Particles, Sources, and Fields

Volume I1

ADVANCED BOOKCLASSICS 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 Feynman, R., Statistical Mechanics td Processes Feynman, R., The Theory of Fa Negele, J. W. and Orland, H., Quantum Many-Partick Systems Nozieres, E?, Theory of lnreracting Femi Systemr 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 ~ gM , CISS~~USCBS

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Editor's Foreword

Perseus Books's Frontiers in Physics series has, since 1961, made it possible for leading physicists to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physicswithout having to devote the time and energy required to prepare a formal review or monograph. 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, gravitational 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~versityProfessor, 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~vergtyin 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 rhas taugfit at hrdue University (fW1-43), snd ward U ~ v a s i t y(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 & m e 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 dMedd of Science Awwd for Physics (19641; a IXumboIdt Award (1981); the h e ~ Citta o 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 l u ~ a u s(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 formation, 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 nh 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

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d volme has sprmd over two y m s md two continats. The writing of this to a sabbaticd leave A substantid portion wm p e m d in Tokyo where, from filmad University md additional support from the G u ~ e A e i mFounda. 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 gthe 2971-19315 p e ~ o dwhen X wm Visiting Professor at the U~vmsityof Galifofnia in Los Angeles. This volume is &most exclusively eomemed with qumtum eleetrsdynamics, As s u ~ hit is retrospstive in its subject matt=. But the wnwntration 0x1 this relatively simple dyxla~calsituation is &ire~t&tawwd the exploration and elaboration of v i e ~ o h c and s teh.&ques that should be viable in the damkns of strong and weak interactions, And it is intended that the self-mnt~nedsource theory development, with its significant wnwptuill and wmputatimal simplifications, shall be the face: of quantum deetrodynami~sfor 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 f o r t h c a ~ a g 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 dody after mme time. The alternative to the plan actually followed w s to g e ~ t various e 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,

Mareh 1972

CalVor~ia

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Contents

charged Psrticlk Propagarion Fwdions A Mapetic Moment GaIeuktion Photon Propagatian Function F a m F~ctors1. Sc~ttering F ~ r F&etars n II: Single and D d I e SpeefraI Form Fom Factors III. Spin $ F o m Faet~rsIl? n e D a t e r o ~ W t e ~ n ofg Light by fight 1,b w F r e ~ n c i e s Scattering of 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 mMethods ~l H-Particle Energv Bisplacemnts; Spin Q Relatiuisficmeaty H-Particle E:nerw Displacemmts. Spin 5 Relaiuhtic" Theory I H-Pmticle E~etgyDzsplacewnts, Spin j Relativistic Theory II

Particles, Sources,

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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~pndtingadditions to the appropriate propagation function. The replacement of the initial singtctp a ~ i c l epropagation 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 characterhtics, 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 of s spin O and 4 were considered, particle-antiparticlescattering 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 ~ e ssptrce-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~mitiveinteraction can be characterized as a fwal coupling of the fields. Two charged padicicle fields and the electromagnetic 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 nsnicxxl mdifieation of the pdmitive interaction alters the electrornwnetic properties wsiped to the charged particles. New couplings can appear, and the nonlmdity 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 n e o r y 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~eof an ekctrarnagnetic m d e l 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 w o r n w e 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 r d a t i o n : k.,;&:i

= 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 conjugate 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 is 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

dMZd 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 == M 22 m2, A reXated integral has been encountered, far arbitrar-~rmasws 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, Pat

""

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 i n t e ~ a t i o nover 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 r p m e n t 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 evaluation (4-1.25). On introducing

this contribution becomes

6 h q d prtlcla pro

4-1

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~riptionis

(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 nin. t 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~lgparticles that repre~ntsthe 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 project ion

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

w11ich exhibits the evaluations

This prtion of the vacuum amplitude thus becomes:

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

Note that the r a u l t h g wurce coupling reduces to (4-1 3 5 ) 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 m d f i e d 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~ghere, 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 + m 0. This contrafun~tionis drasticdty madified in the neighborfioad: &cts the phenomenolo@cail basis of the theay, as we dis~ussedextensively 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 ( 6 f . 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 contact interactions, UnXms adstiond physicaJi considerations can be adduced, the contact terms r e m ~ n . mbitraq and may Ise omitted. But, Wl;hen fields replace sources such local interaction t e m s do have physical content ; their existence must be recomizd and related to the accompan~ngphysical requirements. Since contact coupfings in coorsnate space appea as pofynomid functions of rnomenta in momentum space, the camwt f o m 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 neighborhaad of y p + 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))

ij$(- k ) K l ( -

p) err*

= [+l(- P)(p"

+ P") - X,(--

P ) i l L f k f I q , (4-1.62)

When the f u 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"

(M"

m m)

(- P)Xg(')

$2

P I C ~ ~ ~ m mt l c r

Chap. 4

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

where the M 2 integration can be extended up to infinity, but has an infrared sinmlarity as M 2 -+ 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))

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

The i n v e w p w e m of nk must be averaged aver all partitions of the total, ntomentum P, as indicated by the notation do,dok6 ( f

- p -- k ) (%h)-"'

---

I

I-*

m2

((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)-9in the rest frame of BBcontkns 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 ( 4 1 . 6 7 ) now appears effectively as -2

+

M2 m2 (M" mm212 -8-(M

zr

l

+ (Q%/M%)(I - 2%).

(4-1.76)

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 nUis a time-like vector, Xn contrast with (4-1.65), the weight factor of each (p2i .M 2 &?)-lis 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 .

__+

l4

Ctsctrdynrmlcr;I

Chap, I

structure^ such as (4-1.41) and (&1.65), desfibing the mass spctnxm of oneand two-pa~icleexcttations, 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 possibility 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 M 2 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

-

d+(p,g) -"p------

L

X

mnz"- it3

p%-f- (m"

-

2192KO) - 2"e

(4-1.83)

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 ~at t yk@ 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

- @P = QZ.

QBQ, = (%P)%

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 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 ~ ind 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, 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 , 7 V ,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 transitional 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 ( k 3 . 6 7 ) into

dEa 1 &am

82 % ~m ~ log ' 3% n AE

P''&--.----

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 This produces the physically meaningful cross section experimental arrangeme~~t. 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

Chap. 4

Electrodynizmico f

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

- (n- 0) tan 4-0-

cos 8 1 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:

1

,

2a qs m --3n [log +

7

cos9 + (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

V"

"

6s-

"C"

11 E ..--.

2 *+ir5 ~ :

Ur(

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 yarameter 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 ( M . 1 t O ) is

2 1 - v2, for an assigned a.

The new form of

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 1 Po>>m: L =sin (812)+(cm P ) ,

-c

1, and (4-4.1 17)

where

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

which also continues to be quite accurate even at t3 direction,

[

+(COS48 N 1)N 4 sin 48 log

PS

x,

4 2 . Close to the forward

1

2(1

5

- COS 48) + l ] .

(4-4.1M)

The accompanying high energy form of (44.114) 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(PI~- '~ e i) ~ yp ~= ~. , !P! ~ where, it is recalled [Eq, (2-&.W)],

Aecordingly,

and

- ~ 2 ) ~ p T , , ~ @ i ~ s ~ ~ , (4-4*126) ~,.

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/$OjBfactor, the spin 4 analague of (44.121) is immediately obtained: 17 -+ 72 --+ g sin #@+(cosg@)

401

1 -2%

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,

+ The e n e r a l spin

8 formula,

(JE

- 0) tan 4s +

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 scattex-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 it since it violated my selfimposed 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 A N D 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)

where

l&*,

'lirpyuqlrzfl"")

-

')11~2(')1

J

(4-6.1)

Form facton II. Single and double spectra1 forms

C S

73

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

I

i f du,, dwk, do,, dok,(2n)4&Pl

+ k1-

P2

- k2)

Here, using the notation 1

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

Chap, 4

Electrodynamics I

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:

+

m on the left have been discarded. The where expressions containing yPz 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

This psitive function varies from a value of Q at tW2 == m2to unity, for MO The outeome of these calculations is given by

75

>) m2.

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

cI r

Chap. 4

Its space-time extrapolation is stated as

whereL,# 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

1

a eq 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 ~ (4-5*27) ,~frT,f~)l

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 r e f e r ~ dto 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

-lkl,

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

Form f a a r s 11. Single! a d doulbga S P ~ forms FI~I

4-5

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+ mB)kI2+ (h1%)%

(M2 - m%)% (M2

+ m2 + k12)2 - 41MkZ,

(66.42)

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

80

Chap, 4

Etarodynamice

-+

K,.

The vector V, is exhibited as a finear combination of the vectors 2pl kl and 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, We can

where

ROW

+ K,)

=

- (k,2g,, -- k,,k1,)(2-P,

+ k,)"'

recognize that (k5.31) has the field structure

(4-5.48)

4-S

Form fwe~rrC 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 praduced 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 r n can be d t t e n as

where

and a f,,(k) = g,, f (k,kv - g,$%) 2;(

+tnf

dxI ( M P ,kg).

(M.53)

The f a m factor arrived at in this way should be identical with (44.25, 21). Or rather, it si~ouldbe 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 M 2- 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.

Pi

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 ~ ) ~ (+PPzI - 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 i n v a ~ a n t meaning of the PEcomponents is supplied by

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

-

- M1"

mm"KZ$- M,% M, BM,

P -

+

P

(4-5.88)

I

Under thew conditions, the value of the intqraf ( 6 5 . 6 5 ) 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 sI

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.

The effective replacement for P1 (p1

(4-5.74)

+ Pz - 2k in (4-8.63) is of the form

+ pz - 2k) = (p1 + pz)a +

(p1

- Pz)S.

(4-5.76)

+ P2 and P1 - PE, rbpectively, implies the equations K Z + Mls + MzZ + 2mz = (Ke + 2MI2 + 2Mz2)a+ (M1' - M3Z)p,

Multiplication by Pl

- Mz9a + P P .

01:- ( M l g

(46.76)

The result is

where K'+M1'+Me8+2me

1 1+x K81 + 2 x + x V S

3 -

A

(46.78)

We also note that 2KZ

+ M I Z+ 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 ) ) .

We now observe that

(4-8.80)

4-5

Form f a m m 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 m i t e the space-time exbaplalion of (4-5.86) as the dsubfe spectral form

W

Blscfrdmanzlirs 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

-1

M 2- m2

+ 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 m d 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 -k2 4mf4m%

+

+

1 1 (M. 96) ( k g p B i T )2 '

In addition, we shall write

and omit the l/k2term 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/4mPas 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(M3 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

4-6 FORM FACTORS Ill. SPIN

89

4

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

retaining only the field dependent part of,the source:

- E)YJl(x)+

iJ"(€)q(~)lctr. = eq

,

(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, 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

W.8)

90

Chap. 4

Eterodrynamics f

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 X , the ambiguous imaginary part of the logarithm implies ambiguous real terms in crfM). The undefined discontinuity at M == .Mo tells us t o call a halt. For nonrelativistic values of k , the farm factor ( 4 7 . 9 7 ) 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 extrapofation procedure.

Saaerlng of light by light l, Low frquenc9m

4-8

$23

4-4 SCATTERtNC OF LIGHT ELY LIGHT I. LOW FREQUENCIES We have been discussing multiparticle exchange effects that modify skeletal interactians, 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 p~sitronsource :

a,

combination of two photon fields as an effective. electron-

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 1AIA 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 t e r n 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~~AG$

and its formal solution [Eq. (3-12.22)] G: = (1

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

(4-8.21)

This gives 6W(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,)

=

- +iTr 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) is 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) - e2A2)d,].

(4-8 3%)

In particular, W,, = - i& Tr[cqfPA

+ Apjd.,.j4-+ i$ Tr' (eqjpA + 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 - 694

+

where E -+ O is implicit in the integral as a convergence factor, exp(Now the differential expression (&8,3t) reads

and

CS).

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(-

(4-8.37)

ieqA)

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 ma or equF me to a particle Hamiltonian, and formal similarity of 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

[D,,

(29

(-.W

= 2icqFCav,

and therefore

(H.w

I!&) = exp(isDe)D, exp(- islI3 obeys the equation of motion

The solution is, in matrix notation, D(s) = exp(2qFs)lZ = D exp(- 2eqFs),

(4-84

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 ' [ ~ ~exp(~ I I ~i s P ) ] = TrP[& exp(- isR7 &(S)]

- isfl)].

= Tr'[&(s)n, expf

(44.49)

An equivalent form is

Tu,= Tr'[llPnv(s)exp(- i s P ) ] - Trt[[&, K(s)] exp(- islIe)]. (4-8.88) The commutator that appears here is evaluated as

14, nVwl = rn,,nArexp(- ~ ~ ~ I ; S ) I ~ , ~M I exp(F ~ F ~ I I , ,~4-8.51) . and a return to matrix notation gives

T = T exp(- 2eqFs) - iepF exp(- 2eqFsf Tr8[exp(- i s m ] , (443.62) or

We vse this result to evaluate

d ds

i -Tf[exp(- is17131

--

Tr'[P expf- i s P ) ] 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~sideringthe 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 computed as

+ X)-dimensional momentum space is

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) :

Scaaaring of light by lfght I, Low trquanclss

4-43

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 ) = (ErJ1: 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 \ " m

sFfs sin @F'ssin eF"s

2(@$)%@ cos(,~~2) -

--

where we have .finally written just % . ? in place of X-. Here, then, is the spin 0 result:

cos(essyP)

Ilmrotfynamficlr C

132

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 ( k 8 . W ) and use the determinantal expansion

This gives

and or2 1 80 M"

= -- ( 7 F 2

spin O :

The corresponding spin

4

+- 9%)

result is produced by inseding in the integrand of X 4 dimensionaf charge-"pin space:

9 the following trace over the 2

where the substitution is -+s has already been made. The algebraic properties of the spin matrices are such that

‘H@KA~

=~

@fiJ

- gxvgifi - ~ X A B V Y ~ *

K @ ~ A V

(4-8.75)

Therefore,

and the eigenvaltxes of crF are

(oF)' = &

i ~ y+ ,iXy.

(&8,77)

That gives

- g trtctleoshfes~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~grangefunctions. 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 2m, The weigllt factor a(M2j in the structure sf the modified photon propagation function [Eq, (4-3.82)] is inferred as I=.

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 9with 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 couplinf3sthat 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

ScaeesrOng of light By tight #I, IFo~wwdrcaetering

4-9

$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 2A2rthat describe particle creation or annihilation through the combined efforts of two sin~plephoton 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. ( k 8 . 3 3 ) . 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,

(P'

- kg)"

2 +g

ezpez'9" (p' - kg)%+ ma -

I

3 tCC

t f ~ r d y n a m 8 c sI

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 a(M2) = f

-

dz -1

[l -- (4n%jMg)](1 -- 9) - =M%)]z%

a ---vs + vlog-

'vB l-v

(4-9.11)

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 strueture 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~lanizationvector 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%

Chap, 4

Ilwtrodynamlcs 1

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 (dz")d,(x

1

--- 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)S4becomes 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, = I

P; = I

e l 0 e 2= e I e o , when (&f).lO) becomes

A related field structure is

=22

c, e; = O, 1,

Scrttrkrrlng oC l l g k by light 11. Forward m m r f n g

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 g 2 ( x 7 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

is

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." i s

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 8and ) $ , ( x ) 9 , ( x ' ) , Xt is necessary to use the tensor

F2 Now,

= Fg

(X)

F$

(X).

(k9.31)

where, omitting the compensating & i factors, we have

Evaluated in the center of mass frame, and restricted t o 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 ) S z ( 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 )- 2 g 1( 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 t h e farm

so that the relevant term of (4-9.39) reads

Perl~apsit 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 photonphoton scattering plienornena with eitt~erof 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 ~ Fand 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 cornmuting one factor from left to right in the trace. T h e outcome of (be calculation i s expressed by

where .

We a j s ~observe that

Sattarittg of tight by Hght 11, Forward mttertng

4 9

$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 coeffieients 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, e, e;e, * P,

-- e,

r

i'

eZeI* ep C

I

2(e, e,e, e,

+ cl

I

I

e 2 ~ 1ep = el

-- c, e;n,

P,) =

a

*

ei + (el

(c, c;@,* e;

+ e,

X Q;) (aI I

*

X

ej.

I

o,e, e, g

The first of t h e e will be recognized as the polarization vector stmcture of

P 1 ( ~ ) P 2+ ( ~+l(x)%z(x"), 3 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 t e r n that awids a low frequency cantribution from this coupling. The result is

144

EIerodynamlu t

Chap. 4

which has the equivalent form ~ W Ospin P 0, i-

w01,spin f

4-10

(p(k)s(k)

+ g(- k)@(k))

SCAmERING O F 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, Abis 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 A I , 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 f o m

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 i p , 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 r m e s in (610.6) refers entirely to the delta functions. One can easily state the value of this i n v a ~ a nintegral t in the spcial coordinate systern where k,, Ir, occupy the xy-plme and k, coincides with the x-axis,

$41)

ECmf@dynanrfeot

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 (4-10.1l),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 ):

= kzkf - (k,R,)a

- -1 [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 v e c t o ~ .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 photonphoton collision are convenientf y chown as

where the second version refers to the real photon situation, Some quantities derived for this circ~xmstaneeare

where we have used k; =

- (k, + k, +

to evaluate 2Rak,

-+

M: + M!.

-.Q

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),

- A = det

0

- #M:

- +M:

0

- #M:

+ M:)

-M

f (M:

, (4-10.26)

0

- W:

,M:

0

+ M:) - 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 J 1 + Jb, and Je Jb exchange a pair of particles with J 1 J,. We shall make these momenta K , and Kb explicit by introducing the factor

+

+

l

-

- kz - ka)d(Kb kz - kb) (dKa)(dKb)d(Ka

+

h & A n g of tlglrt by tight ill, hubre s

4-1 0

I t is combined with the Pot& momentum delta funetion,

The causal significance of this farm becomes apparent on cansidering the structure

=!=

-

d ~ d : ~ (dx) ; (dt,) (deb)Al ( x ) A,(x 2n 2z P -

- 5,)A,(x - F ~ ) A z (x t a - [a)

According to the causal arrangement of the fields, the vectors 5, and 6, are timelike, with positive time components, We recognize: the appmpriate causal t o m s 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

(k, + k.)"

1

1.

Ma2 X

(kp

+ k,)' + M,'

- ie

.

(6110.36)

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

$!# El f~radmmrics 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 caileulation" 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 fam

+- PE?],

ds sS expf-- is(@%

(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~gexplicit the gauge inr~ariancethat does fiold in the causal arrangement, in order to preserve it after space-time extrapolation. Since our interest in photonpllotoll sratteri~lgis primarily didactic, we sf~aXtavoid tlie relativeIy unrewarding complexities of ar1;itrary polarizations by considering only the simplest polarization assignment. It is tlte one in wl~iciia11 po1arim"tion vectors are parallel to each other, and perpendicular to t l ~ eplane of scattering. In that circumstance the vector potential factors a i (4--10.6)beconlc

wllere the various fui~ctionsare the si~tglenonvaltislring vector components, those perpendicular to the scattering plane. To Itttroduce ficXd strengths, let us consider tlxe product

wl~erekfAA(k)and k A ( k 7 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

1 s

Elaetrdparnlcs I

Chap. 4

h= been perfamed. The three field stmetures in. (&f0,48) can. be expressed gexreraBy as the p r d u c t AI * * A%,multiplied by momentum factors which arc., respective1y,

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

(k'

(k2/M2)(kfg/Me2)

+ 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: (2~14 Sfk

dM@% $M,%

+ + KU')")A (K) *

(k"

A (K""")

* *

"

kf'""")2

(K" kk'y iE(k" + k"')8 + Mba

M?)(K' + km')% +

ie

where

We have elected to use the most symmelricd of the gauge invariant repreentatians 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 foltewing effective substitution,

(R' (k"

+ km""%

kk""")2 M@% - kc: (k"

" "

(k" k@#')% kK""")2 +&a - it.

which is verified by comparing the three terms produced on each side by symmetrizzrtion 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

=.iv+#a(l-v?+fii-v2)(#fl-v%)-1)1og-.

Performing the necessary integration, we deduce that

(zt

f.+v

l-v

(P-10.67)

= 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

+

- 2[(2pJ4Al

-44% (2p,)Tl

+ Tell

(4-10.60)

in which Ti =

2 AA4 tr UFUF

m

A iAa(F2Fb)

+ A2Ab(F,Fa) + A 1Ab(F2Fa)

+ A2“4a(F1Fb)+ A 1AdFaFb) + AaAb(F1F2)

(4-10.61)

"c)

g

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 T 1 is S

I!

El

2 2

'"d

A2 = 0 ,

P4

'CC

&

a

E

+ (k, + kb)yA1

+ S +

Sr,

41" -lCJ 25 -I-

% .

while

2

m

-t

I

U

II

S

- [(KZ + k J 2 + (k2 +

c T1

(4-10.63)

G'a

'CC

CI g

ccl

S

m

m-

-

"&

Tij

"c)

8

Q

Accordingly, the weight factor of the double spectral form is derived from that of spin 0 by the substitution

S

W

)

# . "

+

and spin 4: $2(M,e, Mbs) = +(U,

+ -.- l)"2 + ub - l)-1l2 - (1 - ua)(l(2--ub)(ua U, -u,)~ Ub

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)*~ 2 4 1 ( W ) , ~ o p

+ 2$2(MaStMbS)spinO

(6'2(Ma2*MbB) , +2(Ma8, MbE)*,

a

+ - l)-ll'e

4(~e

ub

(4-10.67) 4

8 41

.M "-4

.M

"c)

cc1

cl*

9

T3

n

i;?"

l

u.?

-4'

Qls

l+v + + 2(a + qb) = c(W)= V1 log l-v .

L- 2(a + 46) cl

v

8

E:

v

U

ri

2

.Clr

-0

23 B

C4

C,

5

h0

G

&l

Q;,

ctJ

*.'

J-,

8

!

0

8

P

c@ F:

Q,

(d

9

and the analogous forward scattering amplitude combination [cf. Eq. (4-9.53)] (4-10.68)

haeriing of tfghr by light ll@. Double spectral forms

4-5 0

t55

Performing the intevation in

we get

and therefore

(Mg) = v + (3 --- 2 ~ 2 )

spin :

(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 ~ xelement 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+@

while at high energies, M angb.

=;;

+(3

+- cos2@),

(4-li0.76)

m, there! is a logarithmic variation with scattering

l36

R l r s c t r d ~ a n l aI

Chap. 4

4-11 H-PARTICLE ENERGY IDCSPLACEMENTS. WON RELATIVISTIC DISCUSSIQN

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 instmctive 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 w b i t r a ~ l ylarge energy, we reach the nonrelativistic form of the propagation function : --.

G(r -- r', t -- 8')

- #')l A,(% 0 exp{i[p (r -- r') - T(p)(t- 8'));). ( 2 ~ ) ~

= Lim(-- gm) exp[im(x0

-

X')

This retarded function is a Green" function of the inhomogeneous Schradinger equation,

The n~ontentumspace version of this equation is

i -aat

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 ( 6 1 1 . 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

Chrp. 4

(4-11.12) The implied field equations are

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

- rt, t - tl)q(rl,tr).

(P11.14)

One uses this solution to find the explicit source dependence of W,

W q * ,1) =

-

=

-

S

(dr)dt q*(r, t)$(r, t )

1

(dr)dt (drl)dtl q*(r, t)G(r - rl, t - t')q(r6,tl). ( 4 1 1 . 1 6 )

The minus sign that appears here, in contrast to the relativistic forms, reflects the sign factor contained in the relation of (4-11.3) ; 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

(4-11.19) and W(?*. ?) =

- j -?*(P.

E)C(p. E)?(p, E).

in which I('

1

E)

5

e x P ( i ~ tr)( ) .t).

v*(

.E)

5

= dt exp(- iEt) v*( ,t). ( 6 11.21)

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 I)

c=j

&--. X).

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

5

t > t' : D(r - r', t - t') = i dm, exp(i[lr (r - r')

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

Chap. 4

Elutrodynamio I

l. Then the photon propagation function can be simsystem, [k. (r - r')l plified to (a factor of c8 is also supplied)

t > t' : 8D(t

- t')

"I

-

i - dRo R0 expt- ikO(t tf)Jl. 332

(4-1 1.26)

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

-

I

c (dr) dt #*p,l ) p A(r, t)$(r, t), m

p = -V ,

(4-11.30)

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), t) Jl(rf. tl)ldf.

-

C

#*,(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

(dr)dl (dr" ddt" qb*,fr, t)p D(r - r f ,t - b")E(r - r" 6 - 8') * pJIZ(r; if)).

8%

=--,.p

G

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

3n m"

dt dl' $*,(p, 1)

l

+

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

+GVG~,

(4-1 1.M)

- CV)-lG = G + GVG + . .

(4-11.41)

=G

which has the formal solution GV

5

(1

A more explicit statement of these first terms is given by

- pl)G(pft - P ) + j dtl G(p,t - t t ) v ( p- p f ) X G(p', t1 - 1') + ,

GV(p,1; p', t*)= 6(p

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). It 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[-

+

I d 1u p [ - ~ ( T ( P ) WoV

-

iT(p)(t tl)]V(p

- p') exp[-

- 4)1 V $ - P') up[--

i(T0')

iT(p1)(t1 - t')]

+ 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

8-24

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 ~ xnotation, 1

E+&--T-

l

E-T

k@+k~'(ko)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~ctrodynamicrI

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~espondinglyredefined by $(X) = [n(-o)T1'v(r)*

4 = [Jdnao)TuS~(r,P ) , %(X*

(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 xlo(x. S) = exp(- X) - 2s X

+ s)/(l - s)]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 @ nwith 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)%

(4-1f .S7)

On evaluating it at tke oridn we get 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 B y ~ a m i c s , U". A. Benjamin, Inc., Mento Park, 1970. The details can be found in the Harvard, 1887, thesis of Michael Liehr. See also Pkys. RIV. "The

174, 2037 (1968).

and therefore ~ n o ( O 8S )

==

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. which again involves only negative terns, is in;&cated effect associated with bV@~,

'&Y 2.W3

+ 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.8233by 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 clyt 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 understanding 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 (4-41,52), which converts (4-lX.f)f, 92) into

There is also the magnetic moment effect displayed in ( H . 1 2 4 ) , 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

and

-- V)$ =. fy5e' v + my@)tf(rt

(&lf*lOO)

+*v' - v+*.

fkltr,ror)

$*M 4- E - 'C")

$*(yJcre

G

+snyof,

These equations are combined in

and, using a, nonrefativistie approximation that neglects E we express the energy displacement (4-11.99) as

---

V relative to m,

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 t o "re dominant logarithm, In states of nonzero orbital a n p l a r momentum and a @venEtotaIangular 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:

Ez,,,,

- E,,,,

l = -- Z4aSRy 3%

f. 5 - %.SIX8 + -

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&Ovisticdiscussion

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 v o e e , 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 cfield the additional mqnetic moment of or/f?n:magnetons was obtained. An inhomogeneous electric field gave just the results that are stated in (4-11.98), 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 contdbutions. By this time (1948) other groups, variously using noncovariant and cov a ~ a nmethads, t 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~antcdculation 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 where t > t':

I

G(r

(dr)

dt' q*(r, t)G(r - r', t

- r', t - t') = - iz b ( r , t)$#, P

Using the source definitions p

=

( )

- t')rl(r',

$

1,

*

t'),

(4-11.116)

t')*,

I

q: = (d.)dt v*@,t)&,(r,t). (611.117)

we find that the vacuum amplitude (4-11.115) is (- iqi)(- q ' P)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 (4-11.41), which gives the persistence probability amplitude 1

J

- i (dr) . dt' $,(r, t)*[V(r) d(r - r') d(t - t') + V(r)G(r - r', t - tt)V(r')]$,(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 E + i e - T(p')E

+

1 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 essentiallyelastic scattering, as characterized by

where R:,, represents the accuracy with which scattered particle energies can be measured. Under the assumption that k&, > m s : %(M%)=l.

(4-13.22)

The c a d transition matrix element is

Its space-time extrapolation proceeds by writing ( 2 ~ ) ~ 8+@p 11

- k2 -p2) - X')]]

.rup[iP(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,

klw-kz,

(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~mitiveinteraction, 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 confipration 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~iclestate is (in simpEfied notation)

where V measures the four-dimensionalinteraction volume. The impliedpersistence 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 (4-13.22) are indeed those stated in Eq. (3-1 2.1M), 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 M 2integration 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--ticEerest 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 undeflected, 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%

Chap. 4

Eloctrodynrmler I

is simplified on specializingto the forward scattering circumstancethat is expressed covariantly by

p

-

(

l

)

4mt

(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),

tr=

M2 - 2m2 M2 - $m2 2 -log -- M 2 - 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 ( 4 t 3 . 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 K g , 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 p a ~ i c l e af s 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"

+ P, - k = PI - k - K,.

.K2

.,

(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 nvacuum amplitude

This is explicitly disp;layed a

Photac~bm particle sattarin#

199

where

I.,

-I

(M) d(k'

X

+p? &(P1-h)' + m2)d((P1- k - KO)' + m? d((PO- k)' +m%) 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 ~ , this refers to the deflection of a particle with given field A," is (p' ~ 5 " ) since 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*, - 2kKa = MMf - W ,

- 2kPa = MS4- m,' - 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, - 2K2P, = M" WM2'- M,' + I 0

C'(-

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

2K,P1,

- 2KzP,

+

Chap. 4

M2 - m2;

21 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 )=

Here, n,

- (do,, e)'/28nael*

*

e,

+ (dw,,

*

0)"'

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 irnmediately 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 ~(4-13.193) 8). The conclusion is that

In the particular situation of back scattering this becomes

while the mdificatian in, the total cross section is log

m

1 - $13- 3

(4-13.196)

which can be an increase or a decreae, depending upon the quantitative relation between ko[kti, 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 r m e t h d 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 i n e and energy were expnded on one such calculation of the 2% r r r ~ f i c a t i o nto 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 m e t h d s 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~lloperator 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 f o m , but that of the photon has been specialized to its causal version. f"Ve shdl remove this rest~ctionthrough the replacement

and 4%be in causat relationship. (-- ze is omitted) and no Xonger insist that 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,

- U),

= su,

s2 = s(1

X(%)

= u [ ( n- k)%

(4-14.9)

dsl ds2 = S d~ d~

produces the form

where

+ 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.

- 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 r n i s included], k

---.

according to the momentum integral (4-8.57). I t is convenient t o carry out a partial integration on "U:

The first term on the right contributes, at 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 t o the substituGion

This yields [for simplicity, p2 is set equal t o zero in the numerator] I

n2+ fm2/1 -

"U)

4%. (614.17)

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.tlmKinemtics 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

we write

="

XO -k XIS

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 &X)]], in which the scalar product of the il! vectors is understood, Consider, for that purpse, the transfomation

m,

[IT, exp(-

where A is an arbitrary constant vector, We shall compare the expansions of both sides, based upan the general commutator expansion

edBe-A = B

+ [ A ,B]+ 1 [ A , [A,B]] +

(614.29)

*,

which e m be ve~fiedthrough successive differentiations of a scale parameter in A .

Thus, up to terns quadratic in A, we have

[m,exp(- kg)] + *[A@, &[m, %] - isifm,[;l.ZTI %]]).

@V(-isg) 3= =p(-

--.

expf

The expansion of Eq. (614.26) presents the ~ght-handside as

- is^)]] (4-1 4.30)

Chap, 4

Etccrrodynamiest

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[--

x expi--- isf(1 - v ) / 2 J w ~ ) ,

In

- ~1x1

(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 -+k cannot be made owing to the nsncommutativity of the substitution K components of 17 in the presence of an eleelronna~eticfield. 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 complementary to k, and write

6 be

the (four-vector) caordinate

The advanlt~eoffered 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 4k through an oprator transfomatisn. Indeed, when the field vanishes and the eompanents of .l2" coxnnnuk, we have exp(- z w f D J / ( h - uf1)exp(ig6fl) = 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-uE

(4-14.44)

and

d -fi = - q F ( x - U€)€, du

(4-14.46)

which is integrated to give

Integration of the differential equation for & then produces

k = k + ul7 - eq

We also note the combinations

du" F(x

- U"€)€

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 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 t e r n 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

(nB +

When the transformation signaled by accuracy X(%)

k X -j- ~ ( --f

-+

@DZ+ 112211, 11 - k

is performed, we have, to the desired -+

( 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 k operator structure

et

+

expf - &[(l v)/2]k2)e ~ ( -

x exp(- isi(I

- v)/2lP6) exp(--

i@[(lf

~)/qPt)

- 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 translations 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 specializationof particle fields indicated by 17%-+ - mVis introduced, we get for the last, tern of (4-14.8),

Chap. 4

Etmrdyrramfcs L

Note that, with p = 0,an infrared sinpliaity a p p a r s 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, m d annuls the ratationally invariant states ( E ' = q, That restricts attention to the last t e r n of (4-14.59), where we encounter

+

expf - &[(l v)/2)k2)expf---izc'pt) f: exp(- is[(l - v)/qkZ)

--P

1 - v2

t2szct p expf- iskzf exp(- isd2[jl 4

After the real particle specialization in l; exhibited, we have

flr;12 --+

- ~a)/4]fl~)~

(&l4.61>

- mz),and with only the term hnear

(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 ifbrand +2, the following :

in which, we have introduced the permissible simplification ( p