ) , we start with the identity

Partial inteeation with respect to v then yields

4-14 H-partfete cnerm dlrplmmcnls, Spin 4 rclrtivOsfic theory I %l

= (k - sn) .exp(- is^) ---. is% g dv Q V expf - isx[(l v)J2]) [ X , 171 -1

which means that, in the context of the k intepatioian, one can make the substitution

Since the commutator IQ is already linear in the field, the weak field simplification pernits the evaluation

where

r-fx, rJ1 = C X ) - Ix, y1.n

The next-to-last step exploits the fact that 2, being constructed from y ( n -- k), commutes with the latter operator. We note here that

and this te rn i s therefore droppd. The k integral of what remains can be evaluated with the aid of our E transformation device. It involves the integral

which, for slowly varying fields, produces the evduation

and then

The vafiaus pieces are put tagether into an operator standing between - z'$,yO and #r2, which, as a supplement to y17 + m may be eatled the mass operator modification. This operator i s

It is convenient, however, to rearrange part of the second term, as indicated by

That repfaces (k18.49) with

W-pafifele anrpm dirplmmcn@. Spin .h rertatfvtstlc thwry C 11M

The last two terms are explicitly quadratic in the field. If they are discarded, and the B integral of the second tern approximated in the manner of (d-14,89j, we get

whem the change of the added conslant from its spin 0 vdue of & comes entirely from the third term of (&16.51),

To examine the nanrelathistic: Emit of (4-16.52) it may be simpler to reintroduce the commutator (4-1 6.331, so that

The last steps involve the nonrelativistic reduetion to a state for which yO" = + l. Note that, in contmt with the spin 0 situatisn, the tern expXicitly quadratic in the potential, V disappears in the nonrelativistic limit to the extent that it will not c o n t ~ b t e a Zce conection, since y only connects states vvith opposite vafues of yO, With the nonrelativistic version stated in Eqs. (4-163,52) and (&18.54), we have arrived, in a unified way, at the dominant parl of the energy displacement where the effective upper limit of pkoton energy is given, as in Eq, (&ll.52), by

Elerdynamtcet I Chap, 4

Again the explicit effect of vacuum polarization is to alter the additive constant,

Now we must produce the Zcr modification to this knom result. Let us first assss the errors introduced in the reduction from (4-18.511) to (4-lfi.52). The path of the spin O discussion is -folXowd in =signing the follovving correction to the second t e rn of (4-16.51) :

where it suffices to use the field at the orie;in, $101, with the property v '= 1. Accordingly,

- ynlF? + Fko?O;fn( f 3- yO) F k o ~ @ , (4-1 6.B)

since neither y e p nor the time component of tbe product contribute:

Pkytm(f + yO)Fngyz --* 0. (4-16.59)

This gives

The last term of (k16.51) also involves the combination (4-X6,68) and gves the energy displacement

The next-to-lat term of (&lB.Bl), on the other hand, contains the product (omitting an inert factor)

where, since

- ?&.YE 6kr l

the second tern effectively doubles the first one. This yields

The sum of these three effects is

The next tit3t of corrections refer to the; enrors made in such replacements as wtting I(@) + dI/d;Zg in place of I(1). As in the spin 0 discussion, we canry out an expansion in powers of the vector potential, We; therefore write xr(u) [Eq. (&16.20)] as

Chap, 4

where [,(g) retains the meaning given in (4-15.52). The linear term in the expansion of I'(Jg) is [cf. Eq. f4-15.M)]

Since this differs in a transparent way from the spin O structure, we can proceed immediately t s the andowe of the matrix element (4-15.59), except that the application ta the first te rn of (616.7) requires the additional factor of X + zc:

Were we to evaluate this expression in the state $(Q), the outcome would be a linear function of A2, which would be annulled by the iZ difference of interest here, As with spin Q, we must consider an iterated field.

Using the xcond order form of the Dirac equation,

we find, as the analopes of (&L5,43),

where W recall that $(Q) represents a state of null spatial momentum and definite parity, yOf = + 1. Xn analogy with (&15.61), this gives

since

@'FoloaOEFa~ - F o P o a (&316,73)

in the state $(O). Introducing the required A difference produces the energy displacement contdbution

b4E = (F, + 6"" + +'")E,

where

and

(616.16) the relation

has been used in this separalion, The part c&& gcE differs from the andogous spin 0 stmcture [Eq. (P16.65)] only in the additional factor of 1 f u. In particular. the multiplicative factorrj arc: i&n(ical, since

Accor&ngly, the mo&fication of Eq. (4-1 5.681 @ves

The momentum i n t e ~ a l of &"E has &W been met bfare, in Eq, (616.72). Suitably chan@w the numeficd factm and %-dependence of (616.73) sappIic;s

Performing the momentum integmtion af the third, part produces

Adding the three pieces, we get

Let us complete the discussion of the first term in (616.7) by cansidering the part of P(R2) that is explicitly quadratic in the potential. For that, we return to the spin 0 result of Eq, (4-15.80) and intrduce the additional spin term:

x expi- z'srZ%22(?r(l - W ) ( p t --. p")zJ. (4-1 6.83)

Then, as the analogue of (4-15.81), we get

(&l@.@]

As in (4-15;,83), but with the additional fatctor of X. + z;~, perfaming the s-integation now produces

H-pot.tlcle cnarm dloplac~mants, Spin fr ralatlvfstic theory C 249

The introduction of the A difference ernplayed for this term has the foUowing effect on the various contributions in (k16.85) :

as in (&15,84), and

1. - v2 (3 , - U)%%% -

(I: - zc + 22%)2 1 1 -H% 4 P " + ~ - ' " . ~ -h- (4-16.87)

B, ffl Do Do2

We shall divide the associated energy shift d,E into two parts that are simply reEated to the spin O edculation,

and a third part that is of direct spin origin:

We adapt (61i5.88) to give

Chap, 4

while (4-15.89) supplies

In these integrals we meet the special situation of Eq, (&15.69), with a = Q:

Here,

d #(z) = ;EE 1%

will be detemined, for our purpasm, by the two properties

I. $(z) #(% --*l) -4- M p (4-1 6.94)

following from

F(x) = (Z - l ) Q z - l), (616.95)

and

#(l> - #(*) = 2 log 2, (4-16.96)

The latter is a particular example (n .= 2, z .=: 4) of the formula

that: is inzplied by the r"-fuplctian multiplication property

ThuS,

and

416 H-partide energy displacaments. Spin ) rdmtirirtic theory I 27l

Turning to the spin term V'$, we perform the momentum integrations to

get

where a partial integration with respect to W has already been effected in the first of the three integrals. The result of these integrations is

and the sum of the three parts appears as

The total effect associated with the first term of (4-16.7) is displayed in

For the second term of (4-16.7). we begin with

and remark, as in the spin 0 discussion, that terms linear in F suffice for the evaluation of the additional commutator with p in IT = p - eqA. As for the additional commutator with A, the sole component, A", selects FokF which has a vanishing matrix element in the yO' = + 1 state $(O). Effectively, then,

and, stated as a matrix element, we have

2 Electrodynamia I Chap. 4

The energy shil"t of interest is produced by a single A difference:

( 4 1 S, 109)

Comparison with the analogous Eq. (4-15.E) shows that, with the usual equivalence,

(4-1 @. 1 10)

these are identical, and therefore

Now far the third term of (4-16.7). Both here and in the fourth term we employ a direct power series, in A, expansion, from which the rdated first approximations, Eq, (616.37) and Eqs. (4-1 fi,39,48), respectively, are to be removed. The linear term of (&16.68), with A = l , is adapted to produce

(4-16.112)

The commutator with y selects only the spin term, and [Eq. (616.18)J

[@FI 41 == &F* ( 4 - 4 ~ . r 13)

Then the associated ene;t.gy displacement is

4-5 6 Wwpat.ticle energy dlsplacsmcsnb, Spirt 4 rsfrtlvtotic thmry I 273

which makes explicit the subtraction of Eq. (616.37) (multiplied by i to ext;ract the energy shift). The evaluation of the matrix element @ves

and

Concerning the term quadratic in A , we note that the y commutator restricts attention in I A Z to terms of the form cxF and uf; - * CF. For the f i s t possibility, ILS modified by (4-16,113), the use of the state $(a), with yO" = + 1, excfudes a finite malrix elexnent for oOk = iyoptk* I t is the rotational invariance of $W), on the other hand, that rer2u;ces dOkFok * * @&FOl to --- .FOk * FOkr which, as a multiple of the unit m a t ~ x , commutes with y. Thus, there is no contribution from l A z t and the full implication of the third term in (&16,7) is conveyed by 07E.

The structure of the double commutator with. fll and y that occurs in (616.8) is already contained in the equations beginning with f&X@,lOB). The appropdaite modification of Eq. (4-li6,108), with A = 1, reads

and the implied energy shift [removing that descllibed in Eq. (&16.39)] is

Since this is just 3a7Et we have

Turning into the hame stretch, we use the substitution f&lBe42) to prduce the finear term

7 EItrcltrodynamicrs I Chap, 4

Something else occurs here that has been omitted. I t is a multiple of

which vanishes in the absence of magnetic charge, The namenturn integral (616.56) is utilized in deriving

(4-t6.122) and then

m4 ybY. ( ( k - uH) .exp(- i s ~ ) ) ~ y ~ ( 2 ~ )

The implied energy displacement, with the necessary subtraction, is

iSBE r=: ShsE + dM9-E, (4-115.124)

where

The f i rs t of these parts is evaluated as (I + v = 2w)

4-16 H-putlck energy dilplwements Spin f dativhtk theory I 275

and, since we shall see that 8"& - 0, this gives

The argument is completed by observing that

for

has a vanishing matrix element in the state $(O). Last of all is the explicitly quadratic term related to (4-16.123). Let us note

first that the vector potentials which supplement the momenta in (4-16.120) do not enter into this calculation since

vanishes in the +(0) state. I t is only the linear terms in the expansion of the exponentials, exp[- i&(1 f v ) ] , that contribute. The factors that appear in this way are

- 2ueqF.(k - up) + 2u(l - u)eqF.p + uqauF (4-16.132)

and

%(R - up).qA - 2 ~ ( 1 - u)p.qA - treqt~F. (4-16.133)

We remark first that the spin terms appearing here can be discarded. It has already been seen that

which is an identity. When the two spin terms are multiplied together we encounter, for example,

where we have only to note that the matrix appearing here anticommutes with yO and therefore effectively vanishes in the state $(O). The same conclusion extends to crF = BkFo,, occurring singly, since y".dk either vanishes, for p = 0, or, for p = 1, involves the three-dimensional vector matrix [yk, yl] whose coefficient must vanish in the rotationally invariant state $(0). Further simplifications accompany the R-integration. Recalling the discussion of (4-16-77), with its

dependence an the characteristic property of a Lorentz gauge, we note again that the redefinition of the integration variabfe produces the ef fective reduction

which involves the vanishing, under inteeation, of a term containing a single factor of k. The outcome of the integration for (4113.136), proportional to yFA =.

y V F , , A yields no contribution in the state $(Q). Terms involving the single R factor of (K - @).eqA also vanish on integration. The two product terms that survive are written out in

x exp(--- icw[ (l -- er) 121) - a s 2 %

x exp(-- istw[(I 4- v)/21)

X C-- 221eqyF.(iZ - zip) + 2 4 1 - u)eqyF,$J

X expf- isc(l -- W ) ] [-- 221(1 -- zc)$.qA] exp{- isc&[(l ---. v) /2 ] ) . (616.137)

Note that, as in Eqs. (4-18.13-161, we have i n t r o d u d the mare symmetrical set of parameters to express the result of expanding the exponentiafs, The k- integration that is anticipated in (4-1 6.137) gives

1 2 ~ ~ l a -- ~I ieeyJ + -- ufkqyF(p' + *")l x exp(-- zsBI) [- %(I - %)(pt + 19'" eeqA], (k16.138)

where, as effectively happens in the discussion leading up to Eq, (P-f6,87), we have made the substitution

In the term

yF(Pf + P") = y0Fo,(P' + pp')* + ykF,($' + p")', (4-18.140)

only the matrix yO can contribute to the expectation value. Its coefficient takes alternative forms depending upon which of the two terms in (4-16.138) is being considered. Far the first one (recall that p" 00)

Fok(p" pf@f" Fr;,,(pU - p?& =; ;Jal (4-1 8,141)

while the other becomes

polc(pY P'')k - Fok(PP - = - ip. (4-18,142)

The two terms of (4-16.138) are then seen to be identical, which we present as

This is the last contribution to the energy shift:

Now that we have seen all oI the contributions, it is worthy of comment that every one of these double: parametric integrals is reducible to ($-15.69), m its specialization (4-1 6.92). f n this example, we gel

afOE = (log 2; - &)zZ2~'

We add the various fiieces, S4E, . . . , cSiloE, as given in Eqs, (4-16.104, l 1 l , 116, $19, 128, 145), to get

Elmradynamlu 1 Chap. 4

and then, recalling 6,E [Eq. (4-lEi.CSB)], we find that the compfete effect, without reference to vacuum polarization, is

This is about 20 percent 'less than the spin O resu'lt of (4-15.98). To finish the calculation, we introduce the appropriate spin 4 structure into the vacuum polarization calculation of Eqs. ('4-15.3-51, so that

which implies the energy shift

(4-16.1413)

The final result, then, is

which has again been sa written that the quantity in brackets represents the effective atld;ition to the list of additive constants that supplement the dominant logarithm.

Concerning the precise magnitude of the additional upward displacement of s-levels, we recaXl that, for n -- 2,

which exhibits the reduced mass effect in $(Q). For hydrogen, this gives:

Added to the earlier theoretical value of Eq. (4-11 .I 141, we now have the folbwing predicted value

4-$7 M-partlcla eaaw dltplwamank. Spin & rdativislfc theory @l

This time, the aveement with the experimental value of 1057.W ZfL 0, X0 MHz is even better than might have been anticipated since t k r e stilt remain vaious secondary effects t a be considered, most notably the modification of relative arder a, as contrastd with. the Zg effects considered here.

Incidentally, the result obtained in this section reproduces that derived quite some time MO by ttlro independent sets of workers (Harvard and Cornell). The first announcements were: R. Karplus, A, Klein, J. Schrwinger, Php~. Rev. 84, 897 (1851) ; M. Baranger, Phys. &v, 84, 868 (1951). The complete descriptions of the methods are found in: Karplus, Klein, and Schwinger, Phys. Rev. 86, 288 (1962); Baranger, Bethe, and Feynman, Phys. Rev.. 92, 482 (1953). There is a genetic relationship between the present work and the earlier one of the Harvard group, but the latter calculation is much more complicated in its details,

4-17 H-PARTICLE ENERGY DISQLAGEIUIENTS. SPIN # RELATIVISTIC THEORY IE

Atomic nuclei carry electromagnetic properties other than charge. Magnetic dipole moments, electric quadrupole moments, and so forth can, and do exist, as limited only by the nudear spin, Correspondingly, the energy spectsurn exhibits an additional, hyperfine structure, For the S-levels of a hydrogenic atom, with their restriction to spin angula~ momentum, only the nuclear mqnetic dipole moment can bt?l effective. I t praduces a doubtet structure in the f sl,% level, whieh splitting has been measured with great accuracy in hydrogen, deuterium, and tritium. Our intent in this section is to review the elementary theory of this effect and discuss its electrodynamic modifications to relative arder a and Zag.

The nuclear magnetic moment is expressed in terms of the spin 8 and the g-factor gs as

where M, is the proton mass. The divergenceless spatial current, from which the moment is computed as [cf. Eq. (3-]i0.59)]

The associated vector patential and magnetic field are

We shall usually be concerned only with the rotational average of the cmr&nate dependence in the magnetic field, which is

Let us begin with the energy displacement cornpate-d from the primitive? interaction, in a nonrelativistic approximation.. Rekrring to the action expression for a spin 1 particle in an electromagnetic field [Eq. (5-10.63) with gp,, = 2, as is appropriate to an ejectran], ure infer the enegy displacement rsssociated with the weak vector potential A(r) to be

where the latter form incovorat.~ts the charge ~ i p m e n t of an electron,

The field $(F) of the bound state omits the time dependent factor exp[- ipOzO]. We explait the Dirac equations

to rewrite (Q-1x6) as

which uses the fact that II = p when the Coulomb field is represented by the scalar potentid A@, and t h t syrnnretrization of the p .A product is Imneeessav, since

In the nonrelativistic agpro~mation, $(r) describes a state of zero orbit& an@= momentum (sliz), and d e f i ~ t e intrinsic parity, v'= + l . This impEes the vanishing of the orbital term

4-$7 W-paelcle eernsrgy dlspfsements. Spin + rela@ivittiic thdrory l1 S1

and permits the mvnetic field to be replaced by the average of (4-17.5). The immediate result is

e @*p. "=:--gssr*S

2MP (4-17.13)

is assigned one of the eigenvalues associated with the total anm1;ar momentum

F .=: S + %CF. (4-17.14)

They are given by

and the splitting of the s state is measured by

The latter fom refers speeifieatlly to the Xs state, and includes the reduced mass effect for a nucleus of mass M.

The measured value of the proton magnetic moment, expressed in the nuclear magneton e/2M,, is

The magnitude of the s state splitting wmputed from (4-17.16) is, then,

which is to be compared with the measured value (many more significant f i g u ~ s are available) :

The major part of the discrepancy of 1.58 MHz i s removed by invoking the a/2n modifieatian of the electwn. moment, This gives the modified theoretical value

Elm$radynanrics i Chap, 4

In connection with the residual discrepancy of 0.07 MHz, we shaU discuss in this section the theoretical modifications of relative orders (Z~)%and Zg2.

The first sf these, - (g@)%, is the purely relativistic correction to the nonrelativistic formula, Perhaps the simplest way to compute it is to use the relativ- islie hydrogenic wave functions in evaluating (&17.Bf, rather than trying to estimate the corrections in (617.9). The solution of the Rirac equation,

that applies to the ground state, is a mixture of s1 ,~ and waves of opposite intrinsic p a ~ t y ,

which are coupled together by the grdient te rn in (617.21). This is described by the pair of equations [yOy = iy5cr]

We exhibit the spin-angle dependence of these two components in

$ S = ) B gp(f.) = - Ys@ ng(*)v. (4-1 7.24)

where n is the unit radial vector

n = fJr , (4-17.25)

and v is an arbitrav unit spinor vvith F'= +- It, This is verified by extracting the p re ly radial equations

the latter one has involved the following operator aigebra [r x Vg = 0)

4-17 H-putlele an- dirplacemants. Spin ) mktlvistic t b r y It 283

The pair of similar equations is effectively diagonalized by writing

= yf(r) , (4-17.28)

and identifying analogous coefficients in the two equations:

This gives (only one root of the quadratic y equation is physically acceptable)

and

2 m - t(Za)%n.

The radial dependence now obtained by solving

f (r) = NY-~" exp(- Zamr). (4-17.33)

The coefficient N is determined (apart from a phase factor) by the normalization condition [cf. Eq. (3-15.33)]

When 2 Z q z 4 9 is neglected, we regain the familiar nonrelativistic normalization constant /+(0) l*,

M e energy shift computed from (4-17.6), or

is given by

Chap. 4

where

The last step above has introduced tlte rotational average, Accordingly, in an eigenstate of a: * S, we have

It is simplest to divide this radial integral by the unit normalization integral of (&17.34),

and then

For small Zac, the relativistic correction factor is

Goasidered by itself, Phis (&fZ effect acts in the wrong direction to resolve the discrepancy bettveen (4-17.19) and (4-17.20). XL would increase the difference to 0.19 M H z .

We therefore turn to the discussion of the mdification of order Za2, that is, of order ZGC relative to the calculation of Eq, (4-17.2-0). With aU the accumulated expertise of the last section to draw on, this is a. comparatively simple task. The first topic is vacuum polarization, which is much more important in this situation. than with the Coulomb field, owing ta the more concentrated nature of the nuclear mape t ic field, The alteration of the vector patenliar AU By &A4 changes energy values by the amount [it is the more- ~feneral farm of Eq, (4-1 7,9)3

where

specifies the nature of the alteration, Perhaps one should observe fimt that, in contrst with the Coulomb field situation, no vacuum polezation modific&iorr of the magnetic field coupling appears when the momentum dependence of the wave function 36 is neglected [this is the use of +(O) only], That is because

vanishes at p =5 0, rNe are therefore interested in the itera;ted field

It implies the enera shift

where we bave reeapized that the tvva cross-product terns contribute equally, and extracted the linear spin depndenee. Thmughout this &%usion, terns quadratic in the spin be discarded since

disdppears in the yO' = + 1 state #(Q). Contained here is the effect of the m w e t i c field an Coulomb fieM vacuum polarization, and the effect of the Ccluilomb field sn rnwetic field vamurn poia~zation. The t w eff eets are equal, fctr both pmduce the tern

286 Electmdynamia I Chap. 4

where [cf. Eq. (4-3.28)]

We shall write all these hyperfine structure energy shifts in units of the nonrelativistic value (4-17.12). which is designated as

Accordingly,

The discussion to follow parallels that of Section 4-16, but is simpler since there is no infrared sensitivity in virtue of the short range nature of the magnetic field. In consequence, we embark directly on the power series expansion, and employ the A device only in the first term of (4-16.7) in order to extract (4-16.12), the nonexplicitly field dependent part. The latter does not contribute to energy shifts. There is just one point to keep in mind. Since this calculation specifically seeks corrections to the description of the electron in which the a/2n addition to the magnetic moment is already taken into account, the two terms that produce the a/2n effect must be removed.

We begin with Eq. (4-16.88), which is repeated here, apart from an additional factor:

Taking the first 1 difference gives the energy shift contribution

(617.67)

and

In the latter, we have removed the term that cont~butes to the magnetic moment in the situation of slowly varying fields [it is equivalent to the first term on the right side of (P16.17)]. Introducing the iterated particle field into (4-17.57) and extracting the linear spin term, we get

or, with the appropriate modification af the structure in Eqs. (&18,72,73),

The double parametric integrals appearing here have been met before, which is the general experience in the present calculation, and give

8"E - .=: (2 - 2 log 2) Z@%* F

(P-17.61)

Turning to (&17.58), we find that

dw duf l + zc)~w(f -- NW) (617.62)

and

Chap. 4

The sum of the two pieces is

which, as it happens, is the negative of the find result (without vacuum polarization).

The linear spin term in Eq. (616.83) leads, as the counterpart of (4-16.85). to

and then to the enerw shift contribution:

(k17.66)

We divide this into two pa&s,

and

= (# -- 2 log 2)Zg'. F

(617.6-9)

which adds to BIE in @ving the complete contribution of the first t e rn (X) in (616.7) :

The dixussion of the seconcf term in the list hgins with Eq, (4-1 6.108), evaluated for A = I,

although one must also remark that the vector potential in 17 does not contribute a spin term. The intrduction of the iterated field produces the energy shift

where one must be careful to note that there are two contributions in each produet, as indicated by

The vector identity

in combination with the properties

17-E =l@, V-EIE = Ot V X E = 0, (617.76)

whi& are applied after partial integration, show the equivalence: of the two terms in (&lL7.73), since the multiplication order is not significant for this matrix element. That @ves

Chap. 4

(617.77)

For the third tern of (4-X6.q we follow the discussion beginning with Eq, (416.112) and leading to (&18.1X4), where the second contribution to the a/2n m w e t i e moment has already been excised. This yields

which will be recognizd as the integral of (4-16.115). Accordingly, we have

GpE/F =. - 3Zaz. (4-1 7.82)

As for the term explicitly quadratic in the field, and linear in the spin, we recall that the double commutator of imllitfy, [exp(- kg) , y33 multiplies the spin term by a, factor of 8 [Eq, (4-16,113)], so that this stmcture can be obtained from the eonesponding one in the first terrn of (&16.7) by the substitution

applied to the il = X fom. Picking out that terrn from (617.67) then gives

=.: (- 6 + 12 log 2)Za2, (6117.83)

and, on ad&ng (417.81), we

SxxrE/i"; = (- 9 + X2 Iag 2)Za2. (4-17.84)

The first three terms [Eqs. (4-27.70, 78, 8411 give

(BE 4- 4, -i- S,,)EIF -; (- (1314) + I% g)xa2, (&I 7.85)

4-17 H-parflcic energy dispfacements. Splm 4 rcilativistie theoq II 294

which is the negative of the particular contribution exhibited in Eq, (4-17.64). One may infer from the remark made there that this is the complete expression. And so it is, since it turns out that the fourth term of (4-16.7) has a vanishing contributian. It would be very pleasant to prove this in one masterful stroke, but we shall have to use more prosaic means, The double commutator term in (4-16.8) has the consequence already stated in Eq, (4-l6,117),

and

Next we consider the linear Eel8 terrn o i Eq. (4-16.123) which, for convenience, is repated here in the form of an energy shift:

The first of' the two pads,

can be inferred from the analogous evalua"ron of (4-17.7 11, leading to (4-1X.77). Making the appropriate substitutions, we get

which we prefer La combine with cfft7E before further evaluation. There is some danger, in discussing the second terrn of (4-17.881, of' being

misled by the notation. First, recall that p' and 9'' are row and eofurnn indices, respectively, in a typical matrix element. 1-f- we then wish to use p ' for representing the momentum in #(Q), a transposition is needed in one of the two terms produced by the insertion of the iterated field, Attentkn must be paid also to the convention adopted in reducing the D, of (615.W) to the D, of (626.62). Thus far, we have

Elmrodynmmic;r I Chap, 4

escaped all this tight-rope walking because, apart from D,, the intepands have been even funetions of the vaxl.ablt; v ; the second tern of (4-I"ZB) is not in that category. I t is now seen that

where we have temporarily written D1(v) to in&cate the D, function of Eq, (&15.64), in which

We shall change the integration variable v to - v, in the first tern af (4-17,81), and employ the reductions illustrated in Eqs. (4-1 6,141, 142). After recognizing that the two tems of (417.81) are effectively equal, we get

4 - Qv$(O) *eqa * IE 1 f - epJ0+(0), (k17.93) P'" & mm" D,

using the equivalence noted in connection vvith (4-1 7-73), Accordingly,

S"", = - 8Z%%F ~ w ( Z W - 1) dg gg(t - g) (P17.94) ( 2 ~ ) P 1

which, added to (&1"1,90), gives

Finally, there is the explicitly quadratic term constmcted from the Enear factors of (616.132) and (4-16.133). Since we want the linear spin dependence, the t ems of interest are obtained by nndtiplying -- " u I ~ ~ Q E ; into the two nanspia t ems

4--l ? fete snrarw dirpilacemsnts;. Spin -B rslattvlatic thsory Itl B3

of (&16,132), and by multiplying the spin terns. The Xatter is an exwptian to the mle, because multiplication with yU is a f s ~ involved. We first note that

since

yP@FyP = 0,

Far the spin product we have, omitting inessential factors and intraducing pemissible partial integrations,

yB.(@F i3, @F) = yk.(a * H ak(- i)py * E + (- z)yOy E 3, a * R)

= +iy0[y % a Hy ta f i l + a&E~yla * Hyk

- Q * Ii. akEgyyyk - y k y t i ) k E I ~ * H]. ( 6 1 7.98)

The next step nctakes use of the null value of V x E, which is the syrnmtry

i t enables one to make the replacement

and to employ the relation [y = iy@y50]

-- Q(rkgmr" y = 4 f ~ g ~ n a ~ t + ~ f ~ a ~ u k )

S,&, + Slm@k - &a#,, (a-17,101)

which follows directly from the anticommutation property

Q(@h# a,) G S,&* (4-117.102)

The ad&tiond fact that V B = 0 leads to the effective evaluation

and then

yv" . (@F @Fly, --. .-.- 4ia BJ@, (4-1 7. ,]LW)

As the countevairt of Eq. (&116,X38), we now have

M Elwtrodynamtcs l Chap, 4

+ U( 1 -- zl)eqyF(Pr 1- p")] exp(-- ksD1) 2ueqcF

- 21% . l iege * H exp(-- isD,) egJO},

and the same reductions used before produce the energy shift expression

dw d2d w(l -. w)d( 1 -- uw)

== (2 - 2 fog 2)ZgZ. (4-17.101)

And, indeed, the sum of the pieces in Eqs. (4-17.87,96, 107) does vanish,

The final outcome, obtained by adding to (417.85) the energy shift of vacuum polarization ofigin, Eq, (617.542, is

6EIF = --- (Q - log 2)Za2. (4-1 7,109)

Taken by itself, it decreases the theoretical value of the hydrogen hyperfine splitting. by 0.137 MHz, But the combination of the relativistic effect in (4-17.42) and the electrodynamic effect just computed results in a comparatively small decreae, measured by

This represents a decrease of 0.023 MHz, altering the theoretical value of (4-17.20) to

4-4 7 H-poefcle enerw dtrgtacements, Spin Q rrrlatfviotfc thbrrzry t l 2%

What shall we say about the residual discrepancy of 0,05 M H z ? Quite simply that we have now moved outside the realm of pure electrodynamics into the domain of the strong interactions that govern the properties of the proton.

It is known from high energy electron scattering experiments that, with regard to its electric and mawetic properties, the proton acts as an object completely distributed aver a, certain spatial volume, fn other words, there are electric charge and mzignetic moment form factors that must be detemined almost. entirely by the nonelectromagnetic interactions associated with the subnuclear particles to which the proton is coupled. I t is clear qualitatively that the abandon- ment of the paint charge, point dipole description must decrease the magnitude of the interaction responsible for the hyperfine structure splitting. That is in the right direction to remove the remaining discrepancy. We shall estimate the magnitude of this effect.

In the nonrelativistic theory, the distribution of nucleon magnetism, pm(r), thus far taken to be a delta function, is integrated over the square of the electron wave function, Accordingly, we now have the replacement

The short range behavior of the wave function is detemined by the nucleon electric charge, If this is distributed in accordance with the function p,(r), we must also take that into account:

note that there is no change once one is sufficiently outside the charge &str;ibution, The implication of both effects is

is an average nucleafl radius. We give it another farm by using the representation. of Eq. (4-15-48),

whence

We have utilized etre principd value inteeal

and the spherical symmetry of the distribution functions, The empiricd data on the protan are represented approximately by

l z P@($) 2 [l + (PP/M@2)11 '

(4-1 7, X 20)

where MO g 0.90 M,. (617.121)

Evaluating the integal in (4-1 7.1 18) gives

wMeh implies the following fractional decrease in the hypedine splitting:

This represents a decxrease by 6.0-5 HHz, whieh, to the aceuraey that has been retained, complete1 y reconciles theory and e x p ~ m e n t .

H a ~ a g broach& the subject of finite nudear size, let us also estimaC;e its effect in the relative &splace?ment of s and p leveb, The chage in the Coulomb intmactioxt e n e r e is

(4-17.124)

md this induces the energy shift

Xf we combine the proprty

(617.126)

with the vanishing af the integral. a t r = 0, it is inferred that

This gives

where

The eneey shift is only significant in s states, and the upward displacement of the ns level is

For the 2s level of hydrogen, this amounts to

which further narrows the gap between the theoretical value, naw adjusted ta

H: E, ,,,, --E, p,IP = 1057.81MHz, (kf7.132)

and the exper;imental value of 1057.80 & 0.10 MHz, But we must again warn that ptentiaHy bigger ef feets than this one have not yet been considered (although it is a fair inference that they are largely coanterbalandng).


Index

Action. See aka Action principle See crleo Lagrange hct ion

addition to, for bound state problem, 281

for spin 0 particlq B9 additional, for arbitrary electromag-

netic field, 126,127 in deuteron description, 1M in non-relativistic dynamics, 164 for slowly varying electromagnetic

field, 130 for spin ; particle, 24

alternative presentations of, 127 contribution to, for non-local coupling

of current and field, 64 describing vacuum polarization, S7 effective, for magnetic moment cou-

pling, 4 for spin particle, 25 for two-photon decay of U-meson, 46

for forward light-light scattering, with parallel polarizations, 138

with perpendicular polarizations, l89 for forward photon-photon scattering,

with arbitrary polarizations, 140 for four photon fields, 124,127 and inhomogeneous Schr ' iger equa-

tion, 157 initial, for spin particle, 26 rep-ting forward Compton scattm-

ing, 232 Action principle

used to improve d d p t i o n , S9 using equal time fields, 310 total, in hyperfine structure, 28l

Atomic excitation energy, in energy displacement formula, Q

BarwLfer, M, 279 Bethe, Hens, 173,279 Bohr radius and energy, 168 Bound system, repeated intemction in,

and noncausal method, 233

canonical transformation, m in operator field theory, 173

Cauchy, principal value, 246 Causal arrangement, 76,82

for deuteron form factor calculation, 109

for a double spectral fonn, l44 in Compton scattering, 198

for energy displacement calculation, 237 non-relativistic, 162 for photon-charged particle scattering,

189 involving photon collisions, 135 of spin $ particle sources, 3

C a d t y , 1 Completenem expression of, for spin $

particle states, 3 Contact interactions, 10. See also Con-

tact terms Contact term& See also Contact interac-

tions for deuteron description, 101 for double epectral fonn, 86

another procedure, 103 and gauge covariance, B3 improved procedure for spin 0,M for magnetic moment calculation, !M non-relativistic, 162 and normalization conditions, 921 for phenomenological g factor, 26 for vacuum polarization, 37

300 Index

Corinaldesi, E., 2l8 Coulomb potential, and logarithmic sin-

gularity, 62 modification of, 33 modified, long range behavior of, 133

Coulomb scattering. See d o Scattdng, Coulomb

of spin + particles, vacuum amplitude for, 55

Coupling constant, of deuteron, 113 Croes section, differential, 182

for deuteron photo-disintegration, 116

for elastic scatteain& l75 for essentially elastic scattering, l77 for low energy modified Compton

scattering, Zl8 modification of, 176 for neutron-proton sea-g, 117 for photon-photon scattering, 155

ecleentially elastic, modification factor for, 18l

for essentially elastic scattering, 65 forward differential, for high energy

Compton ecattering, 196 for low energy Compton scattering, 194

scattering, direct calculation of, 181 total, 175,182

for deuteron photo-disintegration, 116

and forward scattering, 193 for low energy modified Compton

scattering, 218 for low energy photon-photon scat-

tering, 133 modification of, 176 for photon-particle collision, 1W

Cro&ng symmetry, 192. See also Sym- metry, crosgiag

Curreat, effective, W Current vector, of spin 0 charged parti-

cles, 6 l

Decay rate relative, for double pair decay of n-me-

son, 49

for single pair decay of U - m m , 4B for single pair decay of a-m-, more

accurate evaluation of, 49 total, 47 two-photon, of U-meeon, 18

Determinant, diffeamtial popesty of, l26 expansion of, 132 Gram, 146

Deuteron, binding energy of, 108 charge distribution, 114 compoeite, 113 field of, as effective eource, 106 form factor of, 113,120 mass of, 106 non-relatidc d k p t i o n of, ll9 phenomenologid d d p t i o n of, 106 photo-disintegration of, 115 wave function, 114

Diagram, causal, 77 conventions of, 78 for double spectral fonn in Compton

scattering, 198 Feynman, 77 noncausal, 78

Differentiation, to the left, !U Dirac matricee, a beeic property of, 23 Dynamics, non-relativistic, 166

firnit to, 163

Effective range, 116 Effective range theory, 117,118 as precursor of source theory, 118

Electromagnetic vedor potential, as effective eourcs, W, M)

Electron, g factor of, I Energy displacement. See arlao Hyperfine

splitting. c a d arrangement for, B7 frequency unit for, l72 history of formula for, 173 in H-particles, leading contnition to,

166 for hydrogen, expc&w~tal value of, lllL hyperfine, non-relativistic calculation,

280 Za2 modification of.

produced by magnetic moment, 170 produced by vacuum polarization, 811 relative, of 2s and 2p levels, 172,278 spin 0, compared with non-relativistic

calculation, 238 relativistic d o n to, 999 of relative order Zu, 2M Za modification of, 842

spin f , compared with non-relativistic calculation, !M3

of relative order Za, 278 relativistic c o d o n to, m4

for spin ?j particle in Coulomb field, 267 of 2s level, and finite nuclear size, 297

Equation, Dirac, second order form, and iteration, 286

J3quation of motion, proper time, 129 Equations, radial., derived from Dirac

equation, 282 Eulerian conetant, 34 Exchange, multi-pmticle, 1

single-particle, 1 Excitation, spacelike, 211 Existence, of four photon coupling, 124 Expansion, commutator, B3

perturbation, 222 Expectation value, of inveme dhtance

powers,

Feynman,R,rn Field

el-etic, causal forms of, 1s dowly varying, 1%

non-relativktic, action principle definition of, 158

elowly varying, in bound state calculai don, 234

assmlrCe,m epinf,causalstructuresfor,I

local combinations of, 69 two-particle, differential equation for,

306 Field strengths, end vacuum polarhtion,

37 Hne structure constant, 6

value of, S, 172

Flux, invariant, in photon-particle colIi- sion, 194

Form factor, 2,W ~ e , i n c n v r e ~ o n d t e r a ~ 6 2

in energy displacmmts, 6% for* ?j particle8,gO

deuteron, 119, m another calculation of, 119 thresholds in, 120

magnetic, for spin ?j particles, 60 for photcmgpin $ particle emission, 76 scalar, for spin 0 particles, I spin 0, noncausal calculation of, 236 for spin 0 particlq 87 spin ?j calculations for, reaeanbling

0, teneor, for epin 0 particles, 6 8 1

for* f particleq60 in total uoee section calculation, l=

g factor, 36 dynamical &gin o& %B of electron and muon, !M mum-electron difference, improved cal-

culation of, 41 strong interaction a o n t n i o n to, 46 and vacuum polarization, 86,41

of nucleus, !279 of primitive interaction, 81) of win $ particlea, 61

Gamma function, in integral ewehation, U7, no

logarithmic derivative, properties of, no

multiplication pmperty of, 270 relations for, 245

Gauge covahce and contact terms, m Gauge invariance, 56

arbitrariness in enforcing on double apectral fonn, 152

of four photon coupling, 1% maintenance of in photon-particle scat-

tering, 193 in photon source coupling, SO of polarization vactor combinations, #n and space-time extrapolation, lW, 208

302 Index

G m m a ~ g ftulction, for ns wave fme- tions, f iEKl

Green" fwetion. See &o fiopagation fmction

of H-p&iele, 168 h moment= sgam, I(i7 non-relati4stie, convation fw, 1FiS

diM:erential qustion for, lfit integral quation for, 16t2,lM

retmdd, 167 for 8ph O pmtieles, 21 spin i, integetf q a ~ o n for, f 28 thw-&mmsJoneif, 8Mfi.e interpreta-

tion of, M8 t rms fomd , md hdf-plane of re@w-

ity, 157 G y o m p e ~ c ratio, Clf eiatron and muon,

2

W-pdcle, 1w deeay rate of* 270 mergy dkpfacemenb of, 166. See also

Energy &~pfacemmt Grwnk ffundion of, 168 imubigty of, 1 69

Hmold, on meient k t o v , 178 on c a d Cliapms, 77 on the E ~ t o r y of ~ a t t e & g cnrleula-

tiom* "l3 an the impo&llxtce of &en~Ec mem-

aim, 11 8 on mulGple dexliwationg, IO(i on h g l e p&iele exchmge, %X5 on his trmsfamation, ]LW

Hydrogenie a tom, vacurn p l eza t i on in, 34

Hypesfine spfftting. See aho Enerl~y &S-

placement with mdf i ed mapetic moment, m non-reja~*~c* non-relatihtie ( nueEa effect on, W af relative order Er2, %M r e l a t i ~ t i e commtian, 282 r e d u a l dkrepmcy in, 2945

mdfieation of, 279

Xnfhite momentum frme, 5 f n f r m d dvergence, 10 Infrard problem, S3

md photon m=, &l, 87 f n temdsB double p m m e t ~ c , waluation

of, 277 Interaction, repeat&, in. b w d system,

m Xntmaction mergy, mdfied, of &B-

tgbutions, 38 Invhmt moment= intepd, for two

p&icX~, 5 fbration, and w o n d order Dirtxe qua-

tion, 2M and short &smce behagor of wave

fmction, %B

Jwt, R., 218

K m l w , R., g79 K1&, A,, 279

he;rmge frmction. See alsa Action add i~ond , for slowly vmfig e j ~ t r e

magneltie field, 130 effmive, for unstable R - m a n , 47 ima&my part of, for strong e i e t ~ e

field, 1% a d lw&ty, 38 q u d e h d w t r o m a p e ~ e field, 132,133 for slowly vmgng el~tromameGc

~e l& , 3kl h p e m e p o l y n o ~ d , 14% t m b sEft. &e H-peiele, ens@ &-

placementh3 of Geber, Mchael, 3.68 Gght scatter& by Eght, 128 bgar;it&c &nmldty ,

a u l o m b ptenGd, 5;2 h r m t z gauge, M, 89, $3 Lorentz t-rm~fomation, w t ~ x for, 18

Mawe-tie ehmge, a M a p e ~ e mammt,

proton, %ill d e h t m

fam, ifmt cavdmcq in opa to r field the-

ory, 1173 M-

r d u e d , 172 ~ p t r d , for a double s p t r d fom, 1918

MW e&rapola~on, S S

h mapeeic field, 24 M m &ell, 1132 M &&c, trace propdy of, X28 M quations, m d B d by v a w w

pl&mgon, 3113 Muon, g faetor of, 26

N o n a d cdculation, l&@, 203 ad~mtage of for b~d. 8bt@ probl

219 Nom&mtion, of m a c wave fmction,

m Nom&zaGon eon&Gons, md con&&

Nam&zation r w ~ m m l s , a d m m opator , $6

and phenommoXa@eal g factor, 26 in m&&le q-mmn in v a m m polkzatlion, 37

lisuclwn e1eet;ric chmge, &t&bu~on of, m&

Nucl- ma~E?tie &pole moment, 279 V ~ ~ Q F p t a ~ d md m g n @ ~ e field, of,

279 EtTuclw mulGpale momat63, m& sph, 27% Nuclm phyda, 1W Nuclwn ra&w, average, 29fi Nuclmn marnet*, &t&bution of, S 6

tor field thwry, 173 apad t h ~ ~ m , 193 O&hogon&ty, rela~ve to vawm

state, S

p~ob%biEty, 8ph d te~n- dmce of, 88

P d t y , Itl

probabiHty mpEtude, 174 on, 1-Slr an, h q u m t m me-

Pefimbaaon &wry, &pp&& to vacurn p i d m ~ o n , 38 a&, &-wave, h neutron-proton rpcatte*g, S17

Photon, ~ s o t r a p i c propagaGon of, X33 coughg to pmmn, 46

m= of, 6t"t and the Coulomb potenGd, 6;3 . detehble mmgy of, g3

on probabitity forp X4,1@, X9 d n h m detwbbie hqumcy of,

on probabifity of, 64

Photon decay, of neutra) v-rnmn, 443 Photon prapagation funeGon, 27 Photon mmces c h g d w i d e

&am, 1 R-mwn, neutrd, dway into photcma md

Jstron-@trm p&, 46 strong: intmcGan p m m i a Q.f, 441 in v a ~ ~ u n z plariza(Jm, 46

PotenCial, ad&tiond, in non-relatidtic dyn M

e f f ~ G v e ~ r e h ~ d ~ c watte~ng, ]La

tive hteracGon, S appIications of, 72 for deutwan and nuelmas, IQ6 with dectromagnetic field, for daubran,

X09 g faetor of, 26

3M Index

P~&tiivtl intera&on (m&.) m lmal couphg, 2 nnodiification of, 20 non-lwd mdEcation of, 2

&pal value, h G r w k f w c ~ o n inter- preSation, W

htewd invduing, eompfex computation of, 246

hobabitity for pair crmGon by strong e l s t ~ c field,

134 for pe*tmce of vamm* 83 for &ngle pair * " ,S

fiopaga~on, =&tropic, in ee af rnametic: field, 133

fiopaation hetion. 6"ee aka Grm's fmct-ion

c a d fom of, 7 rrf chmgd p d c l a 9 2

for deuteron, 108 in mamet_ie field, 28 for a non-reIati*~c p&icle, 1SJ: of photon, a, 3.2,3k) for photon, mathematid f ~ l u r ~ of,

41 for photon, in &mp]iifd fom, 40 of photon, &pin 0 f a m of, 62 for spb 0 peiclw, 12, X3 for spin $ pMicfes, 7, 9,318,119

m d f i d photon, weight facbr of bSt

of neutron md proton, multipEed, l07 non-relatiGstic, for dmtmon, 3108 n o n - r e l a ~ ~ t i c fom, 3166 pmm&&c (pram h(?)

tion of, 127, ZM of photon, %l photon

dyaae fom of, 1Pis b p E B d fam of, 160

&gle pdeXe, 1 sph i, h inand ordw fern, 98 far sph # p A c l ~ in W& fiefd, ttQ

Ropagatar, S e fiapagaGm hul&ion

Proper h e methd, 1928 h o b n , m p S c d &t~budiam of, $396

*-turn action pAnuple, m Quantum Kimmtiers a& wmnzh, $68,

2232

22-4 m, m ~ w b a ~ o n expm-

%ion in, a 2

Radiation gauge, 169

b t &me, 45 Rha-mwn, coupkg to photon, 46 Rydbr& l@

Scatk&g, 50 Campton, 188 double amptan, B& e@aive p o t a u far, 163

Gdly elmtie, eriterion for, 177 &fferentiaE er- w ~ o n for3 177 non-refatietic mdficatian fador,

188 ultra-rehti*~~ mdficaieion faetgr,

f@ fommd mbi@ty in

m in photon-paicle cowon, 193

tian of, m of Xi&t by Iil5]nt, 123 neutron-proton, 117 non-rel&G~tie, m i h g efmGc md in-

phatan-phahn. &@ %& Xi&t by li&t

diffmen~sd er- W i m for, litid

with parallel polmbations, 131 with peqe!ndicular polarimtions,

188 for pohukat io~~ papondicular to

acattehg plane, lm relation between spinOandqin f

particles, 141 epectral fomm for, l52

of epin 0 particles, vacuum amplitude for, m

Scattering, Coulomb, emx!ntidly elastic, croesJIectionof,63

modified, e r n f- for, m in nomlativistic limit, 65,n relation between spins 0 and f,70 at relativistic enesgies, 68 spin 0 high energy form, 68 ofspinOpartiClea,63 spin t high energy f- n of sprn : particles, 69

Schriidinges equation, inhomogmeous, and action principle, 157

SeleCdedP~onQuntumElecbmly- ncvnics, 173

Shelter Island Conference, 115 soft photon emission, 177

relative probability for, 178 Source

effective, for non-relativistic particle and photon emhion, abmrption, l60

in nucleon pair emision, l06 for photon-particle d o n and ab-

sorption, 18B in photon-spiu 0 particle embion, l1 in photongpin t particle emiseIon, 2,

a,m produced by two photon fields, l2$ of proton-antiproton pair, l19 i n s p i n O p a i r ~ r n , 3 1 , 6 0 inspin ?j pairemisaion,!28,55 of spin 0 particles, 126 as two-particle field product, 136 for two-photon decay of w - m m 47

effective photon, conservation of, 13 electromagnetic model of, &l!& 72 extended, 1

extended photon, m thephoton emit- ter, 133

Source theory, creative principle6 of, l flexibility of, 1% general computatioacll matbod of, 186 phenomenological bads of, 10

Space-time extrapolation, of a c a d vacuum amplitude, 86

far a double spectd form, 147,149 and gauge invariance, 63, In, W8 in photon-particle reaction, I97 in photon-particle scattering, 193 and threaholde, l(#

Space-time uniformity, 1 Spectral domain, for B double spectral

f9 spectral Form,

double, 14,LUS, 92, a08 causal arrangement for, 144 in Compton scattering, 186 in deuteron problem, 111 Mucible, 207 two-particle exchange in, 148

double and single, for photon-photm Bcattelhg, 1 n

rdngle, l4 derived fram double spectral tarn, 91 difficulty in desivation from double

spectral fonn, 91 and forward scattering, a#) supplement to double spectral form,

212 eingle and double, for photon-photua

scattering, referring to spPjn f particlea, 163

S p c e function. 9ce Eula's dilogerithm Spin dependence, of pair d o n pmba-

bility, S!? spontaneous emiEn4iaq 170 Strong interactions, !H6 Structure, hyperfine, 279 symmetry, &g, 1% 208, a 0 &g, of polarhtion vect4n combi-

nations, 201

Tenem, dewmagnetic field, of, i a

Tey, JwpMne, 118 The Daw&r of T h , 118 T k d a l d , momdow9 122

in deuteron fom faetor, 1 s nomd, 1B

Tonypmdy, 118 Total dway rate, 43 Total: m u m a t m , unit factor for, 4, 29

tion m~trix, Ilouf energy fom of, for m d f i d ampton wattekg, 217

WO-p&ie;te mchmge9 for a double qw- t r d form, 148

in a mameie field, 22 betwwn photon murca, %3,31 and v-mmon bstabiEt;y, 47 betwen sph peicle sowe=, 3 vacurn mpgtude for, 4,8,Xf, %l8 32

Unit f;actar, in two-ps6ele ex&mget 4, f%9

U ~ G t y , in neutron-proton scatt 118

Unitwy t rmfomat iw method, 218. See aka Carronicd trmfomation

b u m mpli;tude, for a c a d mmge- mend, 135,344,lsB

for c a d exehmge in m eelectromag- netic field, 484

for causal nuclmn exchange, J,W for a c a d sattefing prmm, "1, Bp

@ , l W for ex&mge of photon md nonrela-

6 * ~ 6 p d c l e , 16;0 for free sflin O p&icEe, n o n c a d cdeu-

lation of, 220 for nsneaud cafculation, spin Q, 21LB

spin +, 2M

for photon-gph + p d c l e for two-pdele excbge, 126

V a w a w&taee prababsty, 33 in prmnce of ~trong e I e ~ : ~ c field, 134

Vacam p i d m g o n , 3a elsQon, cont~bution to g,, db md g faetam, 3fi in h y p ~ x l e gtmetwe, b p r o v d trmtmmt of for mwgy &B-

pfacemc?nt, B@ muon, eon~buGon .t;a g,, 4 Q-mmn, cmtributim to gp, 45 md 8pin 0 energy &pIacment, 239 by spin O p&iclm$ Cil, md ~ p h $ = e r a &phcemmC,

Vdrttionttf methd, 118

Wave he t ion . See d o E i g a h c ~ a n a auklimy, 188 M a ~ ~ r n w the o ~ & , 239 of deutmon, l14 Wac, n o m d m ~ o n of, m gerrerathg h e t i o n fur, 1438 ntm-relati*tie, in b m & a n l m wib,

1137 o h s n g m a c qaation, 170 for Is state, 167 ra&af, of deutwon, IlCi S ger quation for, 239 &o~-&tmce b h a ~ o r

m d iteration, %a for 2-9 statq 169

Wave guide propagation, 118 W&&t hn&ian, of photon propagaGon

heGan, .&o Wekkapf, Victor, 173

X e m r e a t e , complmentq to k, 224 X trm~fomaGon, and K inbgration,

Documents

Advanced Book Classics