richard p. feynman-quantum electrodynamics-westview press(1998)

ADVANCED BOOK CLASSICS David Pines, Series Editor

Anderson, P. W., Baric Notions of Condensed Matter Physics Bethe H. and Jackiw, R., Intermediate Q u a n ~ m Mechanics, Third Edition

Feynman, R., Photon-Hadron Interactions

Feynman, R., Quantum Electrodynamics

Feynman, R., Statistical Mechanics

Feynman, R., The Theory of Ftrndamenral Processes

Norieres, P*, Theory of Interacting Fermi System

Pines, D., The Many-Body Problem

Quigg, C., Gauge Theories of the Strong, Weak, and Electromagnetic Interactions

RICHARD FEYNMAN late, California Institute of Ethnology

Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book and Perseus Publishing was aware of a trademark claim, the designatians have been printed in initial capital letters.

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Cover design by Suzanne Heiser

Editor's Foreword

Addison- Wesley's Frontiers in Physics series has, since 1 96 1, made it possible for leading physicists to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics-without having to devote the time and energy required to prepare a formal review or monograph. Indeed, throughout its nearly forty-year existence, the series has emphasized informality in both style and content, as well as pedagogical clari- ty. Over time, it was expected that these informal accounts would be replaced by more formal counterparts-textbooks or monographs-as the cutting-edge topics they treated gradually became integrated into the body of physics knowl- edge 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 Frmliers in Physics or its sister series, Lecture Notes and Suppkments 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.

These lecture notes on Richard Ft;ynnran8s Caltech course on Quantum Electrodynamics were first published in 1961, as part of the first group of lecture notelreprint volumes to be included in the Frontiers in Physics series. As is the case with all of the Feynman lecture note volumes, the presentation in this work reflects his deep physical insight, the freshness and originality of his approach to quantum electrodynamics, and the overall pedagogical wizardry of Richard Feynman. Taken together with the reprints included here of

vi E D I T O R " S F O R E W Q R D

Feynman's seminal papers on the space-time approach to quantum electrodynamics and the theon, of positrons, the lecture notes provide beginning students and experienced researchers alike with an invaluable introduction to quantum electrodynamics and to Feynman's highly original approach to the topic.

Bavid Pines Idrbana, Elf inois December 2 997

Preface

The text material herein constitutes notes on the third of a three-semester course in quantum mechanics given at the California Institute of Technology in 1953. Actually, some questions involving the interaction of light and matter were discussed during the preceding semester. These are also included, as the first six lectures. The relativistic theory begins in the seventh lecture.

The aim was to present the main results and calculational procedures of quantum electrodynamics in as simple and straightfaward a way as possible. Many of the students working for degrees in experimental physics did not intend to take more advanced graduate courses in theoretical physics. The course was designed with their needs in mind. It was hoped that they would learn how one obtains the various cross sections for photon processes which are so important in the design of high-energy experiments, such as with the synchrotron at Cal Tech. For this reason little attention is given to many aspects of quantum electrodynamics which would be of use for theoretical physicists tackling the more complicated problems of the interaction of pions and nucleons. That is, the relations among the many different formulations of quantum electrodynamics, including operator representations of fields, explicit discussion of properties of the S matrix, etc., are not included. These were available in a more advanced course in quantum field theory. Nevertheless, this course is complete in itself, in much the way that a course dealing with Newton's laws can be a complete discussion of mechanics in a physical sense although topics such as least action or Hamilton's equations are omitted.

The attempt to teach elementary quantum mechanics and quantum electrodynamics together in just one year was an experiment. It was based on the

viii P R E F A C E

idea that, as new fields of physics are opened up, students must work their way further back, to earlier stages of the educational program. The first two terms were the usual quantum mechanical course using Schiff (McGraw-Hill) as a main reference (omitting Chapters X, XII, XIII, and XIV, relating to quantum electrodynamics). However, in order to ease the transition to the latter part of the course, the theory of propagation and potential scattering was developed in detail in the way outlined in Eqs. 15-3 to 15-5. One other unusual point was made, namely, that the nonrelativistic Pauli equation could be written as on page 6 of the notes.

The experiment was unsuccessful. The total material was too much for one year, and much of the material in these notes is now given after a full year graduate course in quantum mechanics.

The notes were originally taken by A. R. Hibbs. They have been edited and corrected by H. T. Yura and E. R. Huggins.

R. R FEWMAN Pasadena, California

November 196 1

The publisher wishes to acknowledge the assistance of the American Physical Society in the preparation of this volume, specifically their permission to reprint the three articles from the Physical Review.

Contents

Editor's Foreword Preface

Interaction of Light with Matter-auantum Electrodynamics Discussion of Fermi" mehod Laws of Quantum electrodynamics

RCsumC of the Principles and Results of Special Relativity Solution of the Maxwell equation in empq space Relativistic partide mechanics

Ref a t i~s t i c Wave Equation Units Ktein-Gcrrdon, Pauli, and Dirac equations Alpbra of the y matrices kuivalence tramformation Relativistic invariance Hamiltonian form of the Dirac equation Nonrelativistic approximation to the Dirac equation

Solution of rhe Dirac huation for a Free Particle Defirtieion of the spin of a moving elecrron Norrnalizatian af the wave functions Methods of obtaining matrix elements Intepretation of negative energy states

Poten~al Problems Itn, Quantum Eleetradynamics Pair creation and annihilation Consewation of energy The propagation kernel Use of the kernel K, ( 2 , l ) Transition probablility Scattering af an electron from a coulomb potenrial Galccllation of the propagation kernel for a free particle Momentum repreenration

C O N T E N T S

Relativistic Treatment of the Interaction of Particles with L i h t Radiation from atoms Scattering of gamma rays by atomic electrons Digression on the density of final states Cornpton radiation Two-photm pair annihilation Positron annihilation from rest Brernsstrahlung Pair production A method of summing matrix elemenrs over spin states E&cts of screening of the coulomb fieid in atoms

Interacdon of Several Electron Derivation of the "mules" of quantum electrodynamics Electron-electron scattering

Discussion and Interprebtion of hr ious ""Coxrecdon" Terms Electroncelectron interaction Electron-positron interaction Positronium fro-photon exchange between eiectrons andlor positrons SeXfeenergy of the electron Method d integration of integrals appearing in

quantum electrodynamics Self-energy integral with an external potential Scattering in an ex ternal potential ResoXut ion of the fictitious "inbred catastrophe" Anocher vproaclx to the infared di&uXcy Egect on an atomic electron Chsed-loop processes, vacuum polarization Scattering of light by a potential

Padi Principle and the Dirac Equation

Replcints Summary of Numerical Factors for Transition

Probabilities, Phys. Rev,, 84, 123 (1951) The Theory of Positrons. Phys. Rev., 76, 749-759 (1949) S p a c e - x ~ Approach to @anturn Electrodynamics.

Phys. Rev., 76,169-189 (1949)

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Interaction of Light ith Matter-

The theory of interaction of light with matter i s called wanturn electrodynmics. The subject i s made to apmar more difficult &m it actually i s by the very many equivalent methods by which it may be formulated, One af the simplest is &at of Fermi. We shall take another starting point by just postzzlating for the emission o r absorption of photons. In this form it i s most immediately appXicabXe.

Suppose a11 the atoms of itbe udverse a re in a, box. Classically the box may be treated a s haviw natural modes de~cribable in terms of a distribution o-f: harmonic oscillatars with coupling between the oscillators and matter.

The trarxsfUan to wanturn electrodynmics involves merely the assump- tion that the oscifladars a re quantum m e c h a ~ e a l instead cif classieal, They then have energies (a +. X / 2 ) b , a = 0, 1 .. ,, with zera-point ene rw 112fiw. The box is considered to be full of phstom with a~ distribution of energies &W. The interaction of photom with: matter causeha the number of photons of' tym n ta increase by & l [emission or absorption).

Waves in a box can be repressxrted a s plane e tad iag waves, spherical, waves, or plane rmning wavtsa exp (iK * X$. One can say there i~ an instan-

t Revs. Modern Phys ., 4, 87 (1932).

4 QUANTUM E LECTRODfxiNAMfCS

tanems Coulomb interaction e2/r l j between all charges plus transverse w v e s only. m e n the Coulomb forcea may be put into the Schr6diwer equation directly. a e r f o m a l means of e a r e s s ion a r e M in Hamiltonian form, field opra tors , etc.

Fermi's technique leads to an infinite self-energy term e2 / r f i . It i s poa- sibie to eliminate this term in suiable caarr3inate systems but then the transverse waves contri bute czn idtni ty (interpretaaon more obscure). This mom- aly was one of the central problem8 of modern q?xmtm eleclr&pamics,

Second Lectuw

U W 8 OF QUANTUM EUCTRODPHAmC8

Without justification at this time the "laws of q u m t m ele,ectrdynamics9" will be stated as follows:

1, The amplitude that m atomic system will absorb a photon dur iw the process of transition from one state to another 18 emctly the same as the amplitude that the same transition will be made under the influence of a p- tential equal to that af a classical e lec t romwet ic wave representing that photon, provided: (a) the classical wave i s normalized ta represent an energy density equal t a b times the probabilty per cubic centinneter of finding the photon; @) the real classical wave is split into two complex waves e - hU" and e* ' w t , and only the e- ""tart is kept; md (c) the potential acts only once in mrturbatlon; that is, only terms to first order in the electromagnetic fietd strength should be retained.

mplacing the word "absorbedM by 'kmit? ' in rule X r s q d r e s only that the wave represented by exp (+iut) be kept instead of exp (-iwt).

2. Tbe n u m b r of states ava i lh ie p r cubic centimeter of a given po2ar- izatlon Is

Note this ia exactly the same as the number of normal modes per cubic cen- tirneter in classical theory.

3, Photons obey Boee-Einstein atatistlcs, That is, the states of a csllese- tion of identical photons must be symmetric (exchmge photons, add m p l i - tudes). Also the statistical w e i e t of a state of n identical photons i s l instead of the elassieal a!

Thus, in general, a photon may be represenbd by a solution of the classical Maxwell equations if properly normalized.

Although many forms of expression a re p s s i b l e i t is most convenient to describe the electromagnetic 8eld in terms of plane waves. A plane wave can always be represented by a vector potential only (scalar potential made zlero by suitable gauge transformation). The vector potential representing a real classical wave i s talqen as

I N T E R A C T I O N QF LIGHT W I T H MATTER

A ZZ a e cos (ut - f?;*x)

We want the nctrmalizatian of A to correspond to unit probability per eu- bic centinneter of findiw the photon. Therefore the average e n e r a density should be 60.

Now

IFf = (l/e) (&A/ a t) = (U a/e) s sin. (w t - K; * X )

for a plme wave, Therefore the averscge energ;y density is equal to

( 1 / 8 r ) f l ~ 1 ~ + 1 ~ 1 ' ) = (1/4a)(w'a2/c') sin3(wt - K*@

Setting this equal to t-icc: we find that

Thus

- - er f m p f-ilut - K * xll .c exp [+ilot - K * X ) 1)

Hence we take the mpUtade LhisLt an atomic system will absorb a photon to be

For emission the vector poLentiaL is the s m e except for a positive exponential.

Example: Suppose an atom i s in an excited s t a b \ti with energy Et and m h e s a transition, to a final state Jif with energy Er , The probability of transition p r second i s the same as the probability of transition mde r the Influence of a vector potential a8 expf-t-ifwt - K-X)] represent i~g the ernit- ted photon. Aecasding to the laws of quantum mechanics (Fermi's golden rule)

Trans. prob./sec = 2n/A /f(potential)i l2 * (density of states)

Density of states =

6 Q U A N T U M E L E C T R Q D U H A M I C S

The matrix element Uf i = /f@otential)f/2 is to be computed from perturbation theory. This is explained in more detail in the next lectws. First, bowever, we shall note that more tjhm one choice for the potential may @v@ the s m e physical results. (This is to jusWfy the possibi&t;y of always chws- in@ Q, = 0 for our photon,)

Tjzzz'rd Lecture

The representation of the plme-wave photon by the potantials

i s essentially a choice of 6"gauge." The fact that a freedom of choice edats results from the iwas imce of the Pauli equation to the; q u m t m - m e c h d c a l gauge t r m f o r m .

The quantum-mechmical trmsformatfon, is a simple extension of the classical, where, if

and if X is my scalar, then the substitutions

leave E and B invariant, h quantum mechanics the additional transformation of the wave function

i s intrduced. The invariance of the Pauli equation i s shown as follows. The Pauli equation. i s

INTERACTION OF LIGHT WITH MATTER

The partial derivative with respect to time introducaes a term ( @ X / B t)Y and this may be included with 4e-jx @. Therefore the substitutions

leave the Paul1 equation mchawed. The vector potential A as defined for a photon enters the PauB Hmit-

tonian as a perturbation pohntial for a transition from s t a k i to @tale f. Any time-dewdent perhrbation wMeh can ba written

rersults in. the matrix sbment U,$ given by

This emression indicates that the prturb2ttrion h a the a m 8 eB@ct ae a time- indepndent prhrbation. U(x,y,z) beween initial and fha1 states whose an- ergies are, respectively, E ~ ~ w ~ and El. As i s well known? the most importa& contribution will come from the states such that Ef = Ef - wR.

Usim the previous results, the probabilly- of' a trmsritioxl per second is

f See, for exmplti3, L. D, Lznndau and E. M. Ufshitz, "@atnhm Meehan- tca; Non-Relativistic Theory," Addison-Wesley, R e A i q , Massachu- setts, 1958, Sec, 40.

Q U A N T U M E L E C T R Q D Y N A M I C S

To determine U f i , write

Because of the rule that f i e potential acts only once, which Is the s m e as requiring on& first-order terms to enter, the term in; A * A does not enter this problem. Making use of A = aa exp [-i (a t - XC X)] and the two operator relations

where K-@ = 0 (which follows from the choice of gauge and the Mmtvell equatiom), we may write

This result is exact, It c m be simplified by uai% the ajlo-calked '4dipolet" approximation. To derive tMs approximation consicler the term (e/2mc) (p .e ef ' " X ) , whiclr is Ehci or&r of the velocity of an electron in the atom, o r the current. The aponent can be exwaded.

K. x i s of the order ;ao/ic, where Q = dimension of the atam and h = wave- Xewth, I[f +/A<< I, all terms of higher ordfits than the first in ao/h msby be neglected. To complete the dipole appro~mation, it i s also necessary to neglect the last term, This is e ~ i l y done since the last term may be t&en as the order of fiK/mc) = (2i~c/rnc') * (mv2/2mc2). Although such a term i s negligible even this i s an overestimate, Wore correctly,

IPJTERAGTXON OF L I G H T W I T H M A T T E R

(efif/2mc)cr . (K x a) e+ iK " X v i e x [martsix element of

me matrix element i s

A g o d appro&xnatian allowa the separation

Then to the accuracy of this apprsximiation the integral is

l$ *W +I Uf*(a = (K x p))~ I d v o l = o

since the @taws are orlthwoml. For the present, the dipole approximation i s to be used. Then

e Prr * "f UZi =" -a--- - e m

Using o p r a t o r algebra, pf l /m = Rwff X f t , SO that

Pft dS1 = a2 [e2o'/(2n)'1(e xff )' da

where Xfi = l $r * X @$ d vol. The total probability i s obtained by inte- grading Ptf aver din, thus

e2u4 Total prob./sec = j a 2 7 (e lf )' dS2 (2x1

10 Q U A N T U M E L E C T R O D Y N A M I C S

me term s xfi i s resolved by noting ( F i g . 3-21

ixti el = lxfil sin. 8

FTG. 3-1

Substituting for 2,

4 e2 k3 Total prob./sec = 5 G 3 1 1 x ~ l 2

&eorptioa of lLig.ht, The amplitude to go from state k to stab Z in time T (Fig, 4-1) i s given from perturbation t;heoxty by

I H T E R A C T X O N OF LXGHT W I T H M A T T E R

where the time dependence of l& l (t] i s inacated by writing

U, (g) = ur, 8- i w t

(h accord with the mles of b c h r e 2, the a r p m e n t of the exponential ia minus and only terms which a r e lin,@m in the potential are included,) Using &is time dependence md p r fmming the intt;;grat;fon,

the transition probability i s given by

This is the probability that a photon of frequency w traveliw in direction (0, $B) will be, absorbed. The dependence on the photon direction i s contained in the matrix element ujk* Far exmple , see Eq, (4-1) far the directional dependence in, the dipole approximation.

If the incident radiation contains a r a w e of fremencies a d directians, that is, suppose

probabilit;y that a photon i s present with fre- ency w to w -t dw and in solid aqXe dsZ

about the direction f@,@ci3)

and the probability of aboorption of any photon travicsling in the ( B , @ ) direction is desired, i t is necesrJary to integrate aver all fr-equencisa, Tbis ah- sorption probability is

when T is large, the faotor a i n ? ~ ~ / z f i ) has an appreciable value only for Ew near El - Ek, ancl P(@, @,Q>) wf l l km substmtially eomtant over the small r a e in o which contributes to the integral so &at i t may be t&en out af the integral. Similarly far ulk, ao that

Trans. prob. = 2~(%)- ' lulr 1' $)dQ

where

12 Q U A N T U M E L E C T R O f ) ' Y N A M P C S

This can also be written in terms of the incident iatensity (energy crossing a unit area in unit time) by noting that

Usiw the dipole approximdion, in which

the total probability of absorption (per second) i s

It is evident that there is a relation btween the probability of sgontane- ous emission, with aceompaying atomic transition from etate 1 to atate k,

Probability of spontaneous = 2n(1i)-'(2nc)-~ j ~ ~ ~ / ~ ulk2

emission/sec

and the absorption of a photon MUI accompan9ng atomic transition from 8 t ak k to state l, Eq. @-l), although the initial and final states a r e reversed since /ulkf = i/uklf . This relation may be stated most simply in terms of the concept of the probability n(u, 8, (13) that a pmtieular photon state i s accupied, Since there are (2trepo2dw df2 photon stales in frequensy range diw and solid angle dBZ, the probability that there is some photon vvit-hin this range is

Ewreessiw the probability of absorption in terms of n(w, 8, $1,

Trans. prob./sec = 2n (~ )* lulk nfw,~,(13)(2nc)-~ wk12 d o

(4-4)

This equation may be indewreted a s follows. Since a(w,@,cp) is the probability thiit, a phdon state i s occupied, the remander of the h r m s af the rfght-hmd side must be the probhili ty per second that a jphoton in a& s t a b will be absorbed. Comparing Eq, (4-4) with. the rate of spontanems emission s h w ~ &at

I N T E R A C T I O N Q F LIGHT W I T H M A T T E R

R o b ./sec of absorptictn prob. /see of spontaneous of a photon f r m a state emission of a photon into (per photon in that stale) that state

h what follows, i t will be shown that Eq. (4-4) is correct even when there is a possibility of more &an. one photon per: state provided nfo,8,$) is t&en as the mean number of photons W r state.

Lf the initial state consists of tulo photons in the same photon state, i t will not be possible to a s t i nwi sh W1em md the statistical weight of the initial s t a b will be 1/2 ! However, the m p l l h d e for ab~orption will be Mice that for one photon. TaXcjng the statistical weight times the square of the mpU- tude for this groesss, the transition prabrtbility per second i s found to be twice that for only one photon per photon state. m e n there3. a r e Lhree No- tons per initial photon state md one is absorbed, the fo l l~Mng six proceases (shown on Fig. 4-22) can occur.

Any of the *ree incident photons rnw be absorbed and, in addition, there is the possibility that the photons which a r e not absorbed may be interchanged. The statistical wei&t of the initial state is 113 !, the statistical weight of the final state i s 2/21 , and the amplitude for the process is 6. Thus the t rmsi- tion probability i s (1/3 1)(1/2 1) (6)' = 3 times that if them were one photon p r idt ia t state, In general, the transition probability for n. photons per initial photon state is n times that for a s iw le photon per photon state, so Eq. (4-4) is correct if n(cr/,@,$yj) is t&en as the mean nurnbr of photons per state.

14 Q U A N T U M EILECTRODUMAMICS

A transition that results in the emission of a photon may be induced by incident radiation. Such a process (involviw one incident photon) could be indicated diagrammatieztlly, a s in Fig. 4-3.

One photon i s incident on the atom and two indlstinwiahable photons come off. The statistical weight of the final state I s 1/2f and the amplitude for the process i s 2, s o the probability of emission for this process is twice that of Eipontanems emission. For n incident photons the statistical weight of the, initial s t a b is f[nI, the statistical weight of the final s t a k i s l/(n + 1) f , and the a p f i t u d e for the process is (n + 1) f times the amplitude for spontaneous emission. The probability e r second) of emission is then n+ l times the probability of spontaneous emission. The n can be said to account for the iduced part of the transition rate, while the 1 is the spontaneous part of the transition rats,

Since the gotentials used in computi ng the transition probability have been normalized to om photon Wr cubic centimeter and the trmsition prob- ztbiiity depends on the: square of the amplimde of the potential, i t i s clear t b t when there a r e n photonsr p r photon state the eorreet transition probability for abearption wodd ba obhia& by aormaliziw the potential8 ta n pt.lotons per cubie centimeter [amplitude 6 times a s large). This i s the basis for the validity of the so-called semiclassical ~ e o r y of radiation. h 'that theory absorption is calculated as resultiw from the wrturbation by a potential normalized to the actual energy In the field, that is, to energy nEw if there a re n photons, The correct transition pr&&ilily for erni~sion i s not obtained this waly, however, because it is proportional to n + 1. The error corresponds to omitting the spontaneous part of the t ran~i t ion grab- ability. In the semiclassical theory of radiation, the spontmeous part of the emission probability i s arrived at by general arguments, ixzctuding the fact that its inclusion leads to the observed Planck distribution formula. Ein- stein first deduced these re lationshipa by sem iclassicsl reasoning.

INTERACTION OF LIGHT WITH MATTER

%XecLian Rule8 in the f)5pole Appr~~matton. In the dipole approamatlon the appropriate matrix element is

Tlxrs components of of X f r a r e xir, Ygr, zjf and

Selection rules a r e detemined by the conditions that cause tNs matrix element to vanish, For example, if in hydrogen the initial and final shks a re S stabs [spherically symmetrical), X+ = O and transitions btween these slates are "forbidden," For transitions from P to S sktes;, however, X i f f 0 and they a r e 6iallowed. p'

Xn general, fo r single electron transitions, the selection rule is

This may be seen from the fact that the coordinates X , y, and z a r e essentially the Legendre polynomial PI. If the o rb ik l angular momentum of the initial sLak is n, the wave function contains P,, But

Hence for the matrix element not ta vanish, the angular momentum of the final stale must be n r ~ . 1, SO that its wave function will conkin either l?,,$ o r P,."$.

For a complex atam (more than one electron), the &miltonian i s

H =z(1 /2m) P, - (e/c)A(x .)lg + Coulomb terms a

The tmnsition probability is proportional to [pm,/' = 15(pa),,1 where the sum i s over all the electrons of the atom. As k s b e n shown, (Pafmn is the same, up to a consant, a s ( x , ) ~ , , and the transition probability is proportional to

In particulm, for two electrons; the matrix element is

x 1 + x 2 behaves under rotation, of coordinat-ea similarly to Wlc? wave fmction of gome "objectY%iGh unit a w l a r momentum. If the "object7' and the abm

16 Q U A N T U M 1ELE:CTRODYbJAMICs

in the initial state do not interact, then the produet (X* + x2) iEi (xI,xZ) can be formally regarded as the wave bnction of a system (atom -t otsject) having possible values of Ji -c- l, J1 , and Jz - l for total m ~ l a r momentum. There- fore the matrix element i s nonzero only if Jf , the final a.nwlar momentum, has one of the three values JI * 1 or Ji . Hence the general selection rule AJ = a 1, O,

Parity, Pwitg is the prowrty of a wave function referring to i ts behaviar upon reflection of all coordinates, That is, if

parity is even; o r if

parity 18 odd. If" in the matrix elements involved in the dipole approximation one makes

the change of variable of integration x = --X\ the result is

If the pwidy of %cf i s the s m e m &at of -$!l, i t follows that

Hence tlze rule that p a ~ t y must chawe in allowed transitions. For a one- electron atom, L determines f&e gmity; tiherefore, L1 L = O would be forbidden. h mmy-electron atoms, L, does not determine the parity (determined by algebraic, not vector, sum of individual electron m p l a r momfsnta), so hL = O transition8 can occur. The a- 0 transktions a r e always forbidden, however, since a photon always carr ies a m u d t of mlf;ular momentum.

All wave functions have ei&er even or d d parity. This can be seen from the fact that the Hamiltonian (in the absenm of m external magnetic field) is invariant under the parity operation. Then, if EIJlr(x) = E\k(x), i t i s also true &at H@(-X) = f t ; J i ( - ~ ) . Therefore, if the state is nondegeneirab, i t bllows that e i a e r ?P f -X) = Jir (X) o r Q (-X) = -S [X). If the state is degen- erate, i t is possible that ?fr (-X) ;c {X) But then a complete solution would be one of the linear combinations

4? (X) + % ( -X ) even parity

% (X) - @(-X) odd parity

Forbzddelz Llirtcls, Forbidden. spectral lines mzty a p p a r in gases if they a re sufficiently rarefied, That is, forbiddenneess is not ablsolute in all cases. It may simply me= thzt& the lifetime of the state is much longer &an if i t

I N T E R A C T I O N O F L I G H T W I T H MATTER,

were allowed, but not infinite, Thus, if the colliaion rate i s small enough (collisions of the second kind ordbnarify cause de-excitation in forbidden cases), the forbidden trmsition may have sufficient time t s occur.

In the nearly exact matrix element

the dipole approximation replaces emix' at by 1. If Ws vanishes, the transition i s forbidden, as described in the foregoing, The next higher o r quadrupole approximation would then be to replace e-jK' X by 1 - i/K* X , giving the matrix element

For light; moving in the z direction and p la r ized in the x direction, &is ~ G O I Z L B B

met the trsrnsition probabiljity is proportional to

whereas in the dipole approximation i t was proportional to

Therefore the transition prob&ility in the qua&upole approxhation is at least of the order of (Ka)' = a2/%, smaller than in the dipole approximation, where a i s of the order of the size of the atom, and h the wavelengtLh emitled,

Problem: Show that

and consequently that

Note that p,z can be written as the sum

From the preceding prciblern, the f i rs t part of p,z ie seen to be equivalent, up to a constant, to xz, which behaves similarly to a wave function for

18 Q U A N T U M ELECTRODYMAMXCS

2, even pmity, The second pr l i s the operator $, which behaves like a wave function for even parity, Therefore the selecUm rules correspondi% to the Brst part a re Been to be AJ = * 2, +l, O with na parity chawe, This type of radiation is called electric quadrupole, The selection rules far the scseond part of p,z are AJ = rt I* OS no parity clzawe, md the eo r r eapnd iq radiation i s eaUed magnetic dipole. Hots that unless AJ & 2, the two t y p s of radiation cmnot be diat iwished by the change in m ~ l m momentum or pwity. If A J = *l, 0, Lfiey eirn only bbe d i s t i n~ i shed By the polarizatia~ of thr; raaation. Bo& t m a may occur simultmeously, producing interference.

In the case of electric qudrupole radiation, i t is impticit in Lhs rules that 1/2 - 112 and O - X trmasitiona are forbidden (siren though &l' may be & l), since the required chawe of 2 for the vector mmla r nclomentm i s impossible in these caaes,

Continuing to higher approximations, i t is possible by a b i l w reasodng to deduce the veotsr chawe in mmXas momentum, or mp1m momenhmn of the photon, and the selection rules for parity chmge and cbmge of total m- w l a r momentum AJ associated wit;h the varicrus multipole order8 (Table 5-11,

T N L E 5-1. Classification of Transitions and m e i r Selection Rules

Electric m p t i c EL~trie &wetic Ekectric lMultipole dipole dipoh quadmpolt.: qwdrupole octupale

Actually all the implicit selection rules for A J , which become numerous for the higher multipole orders, can be expressed explicitly by writing the selection rule as

where 2' is the multipole order o r 1 i s the vector change in angular mo- m e n t ~ ,

XNTIERACTllOrJ OF L I G H T W I T H M A T T E R If)

Xt h r n s out that in so-eal led parity-favored trmsitions, wherein the p r d - uet of the initial and final parities is ( - l ) l t - and the lowest possible multipole order i s Jf - Ji , the transition probabilities for multipole typea contained witEtin the dashed vertical gnes in Table 5-1 a re rowhl equa1.t In !r parity-unfavored transitions, where the parity product i s (-1) t -ji and the lowest mullipob order 163 I Jf - Jy 1 + L, this may not be true.

Quifibrim of b&%ticm. If a system i s in equiIibrim, the relative number of atoms p r cubic centimekr in two ~statee, say 1 and k, i s giiven by

according to statistical mechanics, when the energies differ by Em. Since the system i s in eqtliubrim, the number of atoms goiw from s t a b k to l per anit time by absorption of photons b must equd the number goi% from l to k by emission. If n, photons of frequency w are present per cubic cen- ameter , then polaabili.t;les of absorption are proportional to n, mci proba- Bility of emission is proportional to n, -t 1. Thus

Thia i s thds Planek black-bdy distribution law. me SGatteri~ af Ugh1, VVe discuser here the phenomena of an incident

photon being sca tbred by an atom into a new direction ( a d possibly e n e r e ) (see Fig, 6-11. This may be considered as the absorpt;ion of the incoming photon and the ernisrsion of a new lpfroton by the atom. The WO photons t&ng part in the phenomenon a re represented by the vector potentials.

The number to !X determined i s the probability that an atom initially in state k will be left in state 1 by the action of thds pert;urba#on LP r=: At + AS in the

"f For nuclei emitting gamma rays this d m s not seem to be true. Far an obscure reason the magnetic radiaiLion predominates for each order of muI- tipole.

Q U A N T U M E L E C T R O D Y N A M I C S

time T. This probability can be computed just aa any transition probability with the use af Alk, where

Ths dipole! approdmation. is to be employed md

where r pins are neglected, In each inbgral defining AXkr each. of the two vector potentials must ap-

par once and only once. Thus, in the f i rs t integral the term p e h of fJ will not appear in Ulk. The p r d u e t A . A = (A1 + A2) - (Ai + Az) will conbibale only its crass-prduct term 2A1&. The second in&gral will have no con- t r ih t ion from A . A , but will be efie sum of two terms. The f i rs t term contains a U1, based on p * Az and a Unk based an p . At. The second h w Ul, based on p AI and Unk cm p *AZ. The Lime sequerzcrzs resuf t iw in these two terms can ba represented schematically as shown in Fig, 6-2.

The integral resulting from We first term vAlf now be developd in de- lai l.

Then the resulting integral is

I N T E R A C T I O N OF LIGHT W X T B MATTER

first

t/ atom

FIG, 6-2

expi-i(E,/fi)ltd - tal - iw exp f-i (Ek/fi)eaj db dt,

The integral is similar to the integrals considered previously with regard to transition probabilities, and the sum becomes

where L& = (EI +&W - Ek - i), and the phase angle c# i s independent of n. A term. wi& the denominator given by (E, - b 1 - Ek)(EI + b 2 - E,) has been neglected, since previws results show that only energies such that El + m Ek + tiwl are important. The final result e m be written

where iM1 I s determined from Alk by integraaxlg over wz and averqing over e2. Then the complete expresssion. Ear the cro8s section cr is

22 Q U A N T U M ELECTRQDYNALMICS

The first term under the summation comes from tlxe "first termm 're- viously referred to and t b second from the ""secand term." The last term in the absolute brackets comes from A * A.

If l k, the scattering is incoherent, m d the result is called the '"man effect,$8 If l = k, the scatbr ing is coherent,

Further, note that if all the atoms are in, the ground state and l r k, f;hen the energy of the atom can only i n c r e a ~ e and the frequency of the Ught ~3

can only decrease, This gives r ise to "Stokes lines ." The oppsaite efhct gives @%ti-Stokes Unes .$'

Suppose wg = (coherent s c a l k r i w ) but further Ewl i s very nearly equal to Ek -E,, where E, is some possible energy level of the abm, Then one b r m in the sum over n bcomes extremely large and domimbs the remainder. The result is called "msomnce sca tkr iw." .B' @ is plo-d agaimt o, then a t such vdctes of w the cross section hais a s b w maximum (see Fig, 6-31.

FIG. 6-3

The @'indexps oaf refraction of a gas e m be obtained by our scattering formula, ft can be obtained, a,@ for other t y ~ ~ of scatterilae;, by consideriw the lf&t sca tbrad in the forward direction.

bJslf-Enem, h o t h e r phenomenon &at must be considered in qum e l s c t s o d p a l c s i s t h ~ pos~ibilft.Y of m atom emitting a phobn and reab~orb- Ing the earne photon. This affects tfie diagonal element Akk. Its @Beat is quivalent; to a sMft of energy of the level. h e find&

where e is the direction of polarization. This integral dive%@@. A more exact relativistic calcuIation also gives a divergent integral. This means that our fomulialian of ebctromametic effecter if; not really a completely satisfactory theory. The modifications r e q ~ r e d to avdd this difficulty of the infinite self-energy will be &scusaed later. The net result i s a very @mall sMft bE in position of energy. levels. mfs shift h m been observed by L a b and &&erford.

the Principles and Results of Special Relativity

The principle of relativity is the principle that all physical, phenomena would appear to be exactly the s m e if all the objects concerned were mov- ixrg uniformly t qe the r at velocity V; that is, xza experiments made entirely inside of a closed space~hip moviw mfornzly rrit velocity v (relative to the center of gravity of the matter in the universe, for exmple) can debrrrine e h i ~ velocity. The pinciple has been verified exwrimentally . Newton's laws satisfy this principle; for they a re urrebmged when subject ta a Gall- lean transformation,

because they i nvolve only second derivatives , The e H qaat ions a re changed, hwevsr , when subjected to this trainsfornnation, and early workers in this fieid attempted to m&e an absoluta determination of velocity of the earth using this featwe fmchelson-Morley exmriment). Failure to detect any effects of this type ultimately led to Einstein" postulate that the Max- well equations a re of the @%me form in any coordinate system; and, in particular, that the velocity of light i s the same in all coordinate syskms. The transformation beween. coordinate system@ wkich leaves the MmweXl equations invariant is th@ Lorentz transf.armatiarr:

where ta& u = v/c. Henceforth we shall use time u d t s so that the speed of light c i s unity. The latter form is written to demonstrate the analam with rotation of a e s ,

X' = x cos 8 i- y sin 6

y' = -X sin B + y cos 6

Successive transforrnatians vl and v, o r ul and u2 add in the sense that a single tran~farrnation va o r u3 will give the same final system if

Einstein postulated [theory of special relativity) that the Newton laws must be modified in such a way that they, too, are unchawed in form under a Lorentz transformation.

An interestin%; consequence of the Lorentz transformation is that clocks appear to run slower in moving systems; that is called time dilation.. In transforming from one coordinate system to another i t i s convenient to use tensor analysis, To this end, a four-vector will be defined as a set of four quantities that transforms in the same way as x,y,z and ct. The subscript p will be used to desimate which of the four components is being considered; for exmple ,

The fallowing wantities are f our-vectors:

a 8 a +.,"-- W.,"-,.. -- W - a (C,) four- dimensional gradient a ~ " y y ~ v t

.lxJ jyJ jzy P (jp ) current (and charge) density

A,, $ 8 A,* V (A,) vector (and scalar) potential

P,# Pyr P z r E (p,) momentum and total energy f

p--

$ The energy E, here, is the total energy including the rest energy me2.

S P E C I A L R E L A T X V f T U 2 5

An invariant i s a quantity that does not change under a brexrtz transformation. If a u and bp are two four-vectors, the "product"

i s an invariant. To avoid writing the summation symbol, the following summation convention will be used, When the same index wcurs twice, s~ over it, placing minus in front of first, second, and third compments. The h r e n t z invariance of the continuity equation i s easily demonstrated by writing i t a s a ' 4product" of four-vectors V, and j, :

Conservation of chmge in all systems if i t Is conserved in one system i s a consequence of the invaiance of &is 'product,' ' t h e four-dimensional divergence V e j , Another invariant is

p,p, = pep = ~2 - pXZ - pY2 - p,' = E' - = m'

(E = total energy, m = rest mass, mc2 = rest energy, p = momentum.) Thus,

It i s also inlteresting to note that the phase of a free particle wave function ~sxp f f-i/tr)fEt - p - fo] i s invariant since

The invariance of p,p, can be used to facilitate converting laboratory energies to eenter-of-mass energies (Fig, 6-4) in the following way (consider identical particles, far simplicity):

xno-trixy~ sttltiona xy particle padicle

Center-of-mass system hboratarjr system


but

and

The equations of e lec t r&mamie~ B = V X A and E = -- (i/c)(@A/@t) - V Q, are easily written in temor notation,

where use i a made of the fact that (p is the fourth component of the four- vector potential A p e From the foregoing it can be seen that B x , B,, B,, E,, E,, and E, a r e the components of a secoad-ra& tensor:

This tensor i s antisymmetric (F = - F,@ ) and the diagonal terms (Ir = v ) t" f" a re zero; thus there are only six ~ndewndent components ( t h e e components

of E and three components of B) instead of sixteen.

The Mmwell equations V x B = 4n J -t- (a E / B L) and V- E =: 4np are wrltbn

m C

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$3 0,

c5 g

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Q U A N T U M ELE:CTRODVEJAMXCS

80LVTXON OF THE ELL EQUATION ZE-4 EMPTY SPACE

h empty space the plane wave solution of &e wave equation

where el, and k, a r e constant vectors, and k, i s subject to the condition that

This may be seen from the fact that V, operating on e-lk' X has the effect of multiplying by ik, (V, does not operate on er since the coordinates are rectangular). Thus,

Note that in these operations P, AI, actually forme a second-rank tensor, V, (V, A, f a &ird-rank tensor, and then contraction on the index v yields a firat-r& tensor o r vector.

The kp i s the propagation vector with components

and the condition k * k = 0 meam

Problem: Show that lthe LLOmntz codition

implies that k e = 0.

m e n worMq in three dimensions it is customary to t a b the poiariza- tion vector 63 such that; K * a = O and to let the @calm potential cp = 0. But

this is not a unique condition; that is, i t is not relativistically invariant and will, be true only in a one-coordinate system. This would seem to b a paradox attaching some uniqueness to the system in which K s = 0, a. situation incompatible with relativity aeorg . The "paradox, however, is resolved by the fact tbat one can always make a so-called gauge transformation, which leaves the field FP, unaltered but which does change e. Therefore, choosing; fiC* c? =- O in a particular system amounts to selecting the certain gauge.

The gauge transformation, Eq. (1 -31, is

where X i s a scalar. But V A = 0, the h r e n t z condition, Eq, (7-41, will sti l l hold if

This equation has a solution X = cremik * X , So

where a i s an arbitrary constant, Therefore,

i s the new polarization vector" obtained by gauge transformation. h ordinary notation

m u s , no matter what coordinate system, is used,

can be made to v a d s h by choice of the constant a . Clearly the field i s left unchmged by a gauge transformation for


the VpVv X - O,Vp X because the order of ditferentiations is immaterial.

The components of ordinary velocity do not transform in such a mamer that (tan be components of a four-vector. But mother quantity

where

dz, = dt, &, dy, dz

is an element of path of the particle and ds i s the proper time defined by

is a four-vector and is called the four-velocity up . Dividing by dt2 @ves the relation between proper time and local time to be

The components of ordinary velacity are related as follows:

It i s evident that upup = 1, for

The four-momentum is defined

p, = mu, = m/(l- v2)'/ 2t mvx/(l - mvy/(l - v2)'/ ', mv,/(l - v2)'i2

Note that pc = m/(1 - i s the total energy E, so that in ordinary notation the mornentm P is given by

S P E C I A L RIELATIVIT'il

where v is the ordinary velocity. Like the veloeft;y, the components of ordinary force defin~d by d/dt (mo-

menhm) cannot forrn the components of a four-vector. But; the quantity

f, - dp,/ds

does form a four-vector with the components

where FP is the ordinary force. The fourth component i s

power - rate of chmge of energy f4 = - -

4 3

This i s seen from the fact that m/ i s tlre total energy and also from the ordnary identity

Thus tfie refat-ivistic mrtlolfue of the Hevvdon equatiom i s

d/ds (p,) = f, = m d2zp/ds2

The ordinary b r e n t z force i s

anid the rate of chmge of energy is

Then from the prece$iw definition of fow-fmce,

and


Problem: Show that the expressions just given for f and f4 are equivalent to

f, = euP F p v

so that the relativistic analope of the Newton. equation becomes

m dZz, /dsP = e(dz,/ds) FP, ( 8-23 1

Also show that this implies

d/ds [(dz, /dsj2] = 0

In ordinary terms the equation of motion, i s

d/dt (mv/ = e(E -i- v x B)

It em be shown by direct alpplication of f i e h g r w g e equations

d/dt (a L/av, ) - ( a L/ax, 1 = o

that the Lagrangian

leads to these equations of motion. Also the momenta conjugate to x is given by 8L/i3v or

The corr e s p o n d i ~ Hami Etonian i s

H = e 4 + [(P - e ~ j ~ + m 2 ] t / 2 (8-6)

which satisfies (H - e@l2 - (P- ek j2 = m'. It is difficult to convert the Hamiltonian idea to a csvari m t or four-dimensional formulation, But the principle of least action, which states that the aetion

shall be a minimum, will f e d to the relativistic form of the equations of motion directly when expressed a s

S P E C I A L R E L A T I V I T Y

Note that by definition

(ds/da lz = (dz p / da ) (dzP /da f

It is interesting t;h& woeher "aetf an, '' "defined

leads to the s m e result as for S in the foregoing,

P~oblems: (1) Show that the Lagrangiinrr, Eq. (8-51, leads to the equations of motion, Eq, (8-4), and that the correspondiw Hamlltonian i s Eq. (8-6). Also find the expression for P, (2) Shclvv that cFiS = O (va- riation of S), where S is the action just given, leads to the same equations.

Relativistic ave Equation

The following convention wilt be used berrtiafter. We define the unlts of maas and time and length such 011151;

Table 9-1 (top of p q e 39) is given as a useful reference far conversion to a s tomary units,

The following numerical values a r e useful: Mp = rnasrs of proton = 1836.1 m = 938.2 Mev Mass u d t of atomic weights =; 931.2 Mev MH = Mass af hydrogen atom = 1.00815 mztsa u d t s

= Masa or" neutron = 784 kev -I- MH kT = l ev when T =. lI,E3Qli*K. H, = Avogadrops number = 6.025 x N,e = 96,520 caulornbs

Accordiw to relativistic classical mechanics, 'che Hmiltaniain is given by

RELATICVISTXC W A V E E Q U A T I O N

T D L E 9-1. Notations and Udts

Present Customary notati on M e a ~ n g ~ notation Value

m Mass of electron

Energy

Momenhnn

Frequency

Wave number

bwth (Comp- ton wave- liewtt.r)/Zr

Time

e2 Fine-a truetwe constant (dimensionless)

e v m Classical radiue of the e b ~ t r o n

2/me2 Bahr radius

m

mc2 5% 0 '99 kev

mc 1104 gauss ern

me2/%

mc/A

li/mc 3.8625 f @-'%cm

If the qumtum-mecbfctal operator -iV fs used for p , the operation dek r - mined by the square root i s undeaned. %us the reEativistie quantum- mechmieal Plmiltadan has not been obtained directly from the classical equation, Eq. (9- 1). However, i t is goe~ible to define the square of the o p r - atar and to write

where the square of an. owrator is evaluabd by ordinary amrator algebra. This equation waia first discovered by Schrainges as a possible relativistic equrztion. It is usually referred to as the Klein-Gordon equation. h relativistic notation it i s


This equation does not allow for "~spin'>nd therefore fails to describe the fine structure of the hydrogen spectrum. It i s proposed now far application to the .rr meson, a particle with no spin. To demonstrate i ts application to the hydrogen atom, let A =; O and @== -Ze/r, then let \k= X (r) exp(-iEt). Then the equation is

Let E: = m + W, where m, and substituting V = ze2/r,

Neglecting the term on the right in comparison with the f i rs t term on the left gives the ordSnary S c h r ~ i n g e r eqtlation. By using; (W - Vj2/2rn a s a perturbation, potential, the student should obtain the fine-structure? split-tiw for hydrogen and compare with the correct values.

Exercise: For the Klein-Gordon equation, let

p = if**a@/at - e s s * / a t ) - eqb3 ?fi* = charge density

j = -if% V 9 - J1 V Jlr *) - BA* ?k* = current density

Then show (p, 1) is a four-vector and show FP jp = 0.

The Klein-Gordan equation leads to a result that seemed so unreasonable at the time i t was first brozzght to light that i t was considered a valid basis for re j~c t ing the equation, This result i s the possibiuty of negative energy s t ab s . To see that the Illein-Gordon equation predicts such emrgy states, consider the equation for a free particle, which c m be written

where i s the R"A1embertian operatar. Xn four-vector notation, &is ewation has the solution V = A exp (-ipp xp ), where pp pp = m2. Then, since

there results

The apparent impossibflity of negative values of E led Dirae ta the development of a new relativistfe wzlve equation. The Dfrac equation proves ts be correct in predietiw the enerm levels of the hydragsn atom md i s the accepted de~cript ian of the electron, However, c o n t r q to Diracss original

R E L A T I V I S T I C W A V E E Q U A T I O N

intent, Ms equation also leads to the existence of negative enerm levels, which by now have been satisfactorily interpreted. m o s e of the Klein- Gordan equation can also be interpreted.

Exercise: Show if 3 -- expf - iE t )~ (x,y,z) is a solution of the Klein- Gordon equation with constant A and Q1, Lhen 9 = exp f + i E t ) ~ * is a solution with -A and -Q, replacing A and (P. This inaeates one manner in which "negative" energy solutions e m be interpreted. It i s the solution for a particle of opposite charge to the electron, but the same mass.

h s t e d of followiw the original meaocf in the development af the Dirac equation, a different approach will be used here. m e KEein-Cordon equation is actually the four-vector form of the S c h r B d i ~ e r equation. With m analogous point of view, the Dirac equation can be d e v e l o ~ d as the four-vector form of the Pauli eqa t ian , h following such a procedure, the terms invalviw "spixltWill be included

in the relativistic. equation, The idea of: spin was first intrduced by Pauli, but i t was not at first clear why the mametic moment of the electron had to be taken a s Ae/2mcl This value did seem to follow naturally &om the D i r a ~ ewatiorr, md i t is often stated that only the Dirac equation produces as a consequence the correct value of the electron" magnetic marnent. However, this i~ not true, a s further work on the PauE equaaon showed that the s m e value foklows just a s naturally, i*e., a s the value that produces the greatest simplification, Because spin i s present fn the Dirac equation, and absent in the Kleia-Gordon, md beemse the Kleixz-Gordon equ.at;ion was thought to be invalid, i t is often stated that spin i s a relativistic requirement. This is incorrect, since the Klein-Gordon equation i s a va ld relativistic equation for p m ticles without 8pia.

Thus the Schrminger equation is

where

and the Klein-Gordon; equation i s

[(H - e@12 - (-iV - * = m2%

Now the Pauli equation is also H* = E*, where

Thus ( -iV - appearing in the Schradinger equation has been replaced by [o . (-iV- e ~ ) j ~ . Then a possible relativistic version of the Pauli equation, in analow to the Klein-Gardon equation, might be


Actually, this i s incorrect, but a very similar form [wi& M replaced by i( El/@ t)] is eorreQt, n a e l y ,

This is one form of the Dirac equation. The wave Eunctian jEr on which the o ~ r a t i o n s a r e being carried out i s

actually a matrix;:

A form closer to that originally proposed by Dirac may be obtained as follows. For canvellienee, wriW

Now let the function X be defined by (g4 + Q r) @ = =g. Then Eq, (9-5) implies. (n4 - ar t r ) ~ = m@. Thls pair of epa t ians can be

rewritten (only to arrive at a particular conventional fom) by wi t ing

Then adding and subtracting the pair of eqationa for @,X, there r e ~ u l t s

mese two equations may be written as one by e m p l o y i ~ a particular eanvention, Define a ntsur matrix wave ftllnetion as

where the matrix character of 9, and 9b has been shown eql ic i t ly , i.e., achallty

RELATXVISTXC W A V E E Q U A T I O N

Than, if the auA1lmy definitiom a r e made,

(Note: An example of the latter definition is

0 0 0 0

0 0 1 0 0 X,

since crx = P 0

1 0 0 0

yY and y, a r e similar.) The two equations in V, and Yb can be written as one in the form

which is actually four equations in feu wave functions, Then using four- vector notation, the Dirac equation is

that is, show

Y: = l = yy2 = yz2 = -1

40 Q U A N T U M E L E C T R O D Y N A M f C S

A similar f o m for the Qirae equation &&t be obtaned. by a different arwment, by comparison to the Klein-Gordon equation, m u s wit21 H = i(a/a t) = i V, and with ecp = eA4, Eq. (9-3) becomes

in four-vector notation. Usix a similar notation in the Pauli equation, Eq. (9-41, but also using a = y and setting Q = yd arbitrarily (to complete the defiation of a four-vector form of Q), Eq. (9-4) can be written in a form similar to Eq. (9-fa),

This should be compared to Eq. (9-9). Now the Pauli equation, Eq. (9-41, diEers from the Sehr3diwer equation;

in the replacement of the three-dimensional scalar product (p - eh)' by the square of a single wantity * (p - @A;). AnalogcrusEy one might wess that the four-vector product (pI, - eAI,jP in Eq. (9-10) must be replaced by the square of a single quantity y p (pp - @Al,), where we must invent four matrices yi, in four dimensions in analogy to the three matrices a in three dimensions , The resulting equation,

i s essentially ewivalent to Eq, (9-9) towrate on both sides of Eq. (9-9) by yp (iVp - e A,) and use Eq. (9-9) again to simplify the right-hand side).

Ezercise: Show that Eq. (13-11) i s equivalent to

(iVp - e ~ ~ } ' - L eyIIy, F,, .Y = m'+ 2

AWEBM OF THE y MATRICES

In the preced i~g lecture the Dirac equation,

yl,(iVI, - ehI1)% = m%

was obtained, together with a slpt3eial represenLaLion for the y %,

R E L A T I V I S T I C W A V E E Q U A T I O N 41

where each element in these four-by-four matrices i s anof;her two-by-two matrix, that isa,

The best way to defhe the y 'ss, however, is to give their eomrrautatlon relationships, since this is all that is important in their use. The commutation relations&pa do not determine a unique representation for the y %, and the foregoing is only one of many possible representation@. The eornmutation relationships a re

or, in a unified notation,

Note that with this definition of and the rule for forming a scalar products

Other new matrices may arise by forming produets of the matrices af- ready defined. For example, the matrices of Eq. (10-5) are producta of y 's taken two at a time. The matri ces

are ail independlent of y,, y,, y, , y, . (They cannot be farmed by a limar combination of the latter .) Similarly, products of three matrices,

42 Q U A N T U M E L E C T R O D YltJtJlMleS

These are: the only new p rduc t s of three. For, if two of the matrices were equal, the produst could be recluced, thus y, y,y, = -y, y, yy = -yy . The only new p r d u c t of four that can be farmed is @ven a s p c i a l name, yg,

Pr&ucds of mom t b four muet conkin twa eqwl so that they c m be reduced, are, thereforr?, s k b e n linearly independent q eambh.tions of them may involve s h k e n arbitrary cons-b. TMs w m e s with the fact t b t such a eombimtion can be expmesed by a four-by-four =a- trix. (It is maaemarclcally inbmtsting then that all four-b-four matrices can be ewressed in the algebra af tbe y 53; this is called a Cliffod algebra o r hyprcomplex algebm. G simpler example i s that of tvvo-by-two matrices, the sa-cdled algebra of quaemions, wMch i s the algebra of the Pauli spin matrices ,)

and sat

14, i s convenimt to define ano&er y matrix, since it occurs frequently:

Verify &at 0

-@x,y,z

For later uae, i t will be convenient to define

from which it can be shown that


For exmple, the firet may be verified by vvritislg

and, movirmp; the sc;cond faetor to the front, by using the cammutation rela- tionrahlips, Doiw Ghia with the first term, (bp?,) af the secoad factor produces

since y, commutes with itselt: and anticommutes with y,, h, and y, . By performing &is oprat ion on all terms, one obtains

&P = = t Y t I(-&,?, + a,?, + ayyy + a,y,) + 2a,y,l

+ b,y,[(a,yt - a,?, - ayYy - BzYz 1 + 28XYX1

+ byr,[(atvt - a,?, - ay?, - a,?,) + Zayv,I

+ bz?,[(a,v, - &,Y, - ayYy - a,yz) + 2a,yzl

= -1391 + 2(b,a,yt2 + b,a,y,2 +by%?; +bZazy;)

=-g$ + 2 b e a

Exercises: (l) Show that

?,Pl~x " id + 2%,YX

l"I-lYlr = 4

v,&, = -W

y,$#yp = 4 a e b

v,rr%h, = -Z&Plc

(2) Verify by expan&w in power series that

=p Ifu/~)y,y,l = c%3sh ( ~ 1 2 ) + y, y, sinh (u/2)

((@/2)y,yyi = cos (@/2) + y,yy sin ( G / % )

(3) Show that

QUANTUM E L E C T R O D Y N A M f G S

Suppose a n o ~ e r representation for the y @s ifs obt&ned which satisfies the s m e commutatio~ relationships, Eq. (l 0-3); will the form of the Dirac equation, Eq, (TO-I), rem%.in the s m e ? To answer this w e ~ t i o n , m&e the following transformation of the wave function Ji( = S*', where S i s a constant m&rix W&& is msurned to have arz inverse S-' ( ~ 8 ~ ~ = l). The Birac equation becomes

The rp and S commute, since n i s a differential operator plus a function of position, s o &is equation may be written

Multiplying by the inverse matrix,

where y ; = S-' y p S. The transformation y ; = S-' S i s called an equivalence transfomatian, and it is easily verified that the new y 'B satisfy the commutation relationships, Eq. (10-3). P r d u e t s of y 'ss,

transform in exactly the same mmner a s the y 'ss, s o &at eqtjlations involving the y ?s (the earnmutation relation8 spcifically) arc? the s m e in the transform representation. Thfs demonstrates andher representation for the y 'ss, md the Diraa equation i s in exactly the e r n e form as the original, Eq. (10-l), md ie equivalent in all i ts result&.

=UTW1;531"IC NCE

The relativistic invarimee of the Dfrac equation may be demonstrakd by assuming, for the moment, that y transforms similarly to a four-vector,

R E L A T I V I S T I C W A V E EQUATION

That is,

Also a t r a s fo rm s similarly to a four-vector because i t is a combination of two four-vectors 'JI, and AB . The left-hand aide y s of the Dirac equa-

F! tion i s the product of two fow-veetorrr and hence invarrant u d e r h s e n t z transformations, The right-fimd side m is alao invariant. Tr ansforming ylr a s a four-vector means a new representation tor the y 's, but Eqs. (10-11) can be used to show that the new y '8 differ fii-@m the old y 'a by an eqaiva- lenee transformation; thus i t is really not necessary to transform the y 's at all. That 18, the s a e special representation can be wed in. all Lorentz eoor&nate systems.. This leads to two possibilities in m M n g h r e n t z transformations:

X . Tmnsform the y % einzilarfy to a four-vector and the wave function remains the s m e (except for Lorentz tramfornation of cooranahs) .

2. Use the stmdard representation in the Lorentz-tansformed coodirta& system, in which c w e the wave funetion will differ from that in (It) by an equivalence t rmsfomation.

HAMLTONXAN FORM OF THE D m C IEQUATXON

To show that Ght? Dirac equation reduces to the SehrMinger equation for low velacities, i t is convenient to write i t in I fa i l ton ian form. The original term, Eq, (1 0-11, may be written

MultipI,ying by cy, and r e a r r a q i w b r m s gives

By Eq. (10-5), B is written

- where p = 7, , a ,, ,Z - yt y x , g , z , Eq, (10-S), and the cr 's salis& the follw- utation relations: a X2 = ayZ = ( Y ~ ~ = p2 = 1 and all pairs anticommute.

It will be noted that a,@ are Hernitian matrices in our special regresen- tation, ~o that in this r-epressntation H i s H~smitian,

Q U A N T U M EEEECREtOYNAMAnCS

Exerczse: Show that a probability denaity p = 9 * ?Xr a d a probability current = @*a% satisfy the ~ontfnuity eqmtiorr

Note: \Xr i s a four-component wave function. and

Xt should be noted that p mid a are Nerxnitim only la certain represenrla- tions . h particular, they are Hermi tian in the repreeentaaon employed thus far; this wilt be called s t d a r d representation md expressions in it will be lab led S.R. when appropriab . The Hermitim property of cli! and i s necessary in order to get

,j =:**Q* S . R. (If -l)

ahs f ie expressions for chage and current density, Hence they are not true In all representations. The Dirac equation is (wStb E, c restor&)

t It i s noted that the Hamilbnian found in Schiff ("manturn M e ~ h n i c s , ~ ~ McGraw-Hill, New York, 1949) differs fmm this one by negative s i g s on all but the et$ term, Also the connponenb tXTZ, %V"3, ?ka of the waw function used in Schiff correspond, respectively, to -i&bf, -9b2p IYp2 here, All this is the result of an equivalence transformation S2 = ipa,aya, between the representations used bere and in Sehlff. It is easily verified that 5" -1 hence S-l = -S and

R E E A T I V I S T X C W A V E E Q U A T I O N

The expected value of x is

remembering that O now is a four-component wave hnction, Similarly it may be verified as an exercise that

< a > = J**(YY dvol

Also matrix elements are formally the same as before. For exmple,

91f A is any operator &en its time derivative is

For X the result i s clearly

since x cornmutee with all terma in W except p * cr. But a = I, ao the eigenvalues of a! are a l. Hence the eigexlvelocities of .k are ~ t : speed of light. This r e ~ u l t i s sometimes made plausibb by the a r cise determination of velocity implies precise determinations of position at t;wo times. Then, by the uncertainty principle, the momentum is completely uncertain a d all values me equaftly likely. Wiefi the rehtiviatle relation between velseity and mamentm, &is is seen t~ imply t;ha;t velocitieo near the speed of light a re more probabb, rso t b t in the limit the exwcted value of the velocity is the speed of Light.?

Similarly,

(P - @A;), = i (Hp, - p,H) - i s (HA, - A,$$) - eaA,/at

The terms in A and A,, except the last, expand m follows:

?This argument is not completely acceptable, for k commutes with p; that ia, one should bs able to nsreaaure the two quantities simultansously.


Thihs seen to be the x component of

The first and last t e r m form the x component of E. Therefore,

where F is the analowe of the Lorenlz force, T h i ~ equation i s sometimes regarded as the analowe of Newton's equations. But, since there is no direct comection beheen this equation and 2 , i t does not h a d directly to Newton" equations in the U d t of small veloeitiee and hence i~ not completely acceptable as a suitable analowe.

The followixrg relaaons may be verified as true but their memiw i s not; yet completely understod, if at all:

where in last relation a means the matrix;

so that

o, = --tar, ay, etc.

From analogy to classical physics, one mi&t e x p e t that the anmlar momentum clpemlor is now

P

From previous results for k and ( p eh), the time derivative of L may be written


The last term may be interpreted as torque. For a central force F, this term vanishes. But then it is seen that L 0 because of the f i rs t krm; that is, the angular molnanhtm L is eaneerved, even wl& central forces.

But consider the time derivative of Ule operator a defined as

where c, = -a,ay, etc. The z component i s seen to commute with the 8, e+, and a , terms of H but not with the a , and ay terms, so that cr, = + l ( H a , a y - aXayH) = + ((Y,T,OI,(Y~ - Q,@y(Yx~x + QyTyaxay- @ x @ y @ y l f y ) .

where

But

so that

This is seen to be the z component of -2or X ar . IFinal3,y &en,

and this is the first term of i with negative sign. Therefore it follows that

which vanishes M* central forces. The operator L t- @/2)o may be regarded as the total anguf a r momentum op ra to r , where L represents orbi- tag anwlar momentum and i-fi/2)o intrinsic a-Ear momenhm for spin 112. Thus total, ang~ular momenhm i s conserved with central forces,

Pwblems: (1) h a stationary field $3 = 0, BA/@ t = 0, show tfiat

i s a coastant of the motion. Nob that this is a consequence of the anomalous gyroma@etie ratio of the electron. Xt also me cyclotron, frewency of the electron equals its rate of precession in iil

mapetic field. (2) fn a stationary nrr2lgnetic field 4 = 0, @A/Bt = 0, and for a sta-

tionary s tak , show that %fr. vIrz in

are the same as JlrZ in the PauXi eqwtion, Also, if EPHuli ia the Mnet l e energy. in the PauXf equation and EBirsc = W + m i~ the msrt plus kineWc errsrm in the? Erac ewtiont, show ~t

and explain the simplicity of &is relationship.

ltt will be assumed &at all ~ b n t i a l s are etatiomry and statimary states will be considered. This m&es the work simpbr but is not necessary, h &is case

T b t is,

f~gi = (m = a * (p- eA)@ +pm* + e@3

It wilE be recalled wiLh ?k written as Eq, (9-5) and wi& a, /3 as given in b c t u r e 10, the previous equaaon may be writbn aa two eqiaaona (11)-4*),

R E L A T I V I S T X G W A V E E Q U A T I [ Q N 51

where, as befora, r = ( p - BA) a& V =- e$. Simplifying a d solving Eq, (11-5) for ilib gives

It i s noted &at if W and V are 2m, a e n JCrb -- (v/c)O,, For this remon +, and are sometimes referred to as the large and small components of @, respectively. S&st;itution of qb from Eq, (11-6) into Eq. (11-4) gives

and, i f W and V a re negiected in. eompari~on to 2m, the result Is

This i s the Pauli equation, Eq, (9-4) . Now fAe approdxnation will be carried out to h3ec0XTd order, that is, to

order vZ/c2, to determine just what error may be expected from use of the Pauli equation.

Using the results of Leetrrre 11, given by Eqs, (11-6) md (11-7), the low- enerw approximation (W - V) << 2m will be made, keeping terms to order v'. Thus

Then Eq. f 11-71 bcomes

while the normalizing requirement + ObZ) d v01 = 1, becornea

By use of the substitution

the normalizing integral can bs lsinzplified to read (to order v2/cZ)

Q U A N T U M E L E G T R Q D Y N A M I C S

This 8ubstitut;ion also allowe easier interpretation sf Eq, (12-2). Rewriting Eq. (12-21,

[l + (o g ~)~/(8rn~]l (W- V) l1 + (cr.

Then applying Eq. (12-.4f and dividing by 1 + (cr*w]2/(8m2), there results

(W-V)X = (1/2m)(cr*~)\ - (1/8rn~,(@.r1'~

The techniques c>E o p r a l o r akebra may be used to convert Q. (12-5) to a form more easily interpreted, In particular ane should recall that

A% -- 2ABA't. BA' = A(AB - BA) - (AB - BA)A

Then, since rr = ( p - ~ f l 3 c ) , and since

there result8 fwiGh cr. r = A and (W - V) = B in the foregoiqj,

f a i n ~ e V x E - 8B/at = 0 here), s o Eq, (12-5) can be expanded as

WX = VX +- (f/ltm)(p -- eA).(p - sA)x - (c?/rztnr)(rr*B)~ (11 (21 1-31

- ( 1 /8m ' ) (~*p~~x (4

+ ( e 2 / 8 m 2 ) ~ - ~ -+ 2 0 . (p -eAf x Ef X (12-6)

(51 (6)

In this form the wave eqwtion may be interpreted by considering each term of Eq. (112-6) separately,

Term (1) give8 the ordinary scalar potential energ.y a s i t has a p p a r d before.


Term (2) can be interpreted a s the kinetic e n e r a . Term (31, the Paull spin effect, ilr~ just as i t appears in the PmI equa-

tion. Term (4) is a relativistic correction to the kinetic energy. The correc-

tion. derives from

The last term in tlris expansion i s e?quivalent to term (4). Terms (5) and (6) express the spin-orbit coupling, To understmd this in-

terpretation cornides the part of term (6) given by a * (p x E). In an inverse- square field this is proportional to o + (p x r)/r3. The factor p x r can be interpreted a s the angular momentum L to get fa* u/r3, the spin-orbit coupling. This term has no effect when the electron i s in a s-state (L = 0). On the other hand, (5) reduces to V E = 4nZ&(r), which affects only the s-states (when the wave funetion i s nonzero at r = Of . So (5) and (6) together result in a continuaus hnction for spin-orbit coupling. The magnetic moment of the electron e/2m, a p p a r s as the cmfficient of term (3 ) , a d again of terms (5) and (G), i.e., (e/2m)(1/4m2),

A classical a rgmen t can be made to Interpret term (6). A charge moving &rough m electric field vvi& velocity 'tt feels an effective magnetic field B = v x E = ff[m)(p - eA) x E, and term (6) i s just the energy (e/2m) x (o B) in W s field. flVe get a factor 2 too much this way, however. Even, before the development of" the Dirac equation, Thornas showed that tMs simple classical armrnent i s incomplete and gave the correct term (6). The s iha - tion is d;ir%eread for the anomalou~ morrxenh intraduced by PauU to describe neubons and protons (aee Problem 3 below). In PauIi's mmodified equation, the momalous moment does appeas w'rth the factor 2 vvhn multiplying terms (5) and (6).

Problems: (1) Apply Eq. (12-6) Lo the Wdrogen atom and correct the energy levels to f i rs t order. The resu lb should be compared to the exact results.? Note the difference of the wave functions a t the origin of coordinaks, This difference actualIy is too restricted in space to have any imporbnce. Near the origin the correct solution to the Dirac equation i~ praportioml to

for the hydrogenie aloma , while the Schrbdinger equation gives 0 - conatmt a s r -+ 0,

t gchiff* "Wanturn Mechanics, McGraw-Hilt, New York, 1949, pp. 323fif.

(2) @uppose A iuld $I depend on time. b t W = iB/a t and follow tfirawh the procedures of this lectme to the s m e order af approa- mation.

(3) Pauli" e d i f i e d eqwl;ion can be applied to neutrons a d protons. It ilts obtained by adang a term for anomalous moments to the Rirac equation, thus

Multiplyiq by P , this may be written in the more familiar "LHrzmil- toniarr" expression

i(@/at)@ = PI,,,, 9 -+ p@ (P E) -cl! * E)*

Show t h d the s m e appsaamation which led to Eq. (12-6) will naw p r d u c e the b r m s

far protans, and a similar expree%sian for neutronw, but Mth e = 0. (4) Equation (12-7) cm be used to inbrpret electran-neutron scat-

tering in an atom. M a t of the s ca t t e r i q of neutrons by atoms i s the ieotropie s c s t t b r i ~ from the nucleus, However, the electrons of the atom also scatter, and give r ise t a a warre which inbrferes with nuclear scattering. For slow n e a r a m , $hi@ eEeet i s experimentally oh- served. lit is interpreted by term (5) of Eq. (12-6) [as mdif ied in Eq, (12-7) vvith e = 01 . Since the electron charge is present outside Ule nucleus, V E has a value different from 0, Term (5) c m be used in a Born approxirnalf on to cornpub tlre mpli tude for neutron-elecdron scattering, However, when the effect was first discovered, it w m @X-

plained by the assmption of a neutron-electron interaction given by the potential e6(E1), where 5 i s the Dirac 6 function and R is the neua Itron-electron distance.

Comwte t-he scattering mplitude vvith ct9(R) by the Born approdmation and campare with f ia t given by term (5). ~ X Z O W t b t

In order to interpret cd(R) as a potential, the averwe p-cltential 5 i s defined as that potential which. acting over a sphere of radius e2/mc2, would p r d u c e the same effect,

Using = - 1.91 35 eB/22hlM, show that the resuiting V agrees with exprimental results within the slat;ed accuracy, i.e., 4400 -1 1100 ev, t -f L, Foldy, Phys, Rev., 87, 693 (1952).

RELATIVXSTXC W A V E E Q U A T I O N

(5) Neglecting terms of order v2/c', show that

Solution of the Dirac Equation for a Free Particle Thirteenth L eeture

It will be co~vttnlent to use the form of the Dirac wuaaan rrlith the Y 'ET when ao2;rriw for the free-particle wave hn~tiom

Using the definition of Lecture 10, = yl, aP

and the Dirac equation may be written

(Recall that the quantity 4 = ypaP i s invariant under a Lorentz transforms- tiaa,)

Xt is neeesgarg to put the probhility density and current into a four- dimensional form. In the smclal repreaenttstion, the prab&iUty deneity m4 current are given by

SOLUTION O F THE DIRAC E Q U A T I O N

If the relativiastic adjoint? of .1XI i s defined

in the stiandard representation, then the probability densitqy and current may be written

To verify this, replace 5 by V*@ and note that p2 = 1 and that flyp = ap.

Ezercises: (1) Show &at the adjoint of 9 satisfies

(2) From Eqs. (13-1) and (13-3) show that V, j = O (conservation of pr ohbility denkrily) . In general, the adjoint of an operator N is denoted by 3, and i s the

same a s N except &at the order of aEE y appearing in i t i s reversed, and each explicit i (not th_ose cokliained in Ule y's) is replaced by -i. _For example, if N = y, yy, N = yyyx = -N. If N = iyg = iy,yyy, y, , then N = -W, y,y,y, = - iyp The EolloMw p r o p r t y tabs the place of the Hiermitian property s o useh1 in nonrelativistie quantum mechanics:

For a free particle, there a re no potentials, s o 85. = 0 and the Dirac q u a a o n becomes

To rsoltve &is, try as a aolution

T't3 iis a four-componsnt column vector,

The adjoint Q i s the four-component row veclor h the

st;ljln&rd repmsenktion, Muktiplicactlon by P c b g e s tke sign of the3 third a& fau&h components, in addition bo c ing \k* from a column vector to a raw vector,

58 Q U A N T U M ELEGTEEODYNAMIGS

.?b is 13 four-component wave Ecrnction m 3 what is memt by US trial soluUon i s that each of the four components is of &is farm, t;hat is,

Thus ul, u ~ , us, and u4 are the components of a column vector, mdl u is called a Dirac spinor. The problem is now to determim what restrictions must be placed on the u% s d p% in order that the trial rJaXutian satisfy the Dirac equation. The VU operation on each component of + multiplies each component by -Qp. so that the result of this operation on produces

so that Eq. (13-5) becomes

Thus the maumed solution will be satisfactory if &a = mu. To rslmplify writiw, it will now be assumed &at pat icte mmes in the xy plane, so LhaLL

Under these conditions, d = y, E - - y,p. in standard representation

By components, Eq. (13-7) becomes

(E - m)ui - (p, - ipY)u4 = O

(E - m)uz - (B, + i%fua = Q

@, - i p , ) ~ ~ - (E I- m)u~ = @

@, + ip,)ut -. (E +- m)u4 = O

SOLUTICQbJ Q F T H E D I R A G E Q U A T f Q N

The ratio ug/u4 can be determined from Eq. (13-98) and also from Eq. (13-9d). These two value8 must agree in order that Eq. (23-6) be a eolurtion. Thus

This is n& a surpridng condition. Il. s~tates that the p, must be &men 80

as to satfefy the relativistic ewation for total e n e r a . Similarly, Eqs, (13-9b) m d (113-$c) can be solved for u2/us giving

which also lsada to codition (13-10). A more elegant way of obtzliaiw exactly the sme condition is to s tmt

directly with Eq. (23-7). Then, by multiplylw %is i~3quation by $ gives

m e former is the same cmdition as obt&ned bfore , and the l a tb r $8 8

trivial solution (no wave knction) . Evidently there we two Xinearly i ndep~den t solutiom of the free-parlick

Dirac equation. Thia is ao because bsubstihtion of the a a s m e d solution, Eq. (13-61, ints the Dirac equation @vs& only a condiaon an pair8 of the U%, ul, U, a d uz, us. It ia conve~ienl; to choose the indewndent solutions ao &at each h w Luro componenb wUch are zero, n u s W u% for Ulris two solutions can be t&en as

where the f o l l o w i ~ notation has been used:


These soluaons are not rrmnndized,

ISEFmITEON OF THE 8Fm OF A MOVmG ELECTRON

m a t do the two linearly independent solutions mean? There must ke some physical qumtity that can still be swcifisd, wkch will u ~ w e l y determine the wave h e l i o n . A ia h w n , for exmple, that in the coordlnab sgs- tern in which the p a r t i ~ b ier staHomry acre me WO passibb spin orlenta- tiane. Mathexnatioally swmw, exjistence of two solutiom to the eigenvalue. equation #U = mu implies the exiabnce of an a p r a t m &At commutes with 6. nis operator will have to be discovered. Qbserve hat yti a~ticom- mutes with $; &at is, = -hI. Aleo observe that m y o p r a t a r [JFJ will anticommuter; Mt;h $ if W e p = O, besause

Ths combination yswof &ese two antieommutfng operators is an op ra lo r which commuke? with that Is,

The eigenvalues of the owrator (iygfl) m w t n07ull be found (tlne i hais, been added to nn&e eiwnvaluee come out real in w b t follows), Deaclting &ese eiganvalues by 8,

To fincl the possible valuss of a, multiply Eq. (13-23) by i y p ,

I[f W * W is t e e n to be -1, the eigenvalues af the op ra to r iyS JW are rt: l. The sigaflicanw of the choice W W = -1 ia as fof lows: In the ~ys tenr in which tXle particle ier at rsct, p, - =p, = O and pl = E. Then

S O L U T I O N O F T H E D I R A C EQUATXOPJ 62

mus , W * W -W * W = -1 or W g W = 1, This staWs that in the emrana t e systei?.m in which the particle is at re&, W is an oranary vector f i t has zero fowth conzwnent) Mth unit Iene~;tX1,

M e n the particle moves in. the xy plme, chmsts p to be y,, so the omrator quat ion for iyg v k c o m e s

Using relationships derived in b c t u r e 10, this becomes, for 8 stationary particle, t

This choice m k e s fdCr the o, oMrator, and the relationship d t h spin is clearly demonstrated. If ws define u to satisfy bath 6~ = mu and iy&u = su, this completely smeifies U. It represents a particle m o v i q wi& momenbm p, and having its spin (in the coordinate syatem moving with the partide) atong We W, axie s i ~ e r positive (s = + X ) o r negative fs = -I).

Exercise: Show that the first of the wave functions, Eq. (13-11), i s We S = -i- l solution and the secand is the a = -I solution.

AnoWer way of obtaiaing the wave Eunction for a freely moving electron is to p r f s r m an eqavalence tran~farmation of the wave function as in Eq. (10-12). If the electron is initially at re& w i a i& spin up or down in the z direction, then the spinor for an electron m ~ v i ~ with a veloeity v in the spatial direction k ies

[For normafization, see Eq, (13-14),] From Eq. {XO-XX), S is gfven by

cosh U = 1/(1 -

'f For a stationary particle y,u = U.

Q U A N T U M ELECTRODYNAMICS

(2m)'/2 cosh (u/2) = [ m ( ~ - v5-li2+ = (E + mjl/2

Writing f = (E+m), a =?,y, aml noting (~~-rn') ' / ' =h, we get

For the case that g i s in the xy pime, t h i ~ just gives the remlt, Eq. (13-XI), with a normalization factor l / n

Noticing that for an electron at rest y,uo = uo , may be written

It is olear that this is a aolutian ts the free-m&lcle Birac equation

for

In namelativistie quantum mechanics, a plme wave is namdized to give unity prob&illliy of andinl4; the particle in a cubic centinneter, t h t is, ik*iE = 1. h malwous normalization for the relativistic plme wave mi@t be some- a i ~ l i b

flowever, 9*4! transforms sfmifarly to the fourth component of a four- vector (it is the fotutth component oaf four-vector current}, so M s normalization would not be invariant, It i e poasibte to m b a rehtivistically invariant normalization by mtting U+ u equal to t b fowthi compomnl of a

S Q L U T I O N OF T H E D I R A C EQfJATXQEJ

sufbbie fom-vector , For exampk, E ia the fow& component of the momentum four-vector pp, s o the wave function could be normalized by

The constant of proportionality (2) Is chosen for convenience In later formulas, Workhg sut ( uyt U) for th@ s = .:+ 1 skte,

The Cl is the normaiizl_ng factor multiplying the wave functions of Eq. (13-11). In order that (uy, u) be equal to 2E, t h e normalizing factor must be chosen (E + m)-'/" ((F)-'~. In terns of (uu), this normalising condition becomes

The same result is obhimd fa r the s = ---.X stab. Thu8 the normaffzing condition can be h b n I ~ B

r, the f o l l o u r i ~ cm kw shorn to km true:

It will be conveniepll to have the matrk elemenb of all the 7% Sheen va- r l o u ~ initkl and f h a l s a t e s , so Table 13-1 has b e n worked out.


TmLE 15-1. Matrh Elements for Particle Moviw in the xy Plane

1 2m F2Fi - Pi+Pz- 0

Y X ~ P X FzPt+ + Pa- F i o

YY 2py -fF2~1+ "(" fP2-F I 0

7% 0 0 -Px*F2 + Pz+Ft

Y t 2E t + Pt * Pz- 0

SQLUTXQEJ OF THE D I R A C E Q U A T I O N

IkzrnzMng cases: To obtain the case &ere 71 is a poa i t ro~ at rest, Lhe table gives mg~a~~u~) if oxre p ~ t s Fi = 0, pl+ = 1 = pi- in the table. For both at reat as wsitrons, the table gives (QzMul) wi& Fi = F2 = O; PI+ = p ~ + = 1.

Fourt~sen fh Lecture

The matrix element af m operator M between i&dial slate uI md final state ug will be denobd by

The matrix element Sa independent of the representations used if ~ e y are related by unitary eqavalence tramfornnatiow , That is,

where Ule groper& 8 = S-$ haa b e n a s s m e d for S. The straightfaward m e ~ d to cornpub the matrix elemen& i s %limply to

write a e m sut in matrix form and carry out the o~rat i ions. l[n W e way the data in Table 13-1 were obtained.

Mar m e a d s may be used, however, sometimes sfmpbr md sometim~ss leading to corollary information, a s illustrated by the following example. By dhs mormiklization canven-tiortr,

s l a ~ e )Ilrr = mu. Similarly,

Q U A N T U M ELECTRODYMAMXCS

(aYppu) m(ayp~)

But also n&e thiat

{ B ~ Y , U) = m(O[yp U)

beeawe Qg3 = g%l =: m& Addiw the Lvvo expreshsions, one obtains

From the relation proved in the i~3xerci~ee sat

it is seen that

h, +?,P =W, Y, = Z

But p, ia just a number, BO it f~ilowtl that

md aince flu = 2m, by normalization

(ilyp~) ZP,

mrt-hermor.~?, the general relation

is obtarSned. norn tM~3 It i s SBESIZ why fjhe possible normalization

P r o b l m : U&% rne&dg andogma to the one just demoaslrated, show that

Xt was found &at a neeeesary condition far solution of the Dirac equation to exist i s

E' = pZ + m'

S O L U T I O N QF T H E D I R A C E Q U A T I O N

The nneaniq of the positive enerw is clear but Llxat of the negative is not. It was at one time suggested by SckS&inger &at it should be arbitrarily excluded as having no mead*. But lit waa found that &ere 858 two hndwen- tal objections to tba excluaian of negative e m r m stsws, Tbe first ie Nysi- cal, &eoretically physical, that is. For the f)irae equaaon yields the result &at starting wiLh a system in a polsiHve enerf3;Y slate &ere Is a probability of induced trasi t ions into mgative e n e r a s tabs , Heme if they were ex- oluded thias would be a coneradiction. The aecod objection is mathematical. That is, excludiw the negative energy s t a k s lea& to an lineomplete set of wave functions, It is not possible to represent an arbitrary h c t i o n a s an expawion in functions of m incomplete aet. This sitxlation led Sehr0Mnger into 'insurnnount&le difficulties.

Prohlem: Suppose that ;for t < O a pmticle is in a positive en- srw stab moving in the x direction with spinup in the z direction fs= +X), Then at t = 0, a constant pokntfal A=A,(A, =Ay = 0) is turned on a d at t = @I' it i s turned off. Find the probability t h t the parLiele is in a negative energy sk& a t t = T.

Answer:

fialiabiliw of beiw in negative enerm eta@ = A~/(A' + m5 sin' [(m2 + A$'/'T] at t = T

Nab &at when E = -m, 1/fl= m , s o Ithe ups apparently blow up. But actually the components of u also vagisfi when E = -m, so that ai Ximitiw process i s invalved. It may bcj avoided and the correct results abbinc3d simply by omitting l/fl and replacing F by zero and p* by 1 in the eomponent~ of: U.

The poaitiva enerm level@ form a conlC-inuw extending &am E = m $0 +m,

and the negative energes if accepted as such form a n o w r continuum from E = -m to -m. Between +m and -m &are are no avai lhle aner,eif;Y levels (see Fig, 14-1). Birac proposed the idea that ail the negative enerw levels a re normalfy fiflsd, Explanations for the apparent obscurity af sueb a sett of electrons in negative enerp;y. statea, if I t exists, usually cozltain a psycho- logical aswct and are not very satisfactory. But, nevertheless, if such a sltua#on is assumed toedst , some of the important consequences are these:

X, Ebctrans in positive enere;y states will not normally be observd to m a b transitions into negatiye energy states b c m s e these s t ake a re not available; they are already k l l ,

2. With the sea of electrons in negative enerm levels unobsemrraible, a "oleH h it prducedt by a trmsition of one of its eleclrans into a positive exlerw state should manifest itself. The rnaPrifestatlion of the hole is regarded as a positron and behaves like an electran with a positive charge.


positive energy levels

+m

--m newtive energy levels normally

FIG, 14-1

3 , The Pauli ernclueion principle is implied in order that the negative sea may be full. That is, if my a m b e r rather than just one electron could oc- cupy a given state, i t would be impoaaibls to fill alI. the nagaX-ive enerm states, It i s in tMs way &at the Dirac aeo ry is sometimes consider& aa? "pr oaf H of the exclusion principle ..

ho-r interpretation of negaave energy s t a b s haa been proposed by t;he prersent author. The furtdmental idea is &at the 'kncsgative en@rmt' states represent the s b b s of electrons movz'w baekwrd in gm@,

In the c la~s ica l equa~on of motion

reversing the dlrectit-ion of p rawr time s amounts to the s m e m reversing the s i c of Iha charge so that the electron moving b a c h a r d in time would look like a. positron moving f award in time.

In elementamty quantum m e c h a ~ c s , the total amplitude for an ebctron to go from xi,h to. x2,tz was compted by smming the mplitudes over all possible trajectories between xl,LI and xz,ta, aseuming t k t the f;rajec- toriea atlways moved forwzurd in time. Thr~tee trajeetortazs might apwar in one anerenslion aa shown in Fig. 14-2. But ~ t h the new pdnt of view, a pss- aible trajectory mi&t b as shown in Fig. 14-3.

Imtzginlng oneself an ob~erver movtw a l o q in time in the ordinmy way, being conscious only of the present and pmt, the sequence of events w a l d apmar m follows:

S O L U T I O N OF T H E D I R A C E Q U A T I O N

X 1 X2

FIG. 14-2

- t only the initial electron present --e the irtitial electron still present but ~omewherfs, else an

eleclrtm-posritrcm pair is formed tr- t, the initial electron, and newly arrived electron positron

a re preerent h% - the positran meets with the injtial electron, both of Wlem

amihilating, leaving only the previously creabd electran t,- tz only one electron presexll

To handle this idea quantum mechdca l ly two rules must be followed:

10 Q U A N T U M E L E C T R O D P N A M Z G S

1, Xn calculating matrix rslementa far gositrom, the gmif;i~w of t b initial and final wave firnctiom must reversed, That is, for aia electron move ing forward ia time from a past state to a firwe state UIez matrix elemsnt i s

But moving backward in time, the electron proceedsfrom +,, to Y,,, so the matrix element for a positron is

2. If the energy E is positive, Ulen e - ' ~ ' ~ is wave function of an eleo- tron with energy p& = E. E E is negative, emBP'~ is the wave function of a positron. with eaerm -E or LE (, and af fow-momentm -p.

Potential Proble

Fifteenth Lecture

PAIR CREATION A m AINNXHIUTION

Two possible pass of an electron be iw scattered between the states 91 and were &seussed in, the last lecture. m e s e are:

Case I. Both @ z states of pocaitive energy, interpreted as 9g electron in ""past," !Ifz electron in "f~$ure. '~ This is electron scattering.

Case 11, Both ?1"$, UF2 states of negative enera interpreted as 4rl positron in "future," "2 positron in ""pst." This is positron scattering.

The existence of negative energy states makes two more t p s of paths pss ibfe . T h e ~ e are:

Case ZfX. The positive energy, @2 negative energy, in te rpewd as Jrl in ''ppa~t," %2 positron in. ""past." Both states a r e in the paat, and no%ing in the futwre. TUs represents pair amiMlat_ion.

Case N. The negative energy, 4T12 posjiitive emrgy, inbrpreted a s Ol positron in "future," "2 electron In "'fuh,rre," This is pair ereation.

Case I Case I1 Case IIX

FIG. 15-1

Gase TV


The EOW cases can be di%rmmed a s shown in Fig, 15-1. Note that in each d i w r u the arrows point from %i to q2, a l a o w h time is increasing upward in all cases, The arrows give the direetion af motion of the electron in the present interpretation af negative sne rm slates. h common lan- @we, the arrows point toward pctsitive or negative time according ta whether 16 is positive or negative, that iia, wheaer the state represented is that of an e l e ~ t r o n or a, positron.

CONSERVATION OF ENEMY

Energy relations for the scattering in case I have been established in previous lectures. It can be seen that identical results hold for case rX. To shaw this, recall that in case I, if the electron goes from the ene rw El to E2 and if the wrt;urbation potential ihs taken proportional to exp(--iwt), tfien this perhrbation br iwe in a positive ene rw w . To see t h i ~ , noLe that the amplibde for scattering is proportional to

=Jexp[(iE2t - iwt - iEtt) dtl (25-1)

AB has been shown, there is a resonance between E2 a& El + w, so that the only contributing energies are those for which Ez Et + w, h case 11 the s m e integral holds but: E2 and El a r e negative. A positron goes from an energy (past) of E,,,, = -Ez to an energy (future) of E = -Et. With the same perturbation energy, the mpfitude is large wain only if Ez = El + w or -Epas, = -Ef,, + W , so that Er., = w + E,,.,; that is, the perturbation car r i es in a. positive energy o, just a s i t does far the elect-ron case.

h thi~! nonrelativistic case (Scbiidinger equation), the wave e w t i o n , in- cludng a perturbation potential, i s written

where V i s the perbrbation potential and He is the unmrbrbed Hmilto- nian, For the free particle, the kernel giving the ampllktde to go from point 1 to point 2 in space and time can be s b w n +m be

where! N ie a normalizing factor depnding on the time interval t z - tl and the mass of the particle:

P R Q B L E N S I @ Q U A N T U M E L E G T R O D Y N A M f G S 7 3

Note that the kernel is defined to be Q for t2 tr. It can be shown t b t W;@ satisfies the epat ion

The propagation kernel Kv(2,1) glving a similar mplituds, but in the presence of the perbbat ion potential V, must satisfy the equation

XL can be shown thrtt Kv can be compukd from the series

]In ease the complete Hmil todan H = H@ + V is fdepndent of: time, md all the e stationary s t a h s #, of the system are known, tf-zen Kv(2,1) may be obtained from the sum

The extension of these idea@ to Lhe rslativistie case (Dirac equation) i s rstraightfomurardl., By chooalw a pat ieular form for the H a i l t o ~ a n , the Dirac equation can be written

Defining the propalyatim kernel a s K ~ , then the kernel is the aoluaon, to the equation

The matrix p i s inserted in the Imt term ia order that the kermll derived from the Hamiltonian be relativistically invariant. [Note the similarity to the nonrelativistic ewe, Eq. (15-G).] lMultiplying &is equation by B, a simpler Eom results:

The equation for a free particle is obtained simply by letting = O, then calfiw the free-psrtiete k e r e l H;, ,


The notation K, replace8 the & of the wmelativistic case, and Eq, (15-10) repfaees Eq. (15-4) a s the defining equation,

Juat aa Kv ean be expanded in the ser ies of Eq. (15-61, s o can be expanded m

Note that the kernel is now a four-by-four matrix, so that all component8 of 3 can be determined. Since this is true, the order of the terms in Eqt. (15-11) is important, The element of integration i s actually an element of volume in four-space,

The potential, -ie$l"(l) can be interpreted as the amplitude per cubic eenti- meter per seeand for the particle to be scattered anee at the paint flf. Thue the interpretation of Eq. (15-11) is completely analogms to that of Eq. (15-6).

P~oblenz: Show that ltA as defined by Eq, (15-11) i s consistent with Eqs. (15-8) and (15-9).

On the nawslativfstic case, the ga%s a l o ~ g which the particle reversed its motion in time are excluded, Pn. the present erne is no lower true. The existence and interpretation of the negative ener&y eigenvaluea of the Birac equation allows the interpretation and inclu~fon of such p.aells.

TaMw t4 2 t8 implies the elciaten~e of v i s a& pa;irs. The section from t4 ta tS repreeellf;~ the motion of a positron (secs Fig. 15-21" h a time-slationary A d d , if the wave hnctiom @, are k n o w for all the

states of the sysbm, then may be defimd by

rr - C I-i~,(tz - tr)l +,(xz)& (xl) neg, energies

P R O B L E M S I N Q U A N T U M ELECTROD'YPJAMIeS

FIG, 15-2

Another aolutution of Eq. (1 5-9) i s

m ~ ~ A ( 2 , 1 ) = C exp[-iE,(t2-tlll$n(~g)$n(~t)

pos. energies

t C exp [- i~,( t l - t l ) ~ # ~ ( x ~ ) ? ~ ( x ~ ) t2 ' neg. energies

Equation (15-13) ha8 an interpretation comiaterrt with the positron interpretation of negative energy states. Thua when the t h i n g i s " o r d i n ~ y p p (L2 > e,), an electron ie presexll, and only positive energy states contribute, W e n the t i m n i ~ i s ""reversed " (t2 <. tlf , a positron is present, and only negative sneref5" %;takes contribute, On the sUler hmd, Eq. (15-13) does not have so satisfactory an interpretation. Although the kernel bA defined by Eq. (15-13) is also a satisfactory mathematical 8ofuLion of Eq. (15-9) (as shown below), the interpretaaon of Eq. (15-13) require& the idea af an e k e - tron in a negative energy state.

To show that both kernels a re solutions of the smcs inhomogeneous equation, note that &eir difference is

for all. t2, This is, term: by term, a solution af the hamqeneouss equation [i.e., Eq. (15-9) with zero right-hand side). The possibility that two such

'76 Q U A N T U M ELEGlPR0DUWAMPC:S

solutions errist results from the fact &at boundary conditions have not been definitely fixed. We shall always use K , ~ .

The kernel K + ~ , defined by Eq. (15-121, allows treatment of case 111 (pair amihilatfon) m d caBe N @dr creation) sham at the b g i a w of a s be - ture. Pn each case, the got,tentiai, -Te&(3), acts a t the fnbrscact;lon of positron and electron paths.

Sixteenth Lecture

U8E OX;" THE KERNEL K,(2,1}

In the nonrelativistic t;heory i t was poasibfe to calculate the wave function at a point xz at time tz from a knowbdge of the wave hnctian at m earltier time tl (see Fig, 16-1) by means of the nomelatfvistic kernel Ko(xz,tz; f t j , t 1 ) r

It might be expc ted that a relativirstie ganc3raXization of this would be

FIG. 16-1 FIG. 16-2

m s turns out to b incorrect, however. I t i s not sdficient, in the re1at;idsr- tic ease, to h o w just the wave hnction ~t an ewl ie r time only b c a u m KJ2,f) is not zero for tg c tie Men the kernel, is det&ned in Wlis m ( b c h r e 15), the wave funetion at xz,tz (see Fig. 16-2) is given. by

P R O B L E M S IN Q U A N T U M ELECTRODYplTAMXGS

The firat term is Lfxe contribution from positive energy states at earlier times and the sacsnd term is the contribution, from negaave energjr states a t later t h e s e mis eq re s s ion c m be generalized to a t ak that i t is me- assary to h o w 9(xltx) on zt faw-d-innensfonat surilaee, 6~urrounid;iq the point xz,tz (see Fig. 16-3):

where W e is the four-vector normal to the surface that encloses xz,tz

The a p l i t u d e to go from a state f to a state g under the action of a potential & is given by an expression similar to &at in nonrelativistic tbeary,

Using the expansion of ~ , ~ ( 2 . 1 ) in terms of K,(2.1), Eq. (16-121, and assuming h a t the amplitude for transition from state f to state g as a free particle is zero ff and g arthogonal states), fAe first-order ampfitude far transition (Born appra&mtion) is

It lie converrierxt to let

Theoe state &at &e particle has the free-particle wave function f just prior to scat ter iw and Lbe free-particle wave hncticsn g juet a f k r s c a t t e r i ~ , and that i t elinninabs any computation of the moti on aist ai free particle, The amplitude for t ransi t fo~, ta first order, mily be written

Q I T A M T U M E L E C T R O D Y N A M I C S

oigdMes integration over time as well as space). The second-order Germ would bbe written

-(1/2) Jjg(4)eA(4)K,(4,3)e8((3)f(3) d ~ s drc

If f(3) ie a negative energ;y statze, then i t represents a positron of the future instead of an electron of the past and t h ~ process described by LMs amplitude is that of pair production.

We shall make use of the theory just presented to calculate the scattering of an electron from an infinitely heavy nucleus of charge %et. Suppose the incident electron has momenhm in the x direction and the scattered electron has momentum in the xy plane (see Fig. 16-4):

FIG, 16-4

The potential i s &at of a stat:fonary chwge 243,

The i ~ l i a 2 and final wave funetism we plane waves:

f(1) = u ~ ~ - ~ P I ' ~ g(2) = u2 e-@zex (four-component wave fune tion)

Thus, foy Eq. (I&-$), the first-order mplitude for transitim from state f to stab g (mamentum pi to miomentm p2) i s

P R O B L E M S IN Q U A N T U M E L E C T R O D Y N A M I C S 79

Separating space and time dependence in the waive funct;ions, this becomes

The first intrzgral is juat V(Q), a three-dirnenoional Fowier trawform of the potential, which was evaluated in nomeXativisUe scatMring theory:

The probability of transition per second is given by

Trans. prob./sec = ~ ~ ( D N ) - ' / M / ~ x (density of final states)

(16-6)

This i s a result from time-dependent ~ r t u r b a t i o n aeory , the only new factor is a normalizing factor (EM)-"vvhieh takes account for the fact t;hat the wave functions a re not normalized to unity per unit volume. The n N is a p r d u c t of factors rJ one for each wave function, o r particle in the initial state, and one for each final wavB fmetion,

for each particle in question. In our normalization, then N = 2E.l The reason far this factor is that wave fmctions a r e nomallzed to

where, m in the comwtation of trmsition probability, tlxey should be normalized in the conventional nonrelativistic manner %*Q = 1 or (Gyt u) = 1 (so N =: 1 for that ewe) ,

The matrix element M, as salculabd in t;ttis nzamer, is relativistieally i nva r i a~ t and in the f u b e the chief interest will be in M, The kansition probrzbitity, bovving M, can be computed from Eq. (16-6).

I)en@lty of etabe, Crosa boaon. For the efsetson scat terhg problem under consideration,

so the transition probability i s

where t h ~ density of f iw l states has been obtained in the following mamer:

Density of states = % = l

hut = + m', so dp2/dE2 = E2/p2 and

Density of states = ~ ~ ~ $ n / ( z r ) ~

When the incoming plane wave i s normalized to one p&ic l e per cubic centimeter, the cross section is given in terms of the tmnsitfoxl probslbillty per second? aa

The essential difference between the relativistic treatment of scatt_ering and the nonrelativistic treatment is c a n b i n d in &a matrix element; ( u2yt ul). From Table 13-1, for a pasLicfe moving in the xy plane and et = + 1, s2 = +l,

where

[E, = E2, conservation of enerw, follows from the wture of the time integral. in Eq. (16-5)], and

( m w ~ t u d e of final momentm equal to mamitude of id t ia l momentum follows from El = E2),

Therefare, vt = p i p t .

PROBLEMS IN Q U A N T U M ELECTRODYNAMICS

When st = + 1, s2 = -1 o r s1 = -1, s2 = -t 1, the matrix element of yt is zero. When si = -1, sZ = -1, the atsaolute value of Gfie matrix element i s the same a s for el = + 1, sz = -t- I. Thus spin doe8 not change in scattering (in Born apgrodmation) a d the cross section is independent of spin,

Ths criterion for validfly of the Born approximation, used in obtaining tMs result, irsl %e2m cc 1. In tha e d r e m e relativistic limit v o. T k i ~ beaornes Z <c 137. Just as; for the aonreliktivistic ease, the scattering can actually be calculated emotly (correct to all orders in the potential) for ttza Coulomb pobntial. T%ia emc t salution of the Dirac equation involves hmrgeomet r ie knctians. Xt was f i r ~ t worked out by Matt and i s a lzed Mott scattering. For moderate ener@ee (200 b v ) there is s m e prohbili ty for change in spin, Polarized eie etrons cauld h produced in this manner.

Problems: (l) Calcdatr? Ule Rutherfod seat ter iw law for the Klein-&&on equttticm (pstrticle with no spin). ResuEl;: same formula ae just given with 1 - vZ sinZ (8/2) replaced by 1.

(2) S o w t h t this scattering forrnuJla is also correct for positrons ( u ~ e ~ 8 i t r o n staites In caloulatiw matrix element),

Seventeenth Lectam

GALGUMTfQN OF THE PROFAWTXOM KERNEL FOR A FREE PARTICLE

As shown in a previous lecture, th@ propagation kernel, when there is no perturbing potential and the Mamiltonian of the system i s conatant in time, ia

For a free parlielie, the eigenfunction~ (p, a r e


a& the sum over n becomes an integral over p. The up is the spinor correspondiw to momrsntm p, poaitive o r negative ene rw and spin up o r down, a s appropriate, Then the propagation kernel for a free particle is , far t2 > tl,

1

spins

for E, = + (p2 + The factor 1/(2zl3 is the density of states per unit volume of momentum space per cubic centFetar. The factor 1/2Ep ar i ses from the normalization u u = 2m o r uy,u = 2Ep used here. The up a r e those for positive energy. For negative energy E, = - (pZ + m2)lh the up are changed accordingly and Ki(2,1) becomes, for t2 tl,

The plculation will be made f i rs t for the case of t2 tl. We first calculate up ~p for p~s i t ive energy, and p in the xy plane and spin up. Under these conditions

Note that U,;, is the opposite order to that usually encountered so that the product i s a msttrix, not a scalar. T h t is,

by Wle wual rules for mzzt;dx multiplieatioxz. But

(p, + ip,)(-p, + ip,) = -pZ = -E' + m2

and the matrix becornens

PROBLEMS. IN QUAPJTUM E LEGTRQDYNANLfGS

By the same process, the result in the spin down ease ia

W Q E + m -p,-ipy 0 upup = (spin dam)

0 P x - 4 ~ ~ -E+m 0 0 O 0 Q

11, may be verified easily dhat the sum of these malrieea for spin up and spin down ia represented by

In Ule genr~ral case when g is in any direetion, it is elear h t t b only chwe is an additioml h r m -p, y, . So, in geukeral,

N N

up + ( ~ ~ u ~ b s p i n down = : : ~ i ; " " ) " r - ~ * ~ + m = : ] p l + m

The sign of the energy was not used in ctbhinlrrg thia r~38ult SO i t i~ the same for ei.f;her siw.

Now put t2 - tl = t and xz - xt = X, For L > 0, the propbmtion b r n e l b- comes

The appearance of p in the form Ep = (p2 + m')'/' in the time part of the exponential makes this a difficult iuxLegraI, Note a t It may also be written in the form

84

where


h this form only one integral insbad of f m r med be dome It may be verified as an exereiac; a t for t 0 the r e s d t is the same except that the sign of t is changed, so t b t putting It/ in place of t in the formula, for I+(L,x) makes it good for all 1,

mis inbgral h s been carried out vvdtfi the followiw result:

where s = +(tZ - x3'I2 for t > X, and -i(x2 - tZ}I/' for t X . & ( s ~ ) is a delta function and ~ ~ ' ~ ' ( m ) is a Hankel function.? Another expression for the foregoing fe

~+(t,xk = -(l/srr2) /oa d a exp I-(i/2)[(m2/(u) + or (t2 - x2)))

Both of these farms a r e too compficabd to be of much. practical use. It will be shown shortly t h t a tremendous simplification results from transforma- tf on ta momenhm represenhtion,

Note t k t I+(t,x) ac-fly demnds only on /xtl not an its dfmtction, In t b time-space diagram (Fig. 27-1) the a p e e axis represents 1x1 and the diag- onat lines represent the surface of a light cone includiw the t axis, that is, the accessible region of I - 1x1 space in the ordinary sense. It; can h 8h0m that the asymptotic form of I,(t.@ for large s i s proportional to e-lmS. W e n one% region of accessibility i s l imibd to thc3 inside of the light cone, large s implies t2 >> /X I ' , s o that the region of the asymptotic approximation lie^ raughly within the dotkd cone around the t axis and is

FIG. 15-1

?See Phys. Rev., 78, 749 (1949); included in this volume,

P R O B L E M S XN QUAPi[TflrM E L E C T E T O D Y N A M I W

The first form i s seen t s be essentially the same a s the p r o m g a ~ o n k e r e l for a free mrticle used in nonrelativistic t;heory. If, a s in the new t k o r y , possible 6gtra~ectorieafl a r e not l imibd to regions within the lig& cone, another region included in this a s p g t o t i c agpro~mat ion is h t &tMn the dotted cone along t6e /X/ axis where large s implies [x12 >> t2. Hence

It i s seen that the distance along fzcf in which this &comes small i s roughly the Campton wavelengW (recall &aL m -- mcfi vvhen it represent8 a len@hv" a s here), ss h t in reality not much of the t - 1x1 h3pee oubide the light cone i s aceessible.

The tranh~formtion to momentum representation will lilow be made, This ie facilihted by use of the inkgral fomula

The ir brm in the denominator is introduced solely to enaure passage around the proper side of the singularities at p: = E: along the path of integration. Passage on the wrong side will reverse the sim In the exponential on the right.

Problem: Work out the integral a b v e by contour integration; o r oaerwise ,

Using the inkgral relation above, d,(t,x) becomes

But E: = p2 + m2 so this i s

where p i s now a four-vector so that d$ = dp4 dp, dp2 dp3, and = pp pp . b r e a f b r the ir term will be omitted. Its effect can h inaluded simply by imagining that m. ha,^ an infinll@simali nee t ive irnagimry part. En &ia form the transformation to momentum representation. i s easily accomplished a s followe? {we actually bb Fourier transform of bath spas@ and time, so I;bis ia really a monnenbm-enerw represenbtion):

where the dummy variable 6 h a ben. substitukd for p in the p intc?gral. But

Hence the 5 integration gives the result

Finally, applying the omrator i ( ip K;" m) to IE,(t,x) gives the propagation kernel (here x = ~2 - x i )

recalling that iJV opra t ing on exp l-i(p X)] is the same a s multiplying by Ifi, From the identity

the kernel e m also be written

By the same process used for X,(t,x), t b transform of K,(2,1) in mornen- turn represexllation is seen to km

T h i ~ i8 the result which was sou&t. Aemlfy &is: tranrsformation could have h e n obtstined in an elegant man-

ner. For K(2,I) i s the Green's anction of (IF - m), t h t ih~,

and id i s hewn ~t ijP i~ 6 in TX~OEXL~II.~UTXL represenbtion and 6(2,1) ihs unity.

P R O B L E M S I N Q U A N T U M E L E C T R O D Y N A M I C S 87

Therefore the transform of &is equation can fie writkn down immediately:

a s before. The fact that Eq, (17-1) for K(2,1) has mare &an one solution is re-

flected in Eq. (17-2) in the fact that (d - m)""' is singular if p2 = m'. We shall have: to say Just how we a r e to k n d l e poles a r i s i q from Ghis source in integrals. The mle &at @elects the particular form we wmt i s &at m h considered as. bv ing an infinibsimal rtemtive imagiwry m r t ,

Eighteenth Lecture

Since the propmtion kernel for a free p r t i c l e i s so aimply expressed in momentum represenbtion,

i t will be convenient to convert all our e w t i a n s to this representation. It is e s ~ e l a l l y useful for problems involving free, fast, m o v i q p r t i c l e s . This requires f o w - d l r n e n s i ~ ~ l Fourier t ran~foms. . To convert the potential, define

Then the inverse transform is

The function a(q) is interpreted as the amplitude that the ptential. con- him the momenbm (q), As an emmple, consider t l ~ Coulomb potential, given by A = 0, p = Ze/r.

SubstiMing into Eq. (18-1) gives

Here the vector Q is the smce part of the mamerttunn. The delta Euae- Lion 6(q4) arise8 from the time demndenee of & ( X ) .

88 Q U A N T U M E L E G T R O D Y N A M 1 C 8

Utrfx Elfamentta, An adivankge of momenttlxkt represenbtioxl i s the aim- plieity of computing ma t rk elements. &call that in space representation the firot-order pert-urbatlon matrix element i s given by the integral

For the free mrticle, this becomes

h momentum representation, this i s simply

where pl ia d~f ined snailogclusly to the three-vector q,

The second-order m t r i x element in s p c e represenbtion i s given by

Substituting for a free mrticle and also expressing the gotenth1 functions as %eir Fourietr transforms by mean8 of Eq. (18-g), this bcscomea

If Eq. (58-2) i s uoed for K,(2,1), this kernel can be writbn

Writing the factors that depnd on 71, this part of the inbgrrbl i s

l exp tip xl) exp (-iql xl) exp (-ipl xl) dr l

= (2n)'b4(p - qg - PI) (18-5)

d e r e the funetion &"X) i s to be lnterprehd a s 6(t1)&x2)6(y3)6(z4). Then the Zntepal over rl i s zero for all ~ except gf = di -t- 41, So the inkmal aver p reduces Eq, (18-4) to

P R O B L E M S f N Q U A N T U M ELEGTROIDVMAMIGS

beegrating over 72 results in another S fueuon [similar to Eq, (18-5)], which differs from zero only *en

Then inbgrating over ddq2 give6 finally

mese reaults can be written down immediately by inswetion of a diagram of the interaction (see Fig. 18-1). T%e electron, enter8 the region at 1 wi&

FIG. 18-1

wave function ui and moves from 1 to 3 as a free particle crf momanhm Xli. A t point 3, it is scattered by a photon of mo f a d e r the action of the potential -ie$(ql)]. bving;5 abfsorbed the um of the phoLon it Ghen movess from 3 to 4 a s a free particle of momentum 6% + & by eonaervation

, At point 4, it i s scattered by a second photon of nnomenwnn 42 [under the action of the pohntial -ie$(q2) absorbing We additional momen-


tun &)l, Fimlly, i t maves from 4 to 2 aa a free prartiele with. wave funetion u2 and momenwm & = & + d¶ + 42. It is also c h a r h a m f&e diaparn that the in&gral need b h k n over qi ody, b c a u m when $l and gi2 are even, io debrmined by d2 = -pjl --&. The law of conservatim of energy requires p12 = m', p22 = m'; but, since the intermediate state is a virtual stak, it is not necessary that (pl + di12 = m2. Since the operator X/(& + dl - m) may be resolved as (fit + dl + m)/((Pt + dl)2 -mZ], the impor- hnce of a virtual state is iaversely grapo~ional t~ the degree to which the consemaaon law is violabd,

m e rasults @ven in Eq8. (58-3') &ad (18-6) may h summrized by the following list of handy ruleat for computing the matrix element M = ( 5 2 ~ ~ 1 ) :

1.. An eketron in a virGual st_ab of momenl-um 6 contrib-uttls the amplitude, i/(jp5 - m) to PS.

2, A wtential containing the momentum q contribuks the amplihde -ie$(q) to N.

3. All indeterminate momenta qi are summed over d4qi//(21r)'. Remembr, in computing the integral, the value of the integral la desired,

w i a the path of inbgration, m s s l m the sinwlarlties in a definite manner. mlts r e p l a ~ e m by m - i~ in the inlegrsnd; then in the solution @lice the limit a s r 0,

For relativistic work, only a few termrs in the pertrxrbiltion series a r e necessary for eompuhtion, To assume that fast electrons (and positrons) i ~ k r a e l wi th tt p h ~ t i a l only once (Born appro&mtfort) Is often, odffeiently aceumte .

Afbr the matrix element is debrxnined, the probabilit;y of trawltioa per second is given by

P = 2n/(n N) J M ~ X (density of final stabs)

where fl N I s t k normalization factor defimd in b c h r e 16.

f8ea Summary of numerical factors for tramition pmkbilitlea, R. P, Fqnnnan, An 0psrator C&eulus, Phys, mv,, 84, 123 (2951); included in thts volum~ti*

Relativistic Treat of the Interaction of Particles ith Light

h b c t w e 2 the rules governing nonrelativistic interaction of particlea with light were given. The mles 8 k b d what ]potentialis were to be used in the calculation of transition probabilities by perdurbation t;heory, mose p- dentiala a re a k a applicable to the relativistic theory if the m l r i x elements a re eompubd a s described in kek r re 18. For absorption of a photon, the potential used in nonrefativiatic theory was

For emission of a photon, the complex conjugab of thle expression i s used, These potentials are normalized to ow photon per cubic centimeder and hence the normalization i s not invariant under b e n t z transformatiom , In a manner similar to that for the normlization of ebctron wave functions, photon pbntiszfs will, in the &&re, kw normalized to 2w photo- per cubic centimeter by dropping the ( 2 ~ ) - ' / ~ factor in Eq. (19-11, giving

A I, = (4~e') ' /~ e exp (ik * X) (19-ltl

This makes any matrix element cornpub$ with Uzeacz pobntials invariant, but to obhin the correct tramition probability in a @ven coordinate sgsbm, i t ie necessary to reinsert a factor (2w)""fsr each photon in the initial and final sk tes . This becomes part of the normalization factor RN, which con- k i m a similstr factor for each electron in the initial and final @&Le%.


h moment- repreat;ntation, the amplitude to absorb (emit) a photon of polarization e s is -i(4ne2)'12d. The polarization vector el, is a unit vector wrpendicular to the p r a ~ ~ t i o n vector. Heam e * e = -1, and e q = 0.

MDUTION FROM A T O m

The transition. prokbili ty per second is

Trans. prob./sec = 2% 1 ~ 1 ' x (density of final states)

where H is the matrix element of the rebtivistic Wamiltonian,

H = cr * (-iV - @A;) S.R.

between initial and final ahtes . T b t ia,

< f/H/i> = (4ne2)1/2J9f+[a! *eexp(ikex)] *i d vol (19-2)

Problem: %ow that in the nonrelativistic limit, Eq. (19-2) reduces to

x ew(fkerz)] d vot

This ia the s a m msult a s was obtained from the h u l i ftqution.

A relativistic treatme& of scattering of photons from e bctrons will, now be given, As an approxi-Lion, camider the electrons to be free (energies a t which a re1ativisd;ic treatment; i s necessary are, p;rtneral,ly, much greater than alaanie binding emr@ea), 'I"his will lead to the B e i n - N i s ~ m f o m u k far the Gomipt~n-effect c rass section,

hobn f (incamin X

recoil electron

FIG. 19-1

I M T E R A C T I O M OF" P A R T I C L E S W I T H L I G H T 93

For the incoming photon take as a potential Alp = alp exp (-iq! X) and for the outgoing photon take Az = exp (-iq2 * X). The light is polarized per- p n d i c u h r to the direction af propgation (see Fig. 19-1). Thus,

et S ql = O eg q g = O

91 g 41 = 9t2 = 0 and q2 q, = q12 = 0

A s initial and fiml state electron wave functions, choose

$2 = ~2 e x p (-ip2 * X)

Conservation of energy and momentum (four equations) is written

U the coordimte system is ehoaen 80 that electron n m b e r 1 i s a t mst ,

d2 = wZ(yt - Y , cos B - yy sin 8) (19-6d)

The l a t k r two eqwtioxrs follow from the fact that, for a photon, the enerm and momentum a r e both equal to the frequency (in units In which c = X). The momentum has been resolved into components, The incoming photon barn can be resolved into two tyms of polarization, which will h designated tym A and tyw B:

Type A has the electric vector in the z direction and type B has the electric vector in the y direction. SinrtiZarly the outgoing photon beam can be resolved into two t yws of polsllrization:


(4" 62 = Y z (B') 4% = y, cos B - y, sin @

Conservation of energy of xnornenbxn diehtes that either the angle of the recoil electron @ or the angle a t which the sca tbred photon comes off @ completely determines the remaining quantities. If the electron direction i s unrimportant, i ts momentum can be elimimted by solving Eq. (19-5) for plz and sqwring the resultiw equation:

= m 2 +o+O+Zmwl-2mw2-2wiw2 (I - cos 8)

where the last line was obtained from the preceding line by u s iw Eqs, f 19-31, (19-41, and (19-6a, c, d). This can be written

~ ( W I - 0 2) - cos 8)

This i s the well-hown formuh for the Campton shift in wavelength (or fie- qaeney) .

DlGREBJON ON THE DENSITY OF FmAL 8TATLS

By the method discussed in the earlier part of the course, the following final state densities (per unit emrgy inrterval) can h obdained. km of t o h l energy E and total linear moment= g disintegrahs into a two- particle fiml state,

Density of states = (2n1-~ dGI

(B-1) - Ej(P0Pl)

where EI = enerlS;V of pdlrticle 1; E2 = energy of particle 2; pg = momenturn, of particle 1; dQt = solid angle, into ~ i c h mrtiele 1 comes oul;; m1 = mass of mrt ic le 1; m2 = mass of gartiole 2; a& El + E2 = E, p1 + p2 = p.

Another useful formula is in terms of the f iml energy of m r t i c b 1 and its azfmul;h $1 [instead of @l). It i8

Density of states = (2~ ) - ' (E1E2/ /p/) dE1 d&

I N T E R A G T f Q N OF P A R T I C L E S W I T H LXQIZT

Special eases: (a) m e n m2 = /E2 = ar: E = m):

Density of states = (2~)""' Etjpl/ d Q f

Density of states = (2n)-3 [ E ~ E ~ dQl /(Et + E111 (B-4)

M e n a Bystern disintegrates into a three-particle fiml state,

Density of states = (2n)-' E3E2

Special ease: m e n ms = 03:

Density of states = (2n)-'E2 lpzl daz pt2 PI (D-6)

The Gonrpton efkct has a Wo-particle final state: takiw particle I to be photon 2 and mrticle 2 to be electron 2, from Eq, (D-l),

W ? day Dens3ity af states = ( 2 v 3 c~ $2

Calculation of /M/'. Using the Compton relation Eq. (19-7) to eliminate B, this bcomes

Density of states = ( 2 ~ ) ~ ~ ( ~ ~ w ~ ' dh2,,/mw

The probabiliiw of transition per second i s given by

iln woreung out the nrr%trix element M, there a r e Wo ways in which the scattering can b p m n : (R) the incoming photon is absorbed by the electron and then the e b ~ t r o n emits the outgoing photon; (8) the electron emits a photon and subsequently absorbs the incident photon. m e a e two procesees are shown diagrammatically in Fig, 19-2.

h momentum represenktion, the matrix element M for the first proeess R i s

bading from right to left the factors in the matrix element a re interprebd as follows: (a) The initial electron enhra wit& amplitude ul; (b) the electron i s first scattered by a potential (i.e., absorbs a photon); (c) bving re-

mived momentum 13fl from the potential the electron travels a8 a free eleetron with momentum gig + &; (d) the electron emits a photon. of pohrization 6,; and (e) we now ask for the amplitude, t b t the electron i s in a @Late ug .

Exercise: Write down the mtrix; element for h e secand process 5, The tob l matrix element is lthe e r n of thte;se two. Ratf onalize these matrix elements and, us iw the: table of matrix elenrrents (TaMe 13-1) work out 1 Mfz,

Twentieth Lecture

Far Lhe R diagram, M was fomd to be

-i4ne2f [l/(& + PI1 - m)] at)- = - i 4 ~ e2 ( G z ~ u ~ )

and as an exercise the: u t r i x e bmant for the S diagram. was f ctu~d to be

-i4%eZ{ &$1[~/(61 - d2 - m)] $2ul) = -i4aeZ( GZ S U ~ )

The complete ma t rk element is the sum of these, so that the eross section bcomes

The p r o b l a now is actually to compute the matrix elements for R and S, First R will be emsidered, Using the identity

the matrices may be removed from the denominator of R giviw

The demomimtor i s seen to be 2mwl from the following relations:

The matrix elements for the various spin and polarizatlorr eombimtion8 can be calculated straightforwardly from &is paint, But cerhin prelimimry manipulations wiZX reduce the hbar involved. Using the identity

it i s seen t b t

But pl has only a time component &nd e1 only a space comp~mnt so pl * el = 0, Wealling that 61ul = mu$, it i s seen &at

and thig is the matrix element of the ffrst term of R. Xt i s also f ie nemtive of the matrix element of the last term of R, so R may be replaced by the equivalexlC


By an exactly similar madpuhtion, the S a t r i x is equivalent to

Substituting & = w - Y,) and dz = w 2(y, - Y , cos B - yy sin 8 ) and bans- posing the 21x1 factor, the complete matrix m;ly be writbn

A till more usehl form is obtaizled by n o t i ~ that pC1 anGeomxxzutes with ql (el *qi = 01 and with qz and &at = 2ez eel -. Pi$z. Thus,

Using this f o m of the matrix, the matrix ebmrsts may be compubd easily. For example, consider the case for polarization: 8; = y,, #2 = yy cos B - y, sin 8. This eorreawnds to cases (A) and (B" )of h c t w e 19 and will be denoted by (AB". The matrix i s

2m(R+ S) = my, (yy cos 6 - y, sin B)[y,(l- cos B) - yy sin 81

since ez m e i = 0. Expanded this becomes

2m(R+ S) = -y,[yyyx cos 6(1 cos 8) + caa @ sin B -c- sin @(l - COS 8)

- y, sin 8

where the antieommubtion of t;he y 'S h a b e n used. IXn the ease of spinup for the incoming particle and spin d o m for tke: outgctiw particb (st = -I), sz =: -11, the matrlx elements

may be found by reference to Table 13-I. But note that in, this prciblenn pi,

= p,g + i ~ y i = O s iwe particle 1 is a t reat, Hence L;ke final ntatrix elennent for this ease, polarization (AB"), spin sg = +l, sz = -1, is

Q U A N T U M ELECTROLZYNAMrGS

2m (F~F~} ' /~( ;~(R + s)u~) = -(l -- C 0 8 @)iF1p2+ - sin 6 Pz+ Fi

The result@ for the other combinations of polarization and spin a re obtained in the same mamer and will only be presented in tbbular form (Table 22-1). They m y b verified a s an e ~ r c i s e .

For any one of the polarization cases listed, 1 I%f2 is the sum of the square amplitudes of the matrix elements far autgoing spin- s b k o averaged over incoming spin ~ k t e s . But this i~ seen to be simply the square magnitude of the noassro matrix element listed under the appropriate polarizatian case, For ex;%mplele, in ease (AA",

By employing the relation

and

the squarre magnitude8 of the matrix elements for the various easea redurn, a&er eorresfderable amount of afmbra, to Lhe expressions given In Table 20-2,

AB" [(@l - ~ 2 1 ~ / ~ i @ 2 1

BB' [(&li - w $ ~ / w ~ w ~ ] + 4 ~ 0 8 ~ 6

XL is clear h t all f w r of these formulae may be writkn simuft;ctmously in as form

Note t b t t h e ~ e fomulas a r e not adeqmLe for circular polarization, m ~ t is, if gil were, for example, 1/a (iy, + yy) , it i s seen thst because of the phas-

INTERACTION OF PARTICLES W I T H LIGHT l01

ing represented by the imaginary part of all the calculations must be carried out before squaring the xnatrh elements in order to get the proper intederence.

Finally the cross section Tor scatterine;: wi lorescr ibed plane polarization af the incomim and outgoiw photons i s

This is the mein-Nfshim formula for polarized light, For unpolarized light this cross section must be averaged over all polartzalions.

It is noted t b t diagram eases such a s Fig. 20-1 Mve been included In

the previous derivation as a resuIt of the generality in the transformation sf of K,(2, X) to momentum represeabtion. In fact, all dfagram eases k v e Been included except higher-order effects to be discussed later, (They eorre- spond to emission and reabsorlption of a third photon by the electron, such a s in Fig. 20-2,)

Twenty -first Leetgre

Dlscussiclg d as f?;latn-Hishim Formula. In the "Thompson limit,'" w i << m. Tnea the electron picks up very little energy in recoil, and wi .:oz. This can be seen from the relation

h this limit, the Mlein-Nishim formula, e v e s

I02 QUANTUM E L E C T R O D Y N A M I C S

which i s the Rayleigh-Thompson scat ter iw eroras section, Note that w is still very large compared to the eigenvalcres of an aloxur, in accordance with our origiml assumptions far Compton scattering.

The same result is obbined by a classical picture. Under the action of the electric field of the photon 1E = East exp (iut), the electron is given the acceleration

Classically, an accelerated c h r g e radiabs to give the scatbred radia- t i on

e (rebrded accelemtfon projected on plane l to & = - - R line of sight)

Tbe scattered radiation polarized in the direction a2 i s determined by the component of the acceleration in this direction. The inbnsity of the scattered radiation of polarhation ez is then (times per unit solid angle and per unit incident inknslty)

The customary fii 's sand c % say be mplaced in Eq. (2 2-11 a s follows (o is an area o r lenglth squared):

m2 = (rne/Ii)' = l eneh aquared

Avemgw 0~8r Folari~ations, It is often desired to have the s c a t k r i ~ cross section for a beam regadless of the incoming o r outgoing polarization. This can be obbined by summing the probabilities over the polarizations of the oulgohg barn and averaging over the incoming beam. Thus, suppose the i n e ~ m f ~ f & beam has polar-ization of t p A. The probbilitiea (or cross sectians) for the two possible types of ouQoing polarization, A,'

and B' can be symbolized as U* and A S . The total probability for scattering a photon of either palarkatton is AA' + AB*. Then suppose the ineom- ing beam is equally Xtkely ta be polarized ass type A or type IS. The resulting probability can b obhined a s the sum 112 (probability i f type A) f 112 @robbility if t;ym B), This ia the situation for unpolarized incomiw beam, and gives

cr (averaged over = (2 /2XMt + GB') + (112XE314" .t BW) pdarizations)

INTERACTION OF PARTICLES WITH LIGHT 103

E, on the other b n d , the polarization of the outgoing beam is measured (still with an uwobrized incorniw barn) , its dependence on frequency mid ~ c a t t e r i ~ a w l e is given by the ratio

Probability of polarization type A' (1/2)[AA8 + BA') Probability of plarization t m e B'

The forward radiation fB = 0) remains unpolarized, but zt cerlain degree af polarization will be found in light scattered through any nonzero angle. X[n

the low-frequency limit (ol wz), the pctlarizgtion is complete a t @ = a/2. Thus an unpolarized beam &comes plane-polarized when ~ca t t e r ed through 90°.P

Tok1 %kcat@riw Cross Sscti~n. l3 the cross section (averaged over polarizations) given in Eq. (21-3) is inkgrahd over the solid angle

the total cross section for sc ia tkr ix through any angle i s obbfrted. So, from Eq, (21-X),

and the variable w2 goes between the limits mwl/(2wl + m) and w l a s eoa B goes from - 1 to + 1. Equation (2 1-31 can be writtein

where the last five terms replace -sin2 B = cos2@ - 1 using Eq. (21-F). Sf mpf e integrations fleldS

h the high-frequency limit (wt- m)

f Cf. Waltrsr Heftler, "Qumtu rXlhet3~ of Rdi&tion," 3rd ed., W o r d , 1954; and B. Rossi m4 K. Greiseen, Phys. Rev,, @Z, 121 (1942).

$ Cf* Heftier, op. eft., p, 53,


Thus CompLon scattering is a negligible effect, a t high frec;fueneie~, where pair produeaon bcomes the i m p o r b t e f f~e t .

From the quantum-eleetradynamleal point of view, another phenomenon completely amlogous to Cornpton seatkr ing is two-photon pair annihilation. Two pfiobns a r e necessary (in the outgoing radiation) La mafnhin conservation af momentum and energy. when pair annihilation k k e s place in the abselllce of an externd potential. T h interaction can be diagmxusnnerf a s shorn in Fig, 21-1. This figure should b compared tie that for Compton scattering (Lecture 20). The only differences a r e that the direction of pha- ton di i~ reversed, and, since particle 2 i s positron, = -(m~mentum of positron), So write

$1 = IE-y, - p,. Y )

where the emrgies E- and E, of the electron and pasitran a re both posl- tive n u m b r s .. The conservation law gives

(just cta for Gompton s c a t k r i q , but the direction of gfi reveraczd), so the matrix element for this ineraction i s

1'1.16 second possibility, fnbistinguishble from the f i rs t by any measure-

I N T E R A C T I O N O F P A R T I C L E S WXTH L I G H T 105

me&, i s oblafned from the first by inbrcbaaging the turo photons (see Fig. 2 1-22); again no& similarity to Compton seatbrlng.

Immediately, the matrfx element is

FIG. 21-2

The sum of the two m a t r h elements 13md the density of final states gives the e m s s section

c (velocity of positron) = 2a/(2E- 2E, * 2w1 a 2wz) I + &12jZ

x (density af sta&s)

In a system where the electron is a t res t and the positron i s moving. The density of final sktes i s

Since p a ~ i c l e 2 is a positron, $2 = --g$,, so the conservation law, Eq. (21-41, gives

gii' pf* = $ i t + Pfi

Then

This reduces to

106 Q U A N T U M E L E C T R Q D Y N A M I C S

Taking the velocity of the wsftron a8 (p+j/E+ the Cross s e ~ a o n i s

@ = ( 2 ~ ) dQ1/12E- * 2 /p+/ 4(2nj3 m(E+ + m)] x I&&$+ ~ , 1 2

From a comparison of the dhgrams, i t is clear t k t the matrix elements for pair annihilation a r e the same as the m a l r k elements for the Compbn effeet if the s i p of (Ifl is changed. In the cross section, this amounts to e b g i n g the s&n of wl . T k n the cro@s section ie

in analom with the mein-Nishina formufa.

TXON FROM REST

The formula for positron-electron amihilation derived in betrxre 2 l di- v e q e s as the positron velocity approaches zero (cr - l/v; this i~ true for other cross sections when a process involves abh-fo~tion of the, irmcoming particle, and Is the well-hown l/v law). To ealculab the positron lzel-fme in m electron density p (recil~ll that the pmcediw cm@@ section ws for an electron density of one per cubic centimeter) as v+ -+ 0, we use

plus -the fact t h e , a s v, -- 0, E,- m and ol -- w2 - m (when the electron and positran a r e btch. approximably a t rest, momentum and energy can be earnerved only with tvvo phobna of mornentra equd in magnitude but opposih in direction). Thus

where 8= a q l e between directions of plarization of lvvo photanrs (cos 6 = ef *ez) The sin2 B dependence indicates that the two photDns have their pohrizations a t sight mgles. TO get the prob&tlity of tmneitton per second for axly photon direction and m y polarization, it is necessary to sum over solid angle ( I dQ = 4n) and average over polarfzations (sin2 B = %/g), giving

I N T E R A G T X O N OF P A R T I C L E S WITH LIGHT

f f a ch r s of e and E r e i n s e d d where required), where ro = elasaieal electron mdiua, and T = mean lgetime.

PrabEeruzs: (L) Obbin the preceding result directly by us lw matrix elements for an electron and positron a t rest. S h w that only the sin- glet state (spins antiparallel) can disinbgrale inb two photn>ns. The triplet state d i s i abgmhs into three photons a d has a l o e e r lifetime (see the nelrt problem),

(H?) Find the mean time required for a positron and electroa to dis- integrab inlo &see pholons (spins must bt? parallel), The following procedure is s u a e s b d : (L) set up formula for r a b of disinbgmtion; (2) wriM M in the simplest possible form; (3) make a table of m a t r h elements ( s m e a s Table 13-1 but with = m%, = (4) find the matrix element at* M fo r eight polarization eases; (5) find the raLe of disintegration for each ease; (6) sum the disinkgration r a k over gokrkatiorxs; (7) ob&in the photan smctrum; (8) obain the b f a l disintegration rate by inkgrathg over photon spectrum and mgle; and (9) compare with Qr r and Powel.?

(5) It is h o w t k t the m a t r h elements should be independent of a gauge traxlsfarmation $" = 6 t. ad, where a i s an arbitrary eonsknd and, 4 i s the momentum of a phobxl whose polarization is p! o r $', Show t b t s ~ b s t i t u t h g 4 for 6 in the m a t r k elements for the Comp- ton effect gives m = 0.

When an electron pmses through the Coulomb field of a nucleus it i s de- flecMd, Assoekbd with this deflection is an acceleration which, wcording to the classical theory, resutLs in radiation. Acceding ta qumtum efectro- dymmics, them is a certain probability t b t the incident electron will m a b a tmnsition ta a different electron sbb with a, photon emit%d, wkile in the field of t h nueleus, ha r ae t i on with the field of tk nucleus i s necessary to aatiafy conservation of enargy ancl momentum, T b t is, the electron eamot emit a photon md m&@ a transition to a different electron s t a b wkle tmv- e l i w along in a vacuum. Figure 22- 1 shows the process and defims awle s t h t arise lawr,

The Coulomb potential of the nucleus will be considered to act only once (Born approximation). The validity of t h h approximation was discussed in Lecture 16. There a r e two (indistinmishabfe) orders in which, the brems- strah;luag proeeaa can occur: (a) the electron interacts with the Coulomb field m d subquen t l y emits a photon, o r (b) the electron first emits a photon and then inbraete vvfth the: Cougomb field. The diagrams for these prac-

TA, Ore and S, L. Pawell, Phys. Rev., 76, 1696 (19.49).

essels a r e shorn In Fig. 22-2. Tfxe inkraetlan with the nucleus gives momentum @ to the electron. Cornervation of enerm and momentum requires

electron 2

J Coulomb field of the nucleus

photon

In hcttnre 18 i t was s b w n that the Fourier t rmsfarm of the Csufomb pobntial was propo&ional to &(Q4), since the potential i s independent of time. This means t k t only traasilions for which Q4 = O occur, o r enera must be conserved amow the incident electron, find eliectmn, a& phobn, Thus El = E2 + W . The transition probability is given by

I N T E R A C T I O N QF P A R T I C L E S WITH LIGHT

Since the nueleus i s to be considered infinlkly heavy,

Notice tfnat there is a, spectrum of phoans; t b t is, the photon energy is not d e b m i n e d (as i t was in the Compton effect, for emmple). Letting "S1Z =

(G2 MU$).

where the f i rs t term comes from Fig, 22-2a and the second term from Fig. 22-2b, The explaimtion of tkm factors in the f i rs t term, for example, is, reading from right to left, that an electron initially in state ul i s scattered by the Coulomb potential acquiring an additional momentum 5& , the electron moves a s a free wrt ic le with momentum 6% + @ until i t emits a photon of polarization gd, We t h ~ n ask: 1s the eletctron in sbtf? u29 For the Coulomb pokntial

(see Momenturn bpresenbt ion , Lecture 18) in a coordinate system in. which the nueleus does mt move, [For pohntial other than Coulomb, use agpro- p r h t e v(Q), the Fourier transform of the space depndenee of the pohntiafi,] 1Rationalkinf;f the denomimlor of the matrk,f

The outgoing photon can h polarized in either of t w directions, and the in- earning and ouQofng electron each have two possible spin sbhs. The vari- s u s m a l r k elements can be worked out using Table 33-1. exactly a s w s d o ~ in d~r iv ing the mein-Mis~na eross section in Eczeture 20, Nothiw new is involved, so we omit the delafls. Mkr (l) summing over phoWn polariza t ion~ , (2) summing over outgoing efeetrsn spin stabs, and (3) averagf% over h c o m h g electron a p h sdates, the following dgferential eross section is obWned:

Q U A N T U M EZ1E:CTRORYNAMId:S

sin @a do2 sin dBt d@

2plp2 sine1 sine2 cos C$ (4E1E2 -GJ2+ 2w2) - 2w2 (p; sinZ e2+ sin2 et) (E2 - P2 f208 @2)(Ei - P1 f2cls 01)

(22-5)

An apgratximale eq r e s s ion with a simple inkrpretation in b r m s of the Coulomb elastic s e a t t e r i ~ cross section can be obbined when the gboan energy is small (small compared da resst mass of a l e c l r o ~ but Zarge compared to electron binbiw energies). WriMng tlnc; matrix (22-3) fn b r m s of 4 insbad of @

using the relationskipa gOPlz = -$z$ + 2~3 * p2 , $14 =: -661 + Ze PI, and neglecting 4 in the numerator, since it i s small, this becomes

where use is made of the fact thatJhe ma_trix element of M between states u2 and u l is to km ealeulakd and uz$z = dtui = mule

The cross section fa r photon emission can then wri tbn

The firat bracket is the probability of transition for elastic scattering (see Lecture 161, so the last bracket may b i n b w r e k d as t k pmbability of phabn ernissian in frequency interval dw and solid a w l e da, if tficlre i s elastic seat ter iw from momentum pi Lo pz,

XMTERACTXQN OF PARTXCLES W I T H L I G H T 11 5

Problem: Calculate the amplitude for emission of two low-energy photom by the for~g-cting meth&, Neglect ql's in the numeratar but not in the denomimtor.

A~lswer: Another faclor, similar to that in the precediw equations, i s obhined for the extm photon.

It. is eas;Cly shorn that a sfwle photon of e n e r n greater t h n 2m camot c r e a b an electron positron pair without the presence of some other means of conserving momentum and energy. Two photons could get together and e r e a b a pair, but t h ~ phobn density is so low t b t this process is extremely ualikely. A photnza can, however, create a pair with the aid of a field, such as that of a nucleus, to which, it em f m p a ~ some momentum. As with brems- s t m h l u ~ , there a re two fncZiatinguirskble ways in which this can hppen: (a.) The incomfw photon c r e a h s a pair and subsequently the electron i nk r - a c b with the field of the nucleus; o r (b) t k pb ton creates a pair and the po~i t ron interacts with the field of the nucleus. The diagrams for these al-

tfveisl are shown in Fig. 22-3. The arlrows in the diagram indicate that

FIG. 22-3

$l i s the positron momenlum and yJ2 i s the electron momentum. Notice that, with respect to the directions that the arrows point f a d without mgard to direction of increasim time), these dhgrams look ercactfy like those for the brsmsstmkjluq pmeess: S t a f i i q with in case (a), the particle i s f i rs t sca tkred by the Coulomb potential and then by the phown; in case m) the order of the events is reversed. The differernce btween pair prduction. and bremastrafilung, when the direction sf tim is taken into account, is {X) i s a positron state (m electron traveliw hckward in time), and (2) t b photon 4 is absorbed rather Lbn emftbd, As a result, the bremsstraMung m a t r h elements can b wed for thia process if 6% i s replaced by --$, and 4 by -4,

122 Q U A N T U M ELlEGTROTZcYNAMICS

The $+ is t b n the positron momentum and i s the momentum of the absorbed photon, The density of f i ~ l skks is different, of course, since the particles in the f iml s t a k a r e now a positron and electmn. Thus

= ( 1 / 2 a ) ( ~ e ~ / ~ ' ) ' e2 @,p- sin 8 , d B , sin 8- d8- d+/w3)

where the bmces a r e the sizme a s for brernsstraMung, Eq. (22-52, except for the follotviw substitutions:

P- fo r P2 -6- for O2 E- for E+ -P+ Pi -8, for 6- -E, for E,

-W for W

Fimre 22-4 defines the angles (rS, = angle between electron-phobn plane and pohsitron-photon ghne) .

positron

A METHOD OF 8UM m EXIEMENm OVER SPm STATES

Ely using current methods af computiw cross sections, one f i rs t arr ives at a cross section for ' ~ o l a r i z e d " eleetrans, t b t is, electrons with definik ineomiw and outgokg spin stabs. In practice it is common that the incorn- 111% barn will b "unpolarized?' and the spins of the ourtgoing paflicles will be unobserved, In this ease, one needs the cross section obkhed from t k t for " p l g r i ~ e d ? ~ eefeetrens by summiw prokbill t les over f iml spin BUL~B and a v e r w i w this sum aver Initial spin st;abs. This is the correct proeeahs since the f lml spin states do not Isrterfere lad these i s equal prokbili ty of initial spin in either direetion. Formally, if

I N T E R A C T I O N O F P A R T I C L E S WITH LLCWT

one needs

1 @ - - C C I ( u 2 ~ u i l l ~

2 spins I spins 2

where mean8 the sum over final spin states for only one s i m of the spins 2

the e n e r n , that is, over only two of the four possible eigenakles. Similarly, is the sum over initial spins for one sign of the energy. The purpose

spins I now is to develop a simple m e t h d for obkining these sums. h aceorknee with t b uswl rule for m a l r h multiplieatioa, the fallowing

i s true:

(G2~uI) (UI~u2 ) = 2 r n ( G 2 ~ ~ u 2 ) all Y

where A and B a re any operators o r matrices, the 2m factor on the right a r i ses from the normlization uu = 21x1, and the sum i~ over all eigenstatas represen;ted by ul, But the shbs u, which we want in Eq, (23-1) a r e nat all shks, just those satisfying IzJiut = mu$. T b t is, they belong the eigenvalue + m of the operator &. Since 612 = rnZF $l also has the eigenvalue -m, t h t is, there a r e twa more solutions 05: &U = -mu wkich, together with the two we wish in Eqi, (23-1) bring the t ok l to four. Let us call the l a t b r '"negative eigenvalue" sslisks,

Mow, !if in Eq. (23-2) the m a t r k elements of B were zero in negative eigenvalue states, this would be the same a s , that is, just over positive eigenvalue s k h s . So consider spins 1

C (;2~u,)(;l@f+m)~uz) = (:2~(dt+m)~u2)2m all U,

But

rV

ui(Iljf+m) = Q for negative eigenmlue slaks

8w

= ui(2m) for positive eigenvalue states

so the p m c e d i q sum also equals

C G 2 ~ul )am(Gf Buz) spins 1

CaneeXliw the 2m factors, this gives

C (Gz~u!)(';l Buz) = (G~AMI + m)Buz) spins 1

(pJi -t- m) is a l h d a proleetion operabr for obviaus reasons. Similarly it f"alX0~8 t h t


spins 2 all U,

where X is again any matrtx. Remembering the normalization &u2 = 2m. i t 18 seen that the last sum i s just the tmce o r spur of the matr"ur. (;plj2+ m)X. Nob that the o d e r of X and -t- m ia immakrfal,

Fimlfiy, when m e wmts

eolleetion and speeklizatfon of the previous results Is seen to give

spim 1 spins 2 spins 1 spins 2

where the last niohtion means the spur of the m a t r k in the brackeds. It is true whether &, &12 represent electrons o r positrons.

Tke followiw list af the spurs of several frequently encombred matrices may km verffied easily:

It is also true that the spur of the product of any add n u m b r of daggerad operabrs i s zero.

I H T E R A C T I O N O F P A R T I C L E S WITH LXCHT I l5

As an example, the case of Coulomb acatt;eriw will be '%treatedw "using this hch ique . The cross secUon for pohrized e lee t ron~ was previously fomd to b

Therefore, since 7, = y,, the cross section for anpolarized electrons is, by Eq, (23-31,

The spur can bt3 evaluabd irnmedia&ty from Eq. (23-5) with m2 = m4 5; O and -g4 = p14 = yt. Anotht?r way is: Since ytdI = 2Ei - ~ % y t , it i s Been t h t

Usirzg ra few af the formulas l i skd p r e&~u~1y , the Spur of this matrix i s seen to be

But p% * p2 = E - p1 a h, p% * p2 = cos 8, and Et = .E2, SO thi8 i s

4~~ -+ 4m2 + cos B

Also rn2 = E' - p', so that finally the cross section becomes

@mpok = 1/2 ( Z ~ @ ' / Q ~ ) [ 8 ~ ~ + 4p2 (cos @ - 111

where v2 = pZ/~Z. This is the same cross section obtained previously by other nnethds.

The cmas sections for the paf r production and brernsstrahlung processes contained tbe factor [v(Q)I', where V(&) i s the momentum representation of the pobntial; t b t 18,

which for a Codornb pobntlal it3

where Q i s the mamenbm tmmferred to the nucleus o r p% - p2 - q.

1 l6 Q U A N T U M E L E C T R O D Y N A M I C S

Clearly V(Q) gets large a s Q gets small. The minimum value of Q oe- curs when all three moxrrenk a r e I l n d up (F&, 28-11:

Pr P a

g

FIG, 23-1

Fo r very high energies E m,

so t h t in this ease

From this i t is seen that Qdn - O a s Er - 00 . This ahows clearly why the cross sections for pair production and brc3mhsstrahXuw go up with e m r m .

From the integral emression for V(Q) i t is seen t b t the main eontribu- tlon do the inkgrai comes when R m 1/Q. So as Q becomes small the im- p o d n t range of R gets I a e e . It i s in this way that screening af the Cou- lamb field Ldcorszes effective. The value of for a conhmgiakd process can be estimabd from the farewinli?; formula, The atomic radius i s given roughly by ao~ - ' / ' , where a. is the Bohr radius. Thus if

or , w h t i s the same,

then sereenhg effect wtll be i m p o a n t , and vice versa for the uppsi* inequalities, E from this estimate sereeniw would appear to b imporknl, one should use the screened Coulomb podential. It gives the result

where F(Q) is the abmie structure factor given by

and nm) i s the electron density as a funcUon of R,

XNTERACTXON OF P A R T l C L E S W I T H L I G H T

P~oblrsm: h d i s c u s s i ~ bremslsptrahlung it was found t d t the cross section for ernksion czf a low-energy pll~tQa can IM approximakd as

where cfo i s the ecat%rfng cross sectlan (neglecting arnlssian). Now consider an energetic Comptoa seatkrfng in which a third, weak photon is e;rnit&d, The three dkgmrwts are shown in Fig. 24-1,

FIG, 24-1.

Show that the cross section for this effect 18 given, by Eq. (24-l), with the Klein-Nisfiina formula r e p l m i ~ m e m e m b r to assume q small ,)

potential region

Interaction of Several Electrons

Even though the Dirac equation describes the motion of one particle only, we can o b h b the amplitude for the inbraetion of two o r more particles from the principles of quantum eleetradywmics [so long a s nuclear forces a r e not involved),

F i r s t consider two electrons n n o v i ~ throu& a region where a pahnth l is present and assume t b t they do not interact with one another (see Fig, 24-2). The amplitude for electron a moving from 1 3, wMle electron b moves from 2 4 i s given the symbol 33;(3,4; U). If i t i6a assumed t b t no inkractian betwem electrons takes place, then K can be written a s the product of kernels ~,(')(3,1) K,(& )(4,2), where the superscript means that K,(') operates only on those variables describing particle a, and similarly for K , ( ~ ) .

A second type of interaction gives a, result indistfngufshable from the f i rs t by any measurement in aceo&nce with the Pauli principle. This differs from the first caae by the i n t e r c h q e of wrt ic les k twesn gossitfana 3 and, 4 (see Fig, 24-3). Hour the Pauli principle says that the wave function of a system compssctd of several electrons i s suck tbat the inbrchirnge of space variables for two pa@icles results in a c b n g e of alw for the wave function, Thus the amplitude (Including both possibilities) i s K = K,(%) ($,l) K , ' ~ ) (4,2) -

1) ~ , (~ " ( s ,2 ) ,

A similar sitwtioa ar ises in the follovving occurrence. Initially, one electran moves into a r e a o n where ra potenaal is prewnt. The potential creates a pair. Finally one positron and two electrons emrge from the region, There a r e two possibilities for this occurrence, a s shown in Fig. 24-4. Again, the total amplitude for the occwrenccz is the digerence btween the amplitude8 for the two possibilltiea.

I -PJTERACTXOM 01F S E V E R A L E L E C T R O N S 119

potentkl region

2 2

The prohbility of this occurrence, or the previous, or any other similar occurrence is @ven by the ab~olute s w r e of the amplltuide times the n m - b r Pv, The Pv is actmlly the prokbility t k t a vacuum remains a vac- u r n ; because of the possiMllty of pair p r o d ~ e ~ o n , i t i a not d t y . The Pv can be camputsd by xnakiw a table of the x>rohbilities of alarting with nothing and endiw with v a r i o ~ n w b e r s of mirrs, a s ie shown in Table 24-1,

TABLE 24-1

The sum of a11 t;hese psohbililies must eqal udty, and Pv is delzernnined from this eqwtion, The magdtude of ]Piv demnds m the pobntial present. So the "probabifities &ken 8s me rely the sqwres of amplitudes (that is, oxnittiw the Pv betor) a r e actwfliy rehtive probabilities for various oc- currences in a @ven pot;e&iaf,

Use of 6 , (B'). For the present, the existence of more than m e possibility for an aeewrence ( t h Pauli principle) will t.le ~ g b e t e d . The tobf ampli- tu& can always be &rlved from one by inkrehanging prowr spaw var- i ab l e~ , maMng W c o r r e s g o n a ~ changes in sim, and summing all t b amplitudes so obkined,

120 QUANTUM E L E C T R O D Y N A M I C S

The nonrehtivisac Born appro&mation to the amplitude for an interae- tion is

where, from earlier k;ctues,

and

Hate Wlat t5 = t6 since a nonrelativistic interaction affects both parLicZes simultaneously, The potential for the interaction i s the Cmlomb potential

Separate variables may be used for tB and t6 , if the fmction 6(t5 - t6) i s ineluded aa a factor, Then

where the differential d~ includes both spa= and #me variables. It is con- ceivable t h t the rektivistie h r n e l couM be obtained by s u b s t i t u t i ~ K+ for K@, and introducing the i&a of a retarded potenLial by replaciw 6(tb - t6) by 6(tS - tg -- r6,@). Nowever this 6 function i s not quite right. Its Fourier transform cmtains both positive and negaave frequencies, whereas a photon has only positive energy. Thus

To corrrsct this, define the fmctian

00

& , ( X ) = exp(-iwX) dw/tr

which contains only positive e n e r g . The value of the fmction i s dekrmined by the ilttegral, mm,

4 + (X) = lim ( l / ~ i ) ( X - i ~ ) c -Q

= 6fm -t (l/ni)(prineipal value 1/X)

Abbreviating tg - ts 3 t and r g , ~ S T, and taking account of the fact that both t5 5 tg and t5 =P t6 a r e ~ a s i b l e , the retarded potential i s

XHTERAGTXQN O F S E V E R A L E L E C T R O N S

Exercises: (11 Show t b t

&fining t2 - r2 as a relativistic invariant, the potential is e2 6, (gS, 2). Another term h i c h must be included i s the magnetic in- t e r adon , proportism1 to -V, * Vb. In the n e t i o n used for the Dirac eqmtion, this prduct i s -a, - orb. Zt will km found emvedent to express this in the equivalent form -(Be), * ( P c u ) ~ , and in & i ~ notstt;ion the retarded Coulomb potential. is proporZ;iomf to P The~c: P '8

come from the use of the relativistic b r n e l . Thus Lhe complete po- tentfal for the interaction becomes

and then Lhe first-order kernet is

Here the superscript on yp indicates on which set of variables the matrix opembs, just as for the superscripts on K, .

The occurrence represenbd by this kerml. can be diagrammed a s in Fig, 24-5. This represents the e x c b q e of a virtual photon be-

FIG. 24-5

tween the electrons. The virtual photon can be polarized in any one of the four directions, t, X, y, z. Summation over these four possibilities is indicahd by the repeatttd index af The integral expression for tke kernel, Eq. (24-2), implies t k t t b amplitude for a photon to go from 5 - 6 (or from 6 - 5 depending on timing) i s b , ( ss , t ) Equation (24-2) can b taken a s another a?&bment of t b fundamen&l laws of qumtum electr&ynamics.

(2) Show th t

Thus, irz momentum space,

From the resu lb of the fasrt lecture, it is evident t b t the laws: of electro- dymmies could b s h b d a s follows: (1) The amplitude to emit (or absorb) a p h b n ie eyp , and (2) the amplitude for a phtobn to go from 1 to 2 is 6 , (91,$), where

in momentum representation. It is interesting to note that 6 , (si, 2 ) 18 the same as I, (sI8$), the quantity appearing in the derivation of the propagation kernel of a free particle, with m, t b particle mass, set eqwl to zero, A more direct comeetion with the MaweXl equations ean km seen by writing the wave equation. 0 k l I , = 47r JP in momentum representation,

We now consider the eomeeti.on witfa the "rules" of quantum electrodynamics @ven in the second lecture. The amplitude for a to emit a phobn whfcb b absorbs will now bt; caletllarted aceordiw Lo those rule^ (see Fig, 25-1)- The amplikde tjRae electron a goea from 1 Lo 5, emits a phobn of pohrizaaon 6 and direction K, tba Ei;aes fmm 5 to 3 is given by

I N T E R A C T I O N QF S E V E R A L E L E C T R O N S

whereas the amplitude that b goes from 2 to 6, absorbs a pbton of potari- zation p! and direction X: at 6, then p e s from 6 Lo 4 is given by

The amplitude t b t h t h these procesees occur, which is equivalent to b ab- sorbiw a% photx>n if t(; t5 is just the product of the individual amplitudes. L1" iz absorbs h% photon, the s i p s of all the eqonentiala in the preceding amplitudes are chnged and t6 must bri? less t b n t5.

To obbfn the amplitude t h t my photon is exckngeb beWc3en a. md b, it is necessary to i nbgrab over photon direction, sum over possible photon palarizations, and inbgrate over t5 m d t(;, subject to the dorementioned restrictions. In summing over polarizations, gf will be replaced by yp and a summation over p will be taken, TMs amomts to summlw over four directions of polarization, sometMng t k t will be eviained fabr, Thus

= 4?re2 C l exp [iK. (rs - r8)] exp [-iK(ts - te)J P

f 24 Q U A N T U M E L E C T R O D Y N A M I C S

Comparing this vvith the msult of the last lecture, i t must b that

This can be written in a form which makes the space-time symmetry evl- dent by using the Faurier t ranshrm

exp ( -iKlt/ ) = i 1 2 i ~ / ( w ~ - K'+ i ~ ) ] exp (-iwt) dw/2r

so that the foregoing egwtioa bcomera

and cornwring this with the result of the last problem of Lect;ure 24 estab- Zisbs that the rules given in Leeturs 2 a r e consistent with relativistic elec- trodywmiics develop4 in the last lecture.

The theory wit1 now be used to ob&in the electron-electron seat ter iw e rass section. The diagrams far the two idistlngulshablo proeesse s a r e s h a m in Fig. 25-2.

XNTERACTXON OF S E V E R A L E L E C T R O N S 125

The axnplitude expressed in momentum represenhtion is obtained a s follows: Write Eq, (25-3) [with the aid of Eq. (25-4)] a s

Since electron s t a e 1, i s a plane wave of momentum and electron s h t e 3 i s a plane wave of momentum $$, i t is clear that in m_omentum representation the spinor part of the f i rs t bracket wJll become (u$Yp us) and the spinor part of the second bracket will become (u4yp uz) Integration over rg and 76

produces thtt consermtion laws given at the bottom of tbe diagrams. Drop- p l w the integration over q puts the photon propagation in momentum represenktion directly. Thus the matrlx element c m be written

The f i rs t term comes from diagram R, the second from diagram S , and the summation over g Is implied. h the eenter-of-mass syskm, the prob~bility of transition per second i s

(see h n s i t y of Fiml g a b s , Lech re 19). The nzethd of Lecbre 23 can be used to average over initial spin states and sum over_final spin states. For ?ample. the sums ?er spin states that result from R by R matrices and R by S plus R by S matrices a r e

By judicious use of the spur rehtions given in h c t a r e 23 the following differential eras8 section is obhined (alternatively, Table 13-1 could be used to calculate M dimetly):


where x = E ' / ~ ~ . This i s called M"ci11edller scattering (see Fig. 25-3).

Problems: (I) Calculate positron-el e c t r scatteriw by the pre - cediq methd.

(2) Y i d the cross section for a, p rneejon to produce a back-on electron, Assume tb t the p meson satisfies the D i r a ~ equation with S = 1/23 and no anomalous moment. Remember t h t the particles a r e diastiwisbblie and hence there i s no fnterchnge of particles

(3) Calculate t b emecbd electron-proton scatkring cross section assumiw the probn has no stmcture but: does k v e an momafoua moment. T b Dirac q w t i o n for a probn i s (see p q e 54)

T h u ~ the pstrl-urbiq po&ntfali cm be k b n a s (see p q e 54)

and the coupfiw with a photon i8

Ths Sum ovsr Four Pokrtewtkow. In ~ l a s s i ca l electmdymmics, longltu- dim1 waves can always be elimimbd in favor of tranwveree m v e s a& an insmhneaus Coulamb interaction. This is tha approach uaed by Fermi (see Lecture I), and it will now be demonstrrz*d that the sum over four polariza- t iam is also equivalient to drmsveree w8ves but plus an fnsbnbneous Gou- lamb hteraction. E h s b a d of chooeing space directions X, y, z, one direction parallel to Q (pbLon momentum) and two directions trmaverse to Q a re hken, the m t r f x element can be writbn

?For the proton p =: 1,7896.

I N T E R A C T I O N OF I3EVERAL E L E G T R Q M S 127

where y9 i s the y matrix for the Q diractlons and y,, represents the y m&- trix in either of the transverse directions, The matrix element of 4 = - is zero in general (from the argument fa r gaugr;. hvariance). t Thus y~ can be replaced by y, with the result

Now l . / ~ ~ represents a Coulomb field in momentum space and yt ie the fourth component of the current density or* ctrarge, so t h t the f i rs t &mm represents a Coulomb ineraelfon white the second term conkins the i nb r - action tbough transverse waves,

f Ln our special ease, i t is easy to eee directly, for emmple,

Discussion and Interpretation of Various "Correction" Ter Twenty -sixth Lectzlre

In many pracesaes the bebvior of electrons in the qumt~m -electrody- mmfc theory turns out 2s be the same a s predicted by slmpIer theories save far small ""earrection" b m a . It i s the purpose of the present lecture to point out and discuss a few such cases.

ELECTRON-ELECTRON mTEMCTIQ;N

The simplest diagrams for the interaction a r e shown in Fig. 28-1. The amplituide for the process h a b e n found -Co be praportioml, in momentum

d.t FIG. 26-1

12 8

representation, to

where q E {Q, Q) and Q i s the momentum e x c h w e d by the two electrons. Also, since 4 =: & - I t follows that

From this identity i t was deduced in the fast lecture that the amplitude far the process a s just given is equivalent to

By taking the Fourier transform of the f i rs t term, i t C- be seen that i t is the momexxtzlm repreaenktfon of a pure, insknmeous Coulomb powntial, The second term then constitutes a correction to the simple Coulomb interaction. In it y, denotes the y's for two directions tmnsverse to the dfrec- tisn of Q,

For slow elleetmns, the correction to the Coulomb potential may be simplified and i n b q r e t e d in a simple mamer. Note t k t in this case

and

s o that v b 2 and q2 in the denominator can be replaced by with small error . (In the C.G. system, = O exacMy.) The correction term h- comes

but

It is recalled that u E , where M, is the large parfi and ub the small

part and that in the nonrelativistic approximation

13 0

Also, since


i t follows that (taken bt;ween positive energy h3%&s)

In free space ];I = p, so the x component, for example, of the fo r ewiw matrix is

where the cornmuation relations for the a's have been used. From this it is. easily seen that the amplitude for the correction ta the Coulomb potentllal may km w i e e n altogether in the form

The f i rs t t e r n s in each of the brackets represend currents due to motion of the electron t rmsverse do Q and the second k r m s represent the transverse components of the nnwet ic dipole of each. So altogether tt appears t b t the correction arises from current-current, ourrent-dipole, and diple-dipole interactions between the electrons. These inbsactions a r e emacted even on the basis cif classicat theory and were de sc r i bd by Breit h f o r e quantum eleetrodymmica, hence a r e referred to a s the Breit interaction.

Cornider tbe dipole-djipole term arising in the correctiart factor. Since Q = p1 - R = p2 - p4 it is

But since crx Q is ze rs w k n Q and $ have the same direction, the sum could a s well be over all three directions and then it is equivalent to a dot product. That is, this term of the correction i s

‘COR3"ZE:CTION" T E R M S 13 1.

By Laking the Fourier transform t this will be seen ta ba the momentum repreeenbtion of the interaction bdvveen two dipoles as was sfaled.

go% tb t the approxinnatian q4 - (v/c)Q used a b v e applies anly liretween positive e n e r n sbt8s. For, if one of the s h h s represents a, positron, then

However, 2rn i s v e q f awe , so the correction is still small. It i f s necessary to redo the a d y s i s ~ v e r t h e l e s s .

It would a;ppesar tb t , since the electron md positron a r e distinguishable, the Paul i principle w u l d not require tfi i n b r e e diagram, leaving as the only one Fig. 26-2.

FIG. 26-2

But it is still p s s i b l e by the same phenamena l~ iw l reasoniing to con- ceive of the d i e r a m in Fig, 26-3, which wwld represent vf s t u d amihllation of the eliectron and p s i t r o n with the photon later c r e a t i q a new pair. It turns out that; it i s necessary to regard m electron-positron pair a s exist- ing part of the time in the form of a vlfiual phobn h order to obhin agreement Wth eqerfment.

t Notice that (tq x Q) (Q X Q) exp I-iQ * X ) , which will appear in tmns- f o m h k g r a l , i s the same a s -(q X V ) * (q X V) exp (-iQ * X ) , w h r e V is the grad operatar. This device enabtes an integration by pads , whichgreatly simplifies t h process and the result, Thus, sigcr?: the trmsform of 1[a2 is l/r, the coupling is -(vt V) * (CQ V )(X/r), wMeh i s the classical energy for inbraetin& mqnet ic dipoles,

Q U A N T U M ELECTRODVNAMXCS

V"j3 FIG, 26-3

From the poht of view that positrons a r e electrons moving backward in time, Fig. 26-3 differs from Fig. 26-2 only in the intercknge of the '6final" skbs &, $d. T b Pauli principle exknded to this ease continues to operate; tlfe amplitudes of the two diagrams must be subtracted, since they dif- fe r only in which outgoiq (in the sense of the arrows) particle is which,

An eleetron and positron can exist for a short time in a hydrogenlike bomd s b t e ba rn a s the a h m positronium, The ground s b t e of positronium i s an S s.tab md may be s i q l e t o r triplet, d e p e d i w on the spin arrange- ment. As has been illdicated in assigned problems, the 'S state can annihilate only in two photons, whereas the '$3 state decays only by three-photon mnihilatlon, The m a n life for two-pfioton mnihilation i s 1/8 10' sec and for three photons it is I l l X 1a6 sec l

P~oblem: Chc-tck the mean life 1/8 1 0 % ~ for two-phobn ami- hibtfon usiw the cross seedion already computed and us iw h,ydrogen wave functions with the reduced =ss for psftronium.

F i e r e 26-2 contrlbuks the Coulomb vtent la l holdiw the positronfum Itogetbr, The correction t e r n @reit% interaction) arising from this same diagram contribubs a dipole-dipole o r @pin-spin interaction t k t is df;f"frennt fn the '5 and '8 skhs (tfie current-current md spin-current interactions are the same fa r b t h sh tes ) . Thus this amounts to a fine-structure separation of the '5 and 'S slates which can Ltc? shown to be 4.8 10"' W,

In view of the fact that a photon has spin 1, and the 'S state of positronium spin Q, conservation of angular momentum prohibits the process in Fig. 26-3 from occurring in the $8 state. It does occur in the ' 8 state, however. The &m ariisiw fmm this d h g m m is small and, therefore, constitutes another fine-structure splitting of the $8 and 'S levels. It can be shown to

amount to 3.7 X XO-a ev in the same direction a s the spin-spin splitting. It i s referred to as spl i t t iq due to the ('new amfhilatfon force* ''

In order to calculate the term arising from Fig, 243-3, one needs to eom- p u b

in this case tq2 C 4m2 (Q = O in the C.G. system), and all matrix elements a r e 1 o r O (regading parEicles a s essentially at rest in the positroniurn), so the result is just a nurnhr , This means that takiw the Fourfer transform one gets a (S function of the relative eoodinrzte of the electron and positron for the interaction in real space, For tMs reason it is sometimes referred to a s the '%sfro&-mnge'"nteraetlon of the electron and positron.

T b combined fine-structure splitting due to the effects already outlined turns out to be represented by

where ac is the fine-atruelure consknt. This amounts to 2,044 X l o5 Mc, using f r q u m e y a s a measure of energy.

There i s still another correction, however, not yet mentioned, a r f s iw from diagrams, such a s Fig. 26-4, where the electron o r positron may emit

FIG. 26-4

and reabsorb f i a own photon. Takiw this into account, the fine-structure splitting in positroniurn i s given by f

t PEEys. Rev., 87, 848 (1952).


having a value of 2.0337 l@ Mc. The experimental value for the positronium fine atructum is 2,035& QA00 Mc, so i t ia seen that this last correction, though of orcfer a amaller t b n the main k m s , i s necessary to abbin agmement with ewerfmenl, It is referred to bath in pasitronium and in hydrogen as the Lamb-shgt correction because of i ts experimenbf observa- tion by Lamb a s the source of the small splitting between the 'glj2 and Z~1i2

levels in hydrogen, In general, i t comes under Ibe head iq of self-action of the electron, to be treated in more dehf2. later,

TWO-PHmON =GUNGE: BETWEEN E UCTRONS A m / OR POSITRONS

It i s easy to imagine t b t processes, indicated by the diagrams in Fig. 26-5, may occur w k r e two photons in;sbad of one a r e exchwed . Although

it k s not been necessary tw, eansider such high-order processes to secure agreement with eqeriment , it may bcarne necessary a s experimenal re- s u l t ~ improve. The amplitude8 for the proceases may be written down easily but their calauhtion is dgficult. The amplfiude for case EE in space-time represenbtion is, for emmple,

o r in momenhm representation it i s

FIG, 28-6

where

(see Fig. 26-61, Thus it i s possible to determine Ifi and Ih, in terms sf eaeh other but not indewndently; that is, the momentum may be s h r e d in any ratio btween the two photons, It ia for this reason that the integral over gI ar ises in the eqress ion for the amplitude.

Twenty- seventh t ectgtiure

SELF-ENEmY OF TEE ELECTRON t In Lecture 26 the fsllavviq idea, was introduced: An electron may emit

and t b n absorb the same photon, as in Fig. 27-1, Then the propagation kernel for a free electron moving from point 3 to point 2 should include Lerrns repsesenliw this possibility. Including only a first-order term (only one photon is emltkd and absorbed), the resulting; kernel is

The correction term in this equation is written down by an inspection of the diagram, followiw the usual procedure for s c a t e r i q processes. In the present case, the initial and final monrenta a r e identical. Therefore the

$.R, P. Feynman, Phys. Rev., 78, '769 F1949); inelrtded in this volume,


nondiagoml elements in the perturbation m a t r k will all be zr?ro. A diagonttl element is one in which the resulting wave functions of a particle remain in the same eie;enstab, For time-idependent perturbations, i t was shown in the development of perturbation theory t h t the only effect on such wave functions is a e h n g e in p b s e , proportional to the time interval T over which the perturbation i s applied, The resultixtg wave function is

Since the perturbation effect (AE)T is small, the second e ~ o n e n t l a l can be expanded a s 1 - i(AE)T -t and higher-order k r m s neglected. It is the second b r m of this expansion which is represented by the inbgral on the right side of Eq. (27-1). The represendation is not yet m equality, since ce rb in normaliziq factors a re different in the two expressions.

To obbin the correct equation proceed a s follows: First , i t is clear that the probability of the occurrence depends only on the interval in space and time btween points 3 and 4, and not a t all on the absolute values of the space and time variables. So suppose a c b g e of variable i s made so t h t d~ represents the elemenL of interval (in apace and time) beween 3 and 1. Then write the inkgral in Eq. (27-1)

where it i s clear t h t the operators K, and 6 , depend only on the intervaf 3-4,

Second, expression (27-2) conbins the time-dependent part of the wave function, exp (-iE,t), because it was assumed that the wave functions used did not conhin Lime factors. In Eq. (27-31, f (31, f (4) do already include the time-dependent part, so i t should be omitted in Eq, (21-2) .

Third, the normalization of wave funetlone i s different for the two ap- p roach@~, For the developnnent t h t led to Eq. (27-21, t b normalization

wals used. For the present development the normalization is

T k s , to eshbl ish an equality, eqressictn (27-3) must tctl3 divided by the normalizing inbgral of Eq, (27-4).

The r e su l t i q expression i s

The integral over d3xs gives a V which cancels with the denominator, and the inkgral, over dt3 gives a T which cancels with the left-hand side, so f imlly

No& t b t the Integral is relatlvistically invariant, FurLher, since p is the same before and after the perturbation and = rn2 + p2, the change in E can be hken a s a ehmge in the mass of t b electron, from

Using this expression, and t rauzs formi~ to momentum space,

The fntcjgrancf may be rewritten; from

using $u = mu and the relations of Lecture 10. Then Eq. (27-6) becomes

This integral i s divergent, and this fact presented a major obsbele to qumtum eleetrd;~r~awries for 20 years. Its solution requires a change in; the fundamental, laws. Thus suppose that the propagation kernel for a photon is


(l/k2)c(k2) instead of just {l/k2), where c(k2) i s so chosen that c(O) = 1 and c(k9 0 as k2 -+W. In space representation the madlffcation t a b s the form

The new function f, differs ~iwif ieant ly from 6, only for small intervals. This i s clear from the fact that if the high-frequency components are removed from the Fourier eqaneion of a function, only the sharl-range de- tails a r e moetifiedl. In the present case the size of the interval over wKeh the function i s mdif ied can Be described roughtg a s foUows: Consider a large number, )lZ, and suppose that so long a s k2 << h', c@) 1. Then {from the exponential term) dzfemnees will occur when the interval s 2 m l/ha. Call

this value a2, and the general behavior of f, i s shown by Fig. 27-2. Thus a2 is sort of a "mean width" of f, . If a2 << 1, as assumed, then when

which is the size of the interval. The significance of the form of f,(s2) can be understood from the following. The original function, b+(s2) differs from zero only when a' = t2 - rZ = 0. That i s to say, an electromagnetic signal can reach a point a t distance r only a t a time t such that tZ - r2 = O or t = r (i.e., the speed of light is 1). This i s no longer true for f,(s2). The departure i s obtained by a measure of 1 - r, But, by Eq. (27-81, for all values of r a this measure is negligible. Thus, depending on A ~ , the laws will be found undfected over any practical dislanee,

Choosing h2>> m2, a practical (and general) representation of c(k2) i s

and the simple form is suggested,

Prom this, obtain the propagation kernel a s

The second t e r n is that for the propagation of a photon of mass h; however, the m b u s sign in front of the k r m has not been explained so far from this point of view.

A convenient represenation for this kernel i s the integral

Introducing this kernel into Eq. (27-6') in place of l/ka gives

which ean be written as the sum of two inkgrals, wMch differ only by having m o r g in the numemtor, t k t is, m or ka (since g = k,y,).

METHOD OF m1C'EaMTION CIF m"1C"EGMU APPEAIRmQ D QUANTUM ELECTRODmAMICS

We shall need to do many integrals of a form similar to the preceding one, A method k s k e n worked out to do these fairly efficiently. We now stop to d e s e r i b this method of inkgration,

EveryLhing will be based on the fallotviryi: two integrals:?

In Eq, (27-XI), to write a little more compactly, we use the rrohtion fl;k,) to mean tbat either X o r k, i s in the numerator, in which case, on the right- hand side the (1;0) i s I o r 0, respeelively. 410 grove the first of these, nok

?R, P, Feynman, Phys, Rev,, 76, 769 (1949); included in this volume. N o b that in the article d% i s equivalent to 4$[d'k/(2~)~1 in our notation.

t k t , if kk, i s in the numerator, the integrmd i s an oc3d function. Thus the integral is zero, With 1 in the numerator, contour integration i s employed. Write the integral

Then for E L + k2, there a r e poles a t w = & [(L + k2)'/2 - it-], and contour integration of w gives

with the contour in the upper half-plane, Two dgferentiratious with respect to L give

Then the remaining integral i s

which proves Eq, (27-1 1). E k -p fs substituted for the variab2e of' integration in Eq. 627-113, the result i s

By diEerentiatlng both sides of Eq. (27-13) with. respect to A o r with respect to p j , tb re foXfows direetly

Further differentiations give directly su~cess ive integrals i n c l u d i ~ more k factors in the numerator and higher powers of (k2 - 2p * k - A ) in the de- namimtor .

SELF-ENERGY mTEaML WTH A% EXTERNAL PQTENTZAL

Last time it was found that the self-energy of the electron is equivalent

" C O R R E C T I O N " T E R M 3

and t k t this could also be expressed in b r m s of integrals,

It was also found that

V s i q the definite inbgraf

the denominattor of the InLegrand of Eq. (28-2) may be expressed a s

so t k t Eq. (28-2) becomes

The integral over k can be done by u s iw Eq. (28-3) with the substitutions xp fo r p and L(l - X) for h, giving

The integral over L i s elernenbry and gives

f I = -2(32n2 i)""' 4 dx (l;xp,) In [(l - x)A? + m2x2/m2x2]

When h2 >> m2, it is legitimate to neglect m2 x2 in the numerator [it i s true that when x m I, ( X - x)h2 is lzot much larger than m"', but the Lnbrval over which this IS true i s so small, for h2 a m2, Chat the e r ro r i s snzarl], so that, wlren the x integration i s performed, t

Q U A N T U M ELEeTROD"YbJAM1CcS

The change in mass is [from Eq, (28-l)]

Since $U = mu and (;U) = 2m, this can be simplified to

Now (ev1271) i s about 10", so that oven if h, is many times m, the fraction c b g e in mass will not be large, The intewretation of this result is a s follows. There is a shift in mass which depeds on h and hence camot be determined theoretically. One can imagine an experimenkl mass and a theoretical mass which am re lahd by

All our measurements a r e of m,xp, that is, self-action is included, and mth, the mass without self-action, cannot be dehrmined, Mare aecurably sbbd,

a theory using mep , 81%~ A. t h e 0 0 u s iw mtt, and e21fic self -action minas

is equivalent to e 2 h c self -action Am a s carnguted for a h e particle S

When the electron is free, the ez/gc seff-action k r m emctfy cancels the Am term and a theory using meXp i s exactly correct. When the electron i s not free, e 2 h c self-action is not qufh equaX to the Am term and there is a small correction to a theory using meXp . This effect leads to the Lamb shift in the! hydrogen atoonn, and, in order to calculate! s w h effects, we s h l l now consider the effect of self-action on the seatkr ing of an electron by an e x b r m l pobntial ,

SCATTERmG I24 AS EXTERNAL PWENTUIZ

The diagram for scatkrine; in an e x k r m l potential is shown in Fig. 28-1, and the relatiomhips fa r this process, excluding the possibiXfty of self- action, a r e a s follows:

Pobntial: d(9) = yt ( B ~ z ~ / Q ~ ) &(Q) lfor Coulomb potential Matrix element: M = -ie(f"uzdul) Conservation relation: = fit + 4

First-order self-action will produce the diagrams shown in Fig. 28-2. The amplitude for process i s obkimed in the usual manner, For example, dlagram I gives

Ratiomliziw the denomimtors and in~erting the convergence factor, this becomes

This ewression also b p p n s to diverge for smaff photon marnenta (k) (a result whiefi has b e n called the; '"drared eat;aslrophe," but which has a

11

FIG. 28-2,

144 Q U A N T U M E L E C T R a D Y N A M . I C S

clear physical interpretation, discussed la%r). Temporarily the k2 ander d4 k will be replaced by (k2 - h2min 1, where h2rnin m2, to make the integral convergent. This is equivalent to cutting off the inbgrai somewhere near k = h ,in , and tk physical interpretation i s left to Lectures 29 and 30. To ftzcilibte the inbgratfon over k, the following identity i s used:

since h2 >> m2 >> hZmin . This substitution produces integrals of the form

To evaluate these inbgrais, we make use of the identity

where $y = y ~ l + (1 - Y)$~ . Performing integrations in the order, k, L, y, and us iw the ztppropriab integrals in Eq. (28-6) gives a s the m a t r h to be taken brjtvveen s t a k s u2 and ul

e2 m 4 - fn-- + @ t a n @ + = 2n X min

where r = In ( A/m) + 9/4 - 2 in (m/Amja) and 4m2 sinZ B = q'. .It is shown in Lecture 30 that diagrams U and EX (Fig. 28-2) prdrtee a

contribution M2 -+ MS = -(e2/2n)r&, which just cancels a similar term in Ms. When q is small, B * (qz)'n/2m, and the sum M1 + M2 + M3 can be approxi- mated by

The (46 - $4) can be writ&n out

But qp is the gradient operator so this can be written, in coordinate repre- serxlatian,

[see Eq. (7-1)I. Reference to page 54 shows that the effect of a particle'^ having an anomalous magnetic moment is to subtract a potential p FP, from the ordinary potential d = ypAp appearing in the Dirac equation. Since this is precisely what the f i rs t k r m of Eq. (28-10) does, one c m say that this part of the self-action correction looks like a correction to t k electron" magnetic moment, so that

Nob that this result [and (28-9) md (28-%@)l doss not depend on the cutoff h, and hence h em now be &ken to be. infinity, f

I t has been shown that w b n a paXlf.i~le i s scattered by a ysobntial, the pri- mary effect is that of g, and that for diagram I (Fig, 28-2) a carreetion term ar ises wNeh is

rdr I FIG. 28-2

R, P, Feynmm, Phys, &v., 76, 769 (1949); included in this volume.

It remains to show t k t the combined e fke t of diagrams II and E1 (Fig. 28-2),

when considered along with the effect of the mass correction, is another correction k r m ,

just cancelling the fast k r m in the preceding expression. It i s recalled that the necessity for considering. the effect of the mass correction together with the self-action represented in diagrams 1, a, and SE is that the theory be iw developed must conlain the emerimenhl mass rather t b n the ''theoretical'' mass.

Suppose that in the Dirac equation

rn lh , the theoretical mass, is replaced by m - Am, where m i s the e q e r i - mental mass; then

The mass correction Am is just a number, so that in momentum representation it is a (S function of momentum, Hence from the form of the foregoing equation, i t is seen to behave like a potential. with zero momentum and In- volves no matrices, Diagrammatiestlly its effect may be represented as in Fig. 29-1, The minus sign i s used because the effect of the mass correction Am is to be subtmckd from the results obhined from dkgrams X, IX, and EX (Fig, 28-2) done, For diagmnn IL the amplitude would appear to be

"CORRECTION" T E R M S

and for diagram IfYFig, 29-X),

But the part of the amplitude for diagram TX. (Fig 28-2) eonbined in the pa- rentheses is just Awnul, so that XI and 11' ssem to cancel, A similar result applied for d i apams m d El" This is an e r ror , however, arising from the fact t k t both of these amplitudes a r e i d in ik , owing to the factor 4 - m in the denomimtor. Hence their difference is i ndebmimte , But by suh- t ract iw them p rop r ly it will be found that their difference does not vanish.

The method proposed to accomplish this subtraction will, in fact, give the combhed effect of the self-action and mass correction of both diagrams 11 arxd Kf and Ifband 1311'. It iia based on the fact t h t an electron is never actually free. An electron's M~story will have alwaya fnvolved a ser ies of scatterings, a s will its future, These scatterings will h conaidered as oc- eu r r iw a t long but finite time intervals, Xt will be sdficient to calculate the effect of seE-action and the mass: correction k tween any t w of these scat- b r i w s , since the result will evidently be the same bdween each pair of them. T k n , the effect will accounkd for simply by r ega rd iq a, eerrrec- tf on, q u a l to that calculated for om of the inbrvals btween scatterings, a s being assochtc3d with the pobntial a t eaeh scater ing (nurnbr of intervals equds number of scat ter iqs) . Then, considering a s iwle seat ter iw event a s here, this correction to the pobntial represents all the effects of diagrams II[, In, B*, and 111"

For an electron which is not quite free, pa .: rn2 exactly, but insbad


by the uncertainty principle, and T is the interval between scatterings. gince T is hrge , E i s a small quantity. Let 6 = (1 + E)&, whem I?(@ i s the momentum af a free electron.

If and 36 a r e the momentum representatives of the scattering poten- t i a l ~ at a and R (any two sca tkr ing~) , then the m a t r k of the amplitude to go from the initial sttzte a t a to the ffml sLaZ;e a t b without axly perturba- t-ions I s

up to brms of order E With the perturbations of self -action and mass correction, this m a t r h is

(a) Without wrturbation (b) With prturbatfon of self-action and maists correction

It is the value of this matrix earnpared to t h t of the unperturbd matrix wucb gives the desired correction tern (see Fig, 29-2).

Problem: Show t h t for two noncammuting (or eornmuting) operators A and B, the following expansion i s truer

Using the result of the precedfw problem, one can wrile.

""CORRECTXOM" T E R M S

so that t b foregoing m a t r h becomes

The f i rs t and last terms a r e identical, up to terms of order E , hence may be cancefed. The integral in the second term h8 already ben . done essentially in eomputiq diagmxn I (Fig. 28-21, except bere replaces 6, &, and g&, so that Ilf = - pfi = Q in this eaae and gives the result

To this order in E the $% in the numerator may be replaced by PJOfs. It i s also noted that since = mu,

s a t b t the foregoing result may h written

This i s just -k2/2a)r times the matrix for no perarbation. Hence the cor- reetion 2erm due to diagrams 11, fII, E ' , and 111' is obhined simply by replacing the s c a t h r i w potential $ by -(e2/2r)r$, a s was slated earlier.

It should be noted t k t the dZfleulity in obtaining the proper subtraction of the self-action and mass corrections just clarified does not represent a ""dvergence " vab l em of qtrmtum eleetrodymmics, It is a tsiczal problem which could as well ar ise in nanrelativistic quantum mechanics if, for example, one c b s e same nomem value at3 a reference of potential, that is, regarded a free electran as m o v u in a uniform nomero pokntial. It may be easily verified that this would givr;. r ise h an "energy correction'9or the free e l e e t r o ~ amlogous ta the -ss correction hvotved here, Then in

computing the anrrplibde for a s ca t t e r i q process where one used ;a "theoretical emrgy" a d subtracted the effect of the "energy correction," the difference of idinite terms would appear i f one used free-electron wave functions.. In this simple ease the i d i n i h term would, indeed, caneel upon proper subtmction but in principle the problem is the same as the present one.

F inally, the complete correction &run a r i sing from self -=lion and rnaas correction is

4 +8bn6+-

tan 28

e2 28 sk + - Mid - dd) -7-- 8nm sin 28

RESOLUTION QF THE FXCTnIOU8 "H ED CATMmOPHEf'

From the correction term just debrmined, it Is seen t b t , to order e2, the eross section for scattcsriw of an electron with the emission of m photons is

where a. i s the eross section for the potential 6 only. This cross section diverges 1ol;~arithically a s h ,in .+ 0, and it i s this divergence wMch was formerly referred do a s the " id ra red c a t a ~ t r o p h e . ~ ~

This result, however, ar ises from the physical fact t k t i t i s impossible to sea%er an electron, with the emission of 7ao photons, When the electron is scattered, t h electromagnetic field rnust c h n g e from t b t of a ehrrrge moving with momentum pi to that for momentum pz. This c h n g e of the field i s met3 ssarily accompanied by radiation, h the theory of brehmsstraMuw, it was s h a m t k t the eross section for

emission of one low-energy photon is

PrclbEem: Show that the integral over all directions and the sum over plarizations of the foregoing cross section i s

where sin2 B = - ~ ) ~ / 4 r n ~ . Thus the probability of emitting any photon between k = Q and k = K&, is

which diverges logarithmically, Therefore, tbe dilemma of the diverging seatbr ing cross section actually

a r i ses from a s k i q an improper question: What i s the c b n e e of s ca t t e r i q with the emission of no photons ? Instead, one should ask: W h t i s the e h n e e of scattering ~ t h the emission of no photon of energy g r e a b r t h 3 For there will always be some v e ~ soft photons emitbd.

Then, effectively, what i s sought In answer to the last question is the chance of seatkr ing and emittlng no plboton, the chance of emitting one photon. of enerm below Q, and the c b n e e of two and more photons below K, (but these terms a r e of order e4 and higher and heme a re neglected).

Each of these k r m s i s infinite, actually, but i s kept finite temporarily by the artifice of the dc m;, . Their sum, however, does not d i v e r s , as may be seen by gatheriw the previous results and by wri l iw

C h n c e of scatbring and emitting no ptrotoo of energy > K&,

+ (terms inde-

of s rder e4)

terms independent

This does not depelld on hmin and hence resolves the " idrared c a b ~ t r o p h e . ? ~ It has been rshown by Bloch and Nordsieck that the same idea applies to all orders.

It i s interesting that the largest k r m in the quantum-electrodynamic corrections to the scal;t;eriq cross section, namely,

may be obbined from classical eIeetrodymmies, since such long wave- len@hs a r e involved. The other terms have small effects. To date, the scattering experiments have been accurate enough to verify the existence of the large term but not accurate enough to verify the exact contributions of the smaller k r m s . Hence they do not provide a nontrivial test of quantum electrodynamics,

These same considerations apply in any process involving the deflection

f F. Bloch and A. Nordsieck, Phys. Rev., 52, 54 (X937).

152 QUANTUM E L E C T R Q D U M A M f C S

af free electrons, The best way tn handle the problem i s to calculate every- thing in terms of the Xmi, and then to ask only questiana which can Izave a sensible answer a s verified by the evenha1 elimiwtion of" the hmia.

Problem: Prepare diagrams and integrals needed for the radiative corrections (af order e2) to the KleiwNishina formula. Do as much as possible and compare msults with those of L. B r o w and R. P, Feynman.?

AHOTHm WPIROAGH TO 1["m W ]EX? DIFFICULTY

hstead sf introducing an artificial mass, assume no weak photons contribute. Thus we must subtract from the previous resulLs the contributions of all photons with momentum magnitude less t b some n u m b r kg >> h. The previous result i s

{l + (eZ/2n)[2 in (m/Xmi, - l)(l- 28/tan 2e)l + t? tan B

The term to be subtmckd Is

We assume ko a pi o r p ~ , and neglect both K and the f i rs t two k2 in this integral. Then using d = 2p, - $', the integral i s approximately

Then

This i s the brrn to b subtracted from expression (30-1). Usiw sin2 8 = q2/4m2, for small q, Eq. (30-4) becomes

Subtracting this f m m Eq. (30-l), also with q small, gives

The last k r m is [ln (M[/2b) + (1 31/24)1.

EFFECT 024 4 9 ATOMIC ELEGTROPiT

Consider the hydrogen atom with a potential V = e2/r and a wave function (R) exp (-iEo t) = qo(xI, ) Take the wave function to be normalized in the conventional =mere The effect of tlre self-energy of the electram i s to s h g t the energy level by an amount

The f i rs t integml fst written down from Fig. 30-1. The second is the free- particle effect a s noted in previous lectures. The kernel i s not well

FIG. 30-1

enough determined to make exact calculation of this integral pos~lble , An approximate ctt.lculation c m be made with the farm

- similar sum over negative energies for t2 < t,

The photon propilt;l;ation kernel can b @ w a d e d a s

6 , (S,,$) = 4% $exp [-ik(ts - t,) + ik(xZ - xl)j d'k/2k(2~)-~

%

= 4n J'exp[+ik(tz -tl) + ik(xz-~$1 d3k/2k(2n)"

tz t1

I f s i q these expressions, Eq. (30-6) becomes

= C +f bp (-iK * R)lon (E, + K - ~ ~ 1 - l (aP exp (iK * R)],@ + D

dSk/4nk - exp (-iK . R)]@, (J E, I + U + E,)-~ --a

[a exp (iK R)lno dS k/4nk - (A m term) (30-7)

This form implies the use of +* instead of and a4 = 1, = a. Another approach to the motion of an electron in a hy$mgen a b m i s the

following. Consider the electron a s a free particle inbrmit%ntly seellt&red by the Coulomb pokntial. The s c a t b r t w s cause p h s e shift in the wave function of the order of fftydbrgfi ) , T hufs the perfod between s e a t b r i q s Is of the order T = tf/Rydbrg. Take the lower limit of the momentum of the "self-action" photons a s very large compared to the Rydberg. Then it Is v e v probable t b t an ernitbd photon will be reaborbed before? two interactions btween the electron: a d the poknthf have taken place; it is very improbable for two o r more s c a m r i w s to take place between emission and absoqtfon (see Fig, 30-2). Then the correction to the potential i s that corn- pubd in Eq. (30-5) for small q @lus anomalous moment correction), This i s

in momentum space, To tmnsfarm to ordimry space, use

s2 P (4a2 - Q') $ M (aZ/at2 - v2) V

Thus the correction i s

This cormetion is of greatest inzporbnce for the s state, since with a Gou- lomb potential V ~ V = 4nze26m), and only in the s states is m(R) different from 0 a t 1R = 0.

The c b f e e of fq is dekrmined by the inequalities m ko >> Rydbrg. A satisfactory value i s = 137 Ryd. With such a h, the effect of photons of k c ko must b included, This wil b done by separating the effect into the sum of t b e e cantrilbutiw effects. It will. be seen that two of these effects

'VGQRRECTJON" T E R M S

probable improbable

FIG. 30-2

are i d e p n d e n l of the potential V and thus a r e ctznceled by s i rnihr terms in the Am correction for a, free particle, Thus for only one situation must the effect be eompuhd. Xn all cases, s i ~ c e k i s small, the nonrelativistic approxfmalian to expression (30-7) may be used,

(1) The contribution of negative energy sLiztes : Neglecting k with respect to m gives

The matrix element for cx4 is very small, and only the elements for at need be considered. Then the sum over negative s k t e s i s

If this sum is continued for +n, a negligible term of order vZ/c2 is added. Thus the sum is approximately

- C J [(aan) (ano)/2m1 k2 dk/k = (a * k2 dW2mk aXf states

This i s il-kdependen-1: of V, and thus i s ~ a n ~ e l e d by a similar quantity in the Am term,

(2) Longitudinal positive energy states (ap -- Q k/k) : As an exercise the reader may show


X@ k/k) (ik * R) lno =: (En - Eg)/k f e w fik . R)fno

and the contribution of these bmms summed over positive e n e q y sbks gives

- (E, - ~ ~ ) ~ / k ~ 1 exp (ik R),, exp (-ik * R),o (E, + k - E,)" dS k/4sk

Writing H = p2/2m (V commutes with the exponent), this becomes

This brm is fnde~nden t of V, a d thus is also eaneeled by the Am eorrac- tfon.

(3) Tramverse p0sitive energy sh tes : Since ko is large campared to the size of the atom, the dipole approximation can km used. t Tbe general b r m in the sum of Eq, (30-7) &comes

wr i ting

(E, + k - E@)-' = l /k - (E, - Ed/(En + k - Eo)k

the term in l[k can be split off from the rest of the inkgra1 as a qumtfty independent of V and tfrtzs canceled by the Am correction. Further, by averaging over direetiom,

in the nonrelativistic approximation. Thus the inbgral of Eq. (30-8) is

Ushg the relation

I. Cf. H, Betbe, Ph.ys, Rev., 72, 339 (1@47)*

and the fact that >> E, - E@, one part of the sum over transverse positive energy s b t e s is

This cancels with the In of Eq, (30-77, leaving the final correction as

-E- m o m a l o ~ s moment correction

This sum has b e n carried out numeri~al ly to be compared with the observed Lamb shift,

GL~IED-1;WP PROCESSES, VACUUM PQLmIZATTON

Another process which is still of f i rs t order in eZ has not been consid - ered in the s c a t k ~ n g by a potential. h a b a d of the potential scattering the particle directly, i t can do s o by f i rs t creating a pair which subsequently annihilates, creating a photon which does the scattering. Wagram 1 (Fig. 32-1) applies to this process; diagram fl applies to a similar process, with the order in time c h g e d slightly, The amplitude for these processes i s

1 1 i4m2 C ( ~ Z Y ~ ~ I ) $ J U rrn Y~ (l

spin states

where u is the spinor pax% of the closed-loop wave function. The f i rs t parenthesis is the ampffbde for the electron to be scattered by the photon; 1/q2 is the photon propagation factor; and the second parenthesis is the amplitude for the closed-loop process which produces the photon. The expression i s inhgrahd over p &cause the amplitude for a positron of mo- m e n b is desired. b the sum over four apin s k t e s of U, two s h t e s take care of the processes of dlagram I and two s k t e s take care of the processes of diagram XI. No projection opemtors a r e required, so the method of spurs may be used directly to give

a form which eorrlains b t h X and E (so aa usual it i s not necessary to make separate diagqams flor pmeesrses whose only difference is the order in time).

15 8 Q U A N T U M E L E C T R O D Y N A M I C 5

This integral also diverges, but a phobn convergence factar, as used in the previous lectures, Is of no vdue because now the integral i s over p, the momentum of the positron. 1x2 the intermediate step, The method which. has hen . used to circumvent the divergence [email protected] is to subtract from ttris integral, a similar integral with m replaced by M. M i s taken to be much: larger

11

FIG. 31-1

t b m, and this results in, a t w e of cutoff in the inkgraf, over p, When this i s done, the amplitude i s f'auxxl to be t

(4m2 + ~ ~ " / 3 ~ ~ + 11/91 (3 f -3)

?See R. P. Feynman, Phys. Rev,, 76, 769 (1949) ; included in this volume,

" C O R R E C T I O N " T E R M S

where = 4m2 sin2 8, which, for small q, becomes

Notice that (GzYp ul) = (G2$u1), SO that, considering only the divergent part of the correction, the effective potential i s

The 1 comes from the theory without radiative corrections, while the e2 term fs the correctian due to processee of the type just descr ibd , Thus the correction can be interpreted a s a small reduction in the effect of all potentials, and one can introduce an experimental charge eeXp and a theoretical c k r g e eth related by

where B @ ) = -(e2/31h) ln ( ~ / m ) ' , in a manner analogous to the mass correction de se r i bd in Lecture 28. This i s referred to a s '"charge renormalization. " The other term,

i s more interesting, since it represents a perturbation 2e2/15r (v2 V). This carreetion i s resp~nsible for 21 Mc in the Lamb shift and the {ln fnn/2(E,- E@)] + (11/24)) term in (50-7" is replaced by (ln (m/2 (E, - Eo)j + (11/24) - (1/5)) . The 115 term is due to the ""palirization of the vacuume?*

One possible process for the scattering of light, and an indiatinguisbble a lb rmt ive , i s indicated by the diagrams in Pig. 32-22. The second diagram differs from the f i rs t only in the direction of the arrows of the electron lines, Reversing such a direction is equivalent to cbnging an electron to a positron. Thus the coupling with each potential would c b n g e siw, Since there a r e three such couplings, the ampiitude for the second process i s the negative of thszt for the first. Since the amplitudes add, the net amplitude i s zero. Xn general, m y closed-loop process of this t m involviq an odd number of couplings to a potentbl (includiw photon), b s zero net amplitude,

P~obEem: Set up the integrals for each of the two diagmms in Fig. 31-2 and show that they are equal and opposite in s i p .

However, the bgher-order processes shorn in Fig. 31-3 can take place. The amplitude for the process is

FIG, 31-3

plus five similar brms resul tfq from permutiw the o d e r of phatans, This integml appears b d i v e ~ e logarfthmlcaly, But when all six albrmtlves are &ken into account, the sum leave8 no divergent Wrm, More eorrrplfcabd closed-loop processes are convergent,

Pauli Principle and the Dirac Equation

fn Lecture 24 the probability of a vacuum remaining a vacuum under the i duence of a potential w%s calculated, The potential c m create and amf- hilate pairs (a closed-low process) htweerz times ti and t2, The amplitude for the ereatlon and ihilation of one pair is (to f i rs t nonvanishing or-der)

The amplitude far the creation and annihilation for two pairs is a factor L for each, but, to avoid counting each twice when integrating over all d q and d ~ ~ , it i s IJ2/2, For three pairs the amplitude is lL3/3!. The total amplitude for a vacuum to remain a vacuum is, then,

where the I comes from the amplitude to remain ai vacuum with nothing happening. The use of minus signs for the amplitude for an odd number of pairs can be given the following justification in k r m s of the Pauli principle, Suppose the diagram for t ti is a s sbm in Fig, 31-4. The completion of this proeess cart occur in two ways, however (see Fig, 32-5)- The second way can be thought of a s obkfnod by the inbrchmgc? of the two electrons, hence the amplikde of the second must be subtracted from that of the first,

FIG. 31-4

PAWL1 PRIEMCIPLE A N D DXRAG E Q U A T I O N S 163

FIG, 31-5

according to the Pauli principle. But the second process i s a one-loop process , whereas the first process is a two-loop p m c e ~ s , so i t can b~ concluded that amplitudes for an odd number of loops must h subtraete.d. The probability for a vacuum to remain. a vacuum is

P ,,-,,, = /c ,I = exp (-2 real part of L)

The real past of L (It. P. of L) may be shown Lo be positive, so it i s clear t h t terms d the? ser ies must a l t e r a t e in sign in order t k t this prokbili ty b not grealter than unity,

We have, therefore, two arguments a s to wfi?J the eq r e s s ion must be e'L, One involves the sign of the real part, a property just of K, and the Dirae equation. The s e c o d involvew the Pzruli princi.ple, We see, therefore, that it could not be consistent to in tevre t the Dfrac equation as we do unless the electrow o k y Fermi-Dirac skt ts t ies , Tberts Is, therefore, same comeetion, btwe?en the relativistic Dirae equation and the exclusion principle. Paul has given a more ebborate proof of the necessity far the exclusion principle but this argument mafices it plausible,

This quesUon 9f the eomection between tire exclusion principle? and the Birac equation is s o interesting that we shall try to give another argument that does not involve closed loops, We shall prove that i t i s inconsistent to assume that electrons a r e completely independent and wave funetions for several electrons a r e simply products of individual wave functions (even though we neglect their Interaction). For if we assume this, then

P robability of vaeuum = Pv

remaining a vacuum

Probabiltty OF vacuum to 1 pair =Pv a 1 23 p&rs I ~ l ~ a i r I '

Probgbility of vacuum to 2 pairs PV C I K ~ IKI pair/'

a i ~ pairs

164 QUANTUM E L E C T R B D Y N A M I C S

Now, the sum of these probabilities i s the prohbili ty of a vaeuum becoming m y tMng and tMs must bit? unity. Thus

1 = Pv [ l -F @rob, of 1 pair) + @rob. of 2 pairs) += a * 1 (31-8)

The probability that an eIeetron goes from a to b md t h t nothfng else b p - pew is P,/K, @,a)/ '. The probability that the electron goes from a to b and one pair is produced is P, [ ~ + ( b , a ) i2 I K ( ~ pair) i 2 , and the probability that the electron goes from a to b with two pairs produced i. P,l~,@,a)/'

/K(2 pair)l'. Thus the probability for an electron to go from a to b with any n u m k r of pairs produced i s

[see Eq. (31-8)j. Now since the electron must go somewhere,

However, it is a p r o ~ d y of the Dirac kernel that

and an inconsiskncy reaults. The Inconsistency can bt; elirnimtad by assurn- ing that elsctrona obey Fermi-Dirae sbt is t ics md a r e not independent, Un- de r tbse c; i reum~Wces the origiml electron md the sfeetran af the pair a r e not independerrrd a d

Probability of electron from a to b plus 1 pair pmdueed < jM,@,a)j"t~(1 wfr)j2

because we should not allow the case that the electron in Ifre pair is in the same sbte as the electron a t b.

For t k kerml sf the Klefnaorbon equation, i t turns out that the sign of the ineqmli+ in Eq, (31-IQ) is reversed, Therefore, fa r a spin-zero pa&f- cle neither Femi-Dirac statistics nor independent particles a r e possfbfe. If the wave f"urmctiions a r e hken symmetric (charges reversed add amplitudes, Elnsbin-Bose sbtfsties), the inquality Eq. (31-11) ia also reversed. h symmetrical statistics the? preeence of a pa&ieXe in a sate (say 6 ) ert- hances the chance t h t another i s created in the same s h t e , So the mein- Gordon quat ion requf reh~ Base statistiee ,

t i q to t ry to sharpn, these a r s m e n t s to show t b t t b [~,@,a)l' db and 1 is quantitatively exactly compen-

sabd for by the exclusion principle, Such a f m h r n e n h i relaaon ought to h v e rt cleat: md simple exposition,

t0. SUBIXHARY OF NUMRmCa FACTORS FOR TMBSITZBH PROBABILImS

The exact values of the numerical factors smearing in the rules of If for cornputing transition probabilities are not clearly stated there, so we give a brid summary here."

The probability of transition per w a n d from an initgaal s h & of enerm E to a final state of the same total egeru (warn& ta be in a continuum) is given by

where is the ddesity of final states p r unit enerw range at enerm E and is the square of the matrix element taken between the initial and find state of the trandtion matrix iTn appropriate to the problem. N is a normiizing constant. For bound states conventioi~alky noxmafized it is 1. For free pa~icle t;lates it is a prdnct of a fachr N , for each prticle in the initial and for each in the hnal energy state. N , depnds on the normdization of the wave functions of the mrticles (photons are considerect m particlm) which is used In wmputing the matrix element of 3n. The simplest mfe (which dms not destroy the apprent covariance of X], isZk N,=2t,, where r , is the enerw of the particle. This corrmpnds to chooliing in momentum space, plane wava for photons of unit vettor potential, e2= --l. For electrons it commpnds to udng (&S)= 2m (so that, for example, if an electron is deviated from initial @I to final h, the sum over ail initiai and fincil spin strxtes of 1 isp~($~3-nrj%Cpx+m)%~ j, Choice of noma- Iization (@rc)-i I resulls in H,= 1 for electrons, The matrix 311 is evaluaM by making the diagmms and foliowing the rules of 11, but with the following debni- tion of numerical factors. (We give t h m here for the spesial case that the initial, final, and intermdiate

*In I and 11 the unfartunate conventian was made thst dab mans d k * f R ~ d k d k t @ n ) ~ for mementurn qace in&pats. The -fusing frrcter (2x1- here serves no usefut purpage so the cm- ventian will be abandoned. In thiogeetion d4k has itrruhal meaning, &&&1dRlaa,.

R C A L C U L U S 123

states consist of fret? particla. The momentum space reprwntation is then most convenient.)

First, write down the matrk directly without numerim1 factors, Thus, eEectron propgation factor is (p--4-1, virtual photon factor is with coupIings r,.. -7,. A real photon of polarization v ~ t o r c, contributes factor e, A potential (timm the eletron charge, c ) A,(%) contxihutes momentum g with amplitude a(q), where @,(p] = JA,(f) expliq. zlfd4sl. (Note: On this point we deviak fran the definition of U in I which b there (h)-% times as Iarge.) A spur is taken on the matrices of a ccimed lwp. Because of the Pauli principle the sign is alter& on conkibutions comapanding to an exchallge of e l ~ t r o n identity, and for each closd imp. one multiplies by (2xj4d*$= (2n)--"dpfd~Jfi&$~ and integrate over all values of any undetermined momentum variable p, (Note: 6 n this point we again differ?@)

The correct numerical value of is then obtained by muftipfication by tfie following factors. (I) A factor (4r)te for each coupling of an electron to a photon. Thus, a virtual photon, having two such couplings, contrihues 4r8, (In the units here, ef= 11137 -proxi- mately and (ha)& is just the charge on an efeetron in hmviside units.) (2) A further factor - 1: for each virtual photon.

For m m n tbeorim the ckanges d i s c u d in 11, Sec, I0 are made in writing m, then further factoa are (1) (4n)bg for each maan-nucleon coupling and (2) a factor -i for each virtual spin one meson, but 3-1' for each virtuat spin zero mmn.

'This s a m s for tmmition probabilities, in which only tfie abwjute quare of IFn is requird, To get X to be the actual phast: shift p~tr unit volume and time, additional fators of i for mch virtual electron propa- pation, and --i for each peential or photon internetion, are necemry, Then, for enwu perturbation problems the enerw shift is the e w t e d value of iERZ for the unperturbd starte in question dividd by the normalization consrant N , belonging to each particle compris- ing the unprturbed state.

The author has profitcsd from diseussio~s with M, Pahkin and L, Brown,

*qn genera!, 1Vi ia the article density. It is N,=(&ynr) far io mrklf hddo and ~ & * d + ( d f l - ~ @ / d 6 ] for 61h. ?L htcer rs t r , ~f tttc field arnp~ttde + ta t a k ss umty.

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P M V S f C A L R E V I E W V O L U M E 7 6 , N l f W B E R 6 S E P T E M B E R 1 5 . 1 9 4 9

The problem of the khavior of positram and electrons in given nrtrrna1 potentkb, nqlwting their mutml inwaction, is m d y d by repking the &mv of holes by rt, reink~retrttian of the solutions of the Dlrac equation. It i s possible to write down a complete slution of tL problem; in terms of: baundary conditions on the wave function, and thi salution contajw ;rutamarialky all the pcrsibilities of virtuat (and real) pair f o r m h n rrnd annihiiaGon rolfether with the ordinary scattering procem, including the w r r s t relative signs of the various terms.

In this saiution, the ""negative energy sbtegB' appfzbr in a form wbieh may be pictured (as by Stiickelberg) in wace-time as waves traveEng away from the external potmtiaf badwards in time. E ~ r i m e n k l l y , such a wave carrebponds to a positran appr-h- in$ the potential and annihilating the electron. A particle moving forward in dae (electron) in a potential may be scatered forward in time (ordinary mtcclring) or backward (pair anni2lilatioian). When moving backward (positron} it may be wattered M w a d

in time (po~tron a t t e r i w ) or fomard (pair produc6on). For such a parricle tfte amplitude for transition from an ioitial to a hnal s a t e is analysed to any ordw in the poten&f by mn&derlng it to undergo a q u e n c e of such scattierings.

The amplitude for a process involving m n y such particles is the product of the transition amplitad= for each prLicle. The mclu&on principte requires that antisymmetric mmbinatiuns of amplitudes be c b m n for these mmplete p r o c m wbjch diifer only by exchaage of partid-. It e m s that a congatent interpre- ~arion is only posJIbk if thr! =elusion pdaciple is adopted, The exclusion principle n d not he &km h t o m u n t in intermediate sitam. Vacuum prAlenrs do not azise for chsrgw which do not i n m t with one another, but these are anaimd nererthelm in mticiparion of application to quantum etatrodynadcs.

The resnlts are also e r & in momentumener@ vdables, kuivafence to the wcond quanthtion tha ry of bletj is proved in arm ~pacia;.

1, INTRQDUCBON as a h a l e rather than breakhg it up into its pieces. THIS & the first of ~t of papers dealing with the I t is as though a bambardier wng low over a road solution of problems in quantum electrdyna&. 'ud"n'Y three an' it '8 when two of

The main prinGpb is to deal direct& with the them come tagetbr and diaP~ar 'gain that he riX&ZtZ3 to the Ham2tonhn differential equations rather than tbt he has Over long in with these equations beaxsefves. Here we treat simpk singXe "". t;he nrotia,sn ejectrons and porjitrons in given over-aa spm'-tim' Point of v'ew 'ads to con-

patentiais, pwr We consider the interactions siderable simplifiation in m n y problems. One can take of &eae particles, that is, quantum elEtrdmamics, into account a t the same time process@ which ordi-

The of chargm in a fixed polential is usually n a ~ l y would have to be considered separately. f i r tr<?atd by the of second quantization of the example, when cansidering the scattering of an electron

electran the d the thmry of holes, by a potential one a u t o m t i a l l ~ ~~ into account the we show that bq. a suitable choiu: and inter- egects of virtuzb pair productions. The =me eqwtion,

prebtion of the solutions of Dirac,s qttatron the Dirac's, rvhich descrihs the deflection of the world line

be equally we,l treated in a manner which Is of an electron in a field, can atso describe the deltectian

fundamentally no more complicatd than SchMinger,s (and in just as simple a manner1 when it is large enough

method dealing with one or more The vaii- to rever* the time-senw of the world line, and thereby

ous and annihilation aprators in c o r r w n d to pair annihilation. Quantum mechanically Ihe dlrclc~on of the world lines is rqlaced by the

tional electron field view are required becaune the direction of propagation of number of particles, is not conxrved, i.e., pairs m y bc ~ b i ~ view is dzerent from tht of the

Or datroyed. On the "lter h'nd charge tonian method which considers the future as developing c o n w e d which su@nts that if we the charge* mntinuovsly from out of &e p s i , Here we imagine the not tbe prticle, the mulls can be sirnplifid, entire space-time hktory laid out, and that we just

the a ~ p r o ~ h a t i o n of chsicai relativistic theory become aware of incrembg portions of it succemively. the crati.an of an electron pair (eleftron A , positron B) rn =tterkg problem this over-rtll view oj the mm- saigbt be rwresentd by the start of two ~ r l d plete xattering p r w w is simikr to the $-matr* view- from the point of creation, 1- ' h e ~ ~ r ~ d grim of &@ p i a t of Heknhrg , The temporal order of events dur- p i t r a n % d l then continue until it annihilat~ aaother ing the sattering, analyEd such detail by eimtron, c, at a world p i n t 2, Betw*n the times 11 the &nziltanian difierential equation, is irrelevant. The and I2 there are then t h r e world lines, before and after rehtion of thae viewpoints will be d k a d more onfy one.. However, the war@ lines of c, B, and A fully In the intduction to the s ~ o n d paper, in which wether farm one continuous line albeit the "mitron the more camplicated interactions are analyzed. partJ' B of thL continuous line is d k e c w backwar& The deveiapment stemmtxi from the idea that in non- in t h e . Follawinp the chwge mtker than the particlm rektivistic quantum mechanics the amplitude for a comespnds to considering this wntinuous world line given procm can be mmidered as the sum of an ampii-

749

tude for each space-the path availab1e.l In view of the (where we write 1 fox xi, tl and 2 for X*, la) in t h i case fact that in cltassical physics positrons could be viewed as electrons procding along world lines toward the 1)=x (bm(xa>@$(xz) exp(- iE,(ba-ti)), (3)

L

pSt (reference 7, the was lnade removes in for le>lI. TiVe shall find it convenient for Is<&r to define the relativistic case, tbe restriction that the paths must = Q (2) ia not valid tr). proceed always in one direction in time. I t was dk- then readily shown that in general K can be defined by c o v e d that the results could be even more easily tht solution of understoob frorn a more familiar physical viewpoint, that of sfattered waves, This viewpoint is the one used <i3/3l$-R~)K(2~ l) = G(2, (4) in this paper. Aflrtr the equations were worked out physially the prwf of the equivalence to ttie sr?cond quantization theory was founds2

First we diwuss the relation of the Hamiltonian digerenth1 equation to its solution, using for an example the Schrifidinger quation, Next we deal in an analogous way with the Dirac equation and show how the solutions m y be interpreted to apply to positrons. The intepretatim w e ~ s not to be conistent unless the electrons obey the exclusion principle. (Charge obying the Klein-Grdon equations can be described in an analogous manner, but here consistency apparently requires Bose statistics.)$ A represenhtion in momenturn and energy varhbles which is uselul for the caiculation of matrix elements is dexribd. A proof of the equivafence of the methd to the theory of holes in second quantization is given in the Appendix.

2, GmBX'S WNCTION T-ATMEET OF SCHRI)DTNGBR% EQUATEOR

We begin by a brief discussian of the rehtion of the non-rektivistic wave eqmtion to i ts solution. The ideas will then be extended to rehtivistic particles, satisfying Dirac's quation, and Plslaitfy in the succading paper to hteracthg rehtivistic particles, that is, quantum ekctrodynamics.

The Schriidinger equation

dwribes the change in the wave function J, in an infinitminzaf t h e At as due to tbe aperation of an oprator exp(-iHdl). One can ask also, if $(xi, 11) is tlze wave function at x, at time t l , what is the wave function a t t h e tz>&7 I t can always be written as

where K is a Green's function for the linear Eq, (l), (We have limited ourselva to a single prticle of cc+ ofd;llEtte X, but the eqmtions are obviously of greater ~ e r a E t y . 1 If II is a cnnst;mt operator having eigenvalues B,,, eigedunctions 4, M, that $(X, Ir) can be expand& C, Cn+n(~) , then $(X, 2%) = ern(-.iE,(tz- h)) XC,rb,(x). Since C,= J@,*(xjarl)Jr(xl, E1)C%72gP one fiads

8 R P feynmn Rev. Mod. Pbys. 2@ 367 (tP@), be 'wuvalen& of the entire ,prmGdure (including photan

inkmtianri) 6th the wmk d Eiebmngcr and Tamaaaga has k m demstrsted by F. 5. Rymn, Phys. Rev. 73, MJIM9). * Thm m W a l a-rampies of the general relabon af $pipin and stortislia d d a d by W, h&, Phys. Rev. 58, ?E6 (1Mj.

which is zero for l%<% where 6(2,1)== b ( 1 2 - 1 ~ ) 8 ( ~ 2 - z t ) X ~ @ S - Y Z ) & ( I Z ~ - ~ ~ ~ and the subserigt 2 on Lfz means that the oprator acts on the vaxiables of 2 of K(2, 1). When E is not constant, (2) and (4) are wIid but K is Iess easy to evaluate than (3),$ Ur, can call 8(2,1) the total amplitude for arrival

at X*, 12 starting frorn XL, I.E. (It results from adding an arnplitude,expiS,for each, space time path loetween these pints, where S is the action along the path," The transition amplitude for fxnding a particlie in state ~ ( x s , t2) at time 12, if at 11 it was in +<X,, t d , is

A quantum mechanical system is dezribert equally well by specifying the function K, or by specifying the &mi;ltonian E from which it results. For some purposes the specificatbn in terms of K is easier to use and visualize. We desire eventurtlly to d k u s quantum electrodpamics from this point of view.

To gain a greater familiarity with the K function and the point of view it suggests, we consider a simple perturbation problem. Imagine we have a partick in a weak potential U(x, 11, a function of psition and time. We wish to cafcufate K@, 1) if U ditfers from zero only for 1 between 9 and to , We shall expand X in i n c ~ e s h g powers of U :

X(2, 1 ) s Ro(2, l)+K(lf (2, 1)+K(2)f2, l)+ * - . . (6) To zero order in U, IC is that for a, free prtkfe, Ko(2, l),' To study the first order correction K""f(Z, I), first consider the case that U digers from zero only for the infinitesimal time interval between s m e t b e ta and la;+dta(lg<k<Is). Then il $(l) is the wave function at xl, it, the wave function at XQ, fa is

since from i8 to t l the particle is free. For the short intemaf &a we solve (I) as

$(ss fa+ bla) = exp(- iEhla)J"(~, fa) = (1 -iH,&a- iUd13rt(x, IS),

4 For 8, non-relativistle free particle, where S$, = enp( ip .x f , E, .I pP/2nr, (31 &v@, as is weiI b w n

Ke(z, 1) -1 e x p l - ( i p - x ~ - i p ~ x ~ ) - i # ( 1 ~ - t t ) r " Z n ~ p ( 2 r ~ - ~ (z&m-g@p-rr))-t exp(#dn(xl- X I ) * ( ~ ~ - ~ I ) - ~ )

for #*>t~, and KB=O fw k <k.

where we put 8-;Be+U, No being the Hamiltonian of a free prdcle, Thus $(X, l i~ibt,) digers from T

what it would be if the potential were zero (namety 1: (1 - iHobta)$(x, l a ) ) by the extra piece

A$= - z'U(xa, l$) #(xa, ta)hta, (8)

which we shag cali the amplitude scattered by the ptentiaf. The wave function at 2 is given by s ~ h a

(01 OR=,@ 19) fbi CI(1DER. E@(@#

$(X., t.)= S K . ( X ~ . 1%; X,, laCbl.)$&~, t.+d.ldh%, Fm. L . The SEhrodiner (md Dim) njuatian a n be visualid W describing the fact that pfane waves are mttered srrawvely bp a potential, Figtlre t (a) illustrsrtm the situation in 5rst &er.

since after It+dla the partick is again free, Therefore Rd.& 3) is rfie am~fitude far a free prtide rarting at point 3 the change in the wave function at 2 brought about by 2 $g:r:p; ~ & a ~ ~ ~ & ~ ~ " " , ~ ~ f the potenthi is (suhtitutc (7) into (8) and f8) into cm*, (Eq , (9)). In (b) is ilfustratd the secmd order rwas the equation for $(X*, In)): (Eq !10)),, the waves mitered I 3 are %atfeted a a i n at P H ~ W -

ever In Dirae one-electron theory K44 3) wenld rqrmat elcc- troni both of poGiive and of nmlive inergkm proceding from

~ $ ( 2 ) iSK0(2, 3) (1(3)Ko(3, 2) ~ ( l ) @ ~ , @ ~ & ~ , 3 to 4 %is is mrdied by chmdng 8 different ~attering kernd K,(4,3), E%. 2,

In the case that the potential exists for an extended time, it may be booked upon tls a sum of effects from each interval &!a so that the total eEfect is obtain& by integrating over 13 ccs we11 as Xa. From the definition (2) of K then, we find

where the integral can now be extend& over all space and time, d7a-d8r&a. Automatically them wilt be no contribution if !a is outside the mnge 18 to ka b a u s e of our definition, Ko(2, 1)=0 for t zS11.

We can undersund the result (B), (9) this way. We can imitgine that a prticle traveb as a free particle fram point to point, but is scattered by the patential U. Thus the total ampiitude for a r ~ v a l a t 2 from l can be considered as the sum of the amplitudes for variaus alternative routes. I t may go directly from I to 2 (amplitude K&?, I), giving the zero order term in (6)). Or (see Fig. I(&)) it may go from I to 3 (amplitude &(St l)), get scattered there by the potential (scattering amplitude -z'U(3) per unit volume and time) and then go from 3 to 2 [amplitude K&!, 3)). This ntay occur for any point 3 so that summing over these alternatives gives (9).

&&in, it may be scattered twice by the potential (Fig. l(b}). I t goes f r ~ m 1 to 5 (K@@, I)), gets sca t t ed there ( - iU(3)) then prmeeds ta =me other pint , 4, in space time (amplitude Re(& 3)) is scattered w i n (--ilZ(4)) and then proceds to 2 (&(;l, 4)). Summing over all pssible places and times for 3 , 4 fuld that the scond order contribution to the total aqlitltde K""(Z, 1) is

was, Qne can in this way obvbusly write down any of the terms of the expansion ((i),6

3. TmflNIBET OR THE D m C EQVATIOPI

We shall now extend the methcrd of the last section to apply to the D h c equation. AII that would =em to be necessry in the previous equations is to consider EZ aa the Birac Hamiltonian, J, as a symbol with four indica (fer each prarlic;te). Then Ko can still be de&n& by (3) or (4) and is mow a 4 -4 matrix which operating on the initial wave function, gives the firm1 wave function. In (10), U(3) can be generafizd to A 43) - aaA(3) where A 6, A. are the scalar and vector potentkg (timm e, the elmtron charge) and a are D k c matrices.

To discuss this we s b l l define a convenient r e C tivistic notation. We represent four-vwtom like x, 1 by a symbol S,, where p= l, 2,3,4 and 2qJra l h real. Thus the vector and scalar potential ( thes e} A, A , isi A,. The four tnat~ces @a, B tan be comidered m trmform- ing: as a four vcrctor r, (our y, digem from Paufi's by a factor i for P== l , 2,3). We use the summation convention &,h,= a&&- atb~-ds-ada= a.6. In pzlrticukr if a, is any four vector (but not a matrk) we write a=afiy, so that a is a matrk with a vector (a wit1 often be used in place ymbol for the vector). The satisfy r p 7 * 3 - . ~ p ~ f i = 226,, where &M= -f- l,

and S,,= 4, Note that ab+ba= 2ct- b and that &=G@@,

[email protected] is a pure number. The sym'boI a/8z, will m a n a/a6 for p ~ 1 . 4 ~ and - a/az, - a/ay, - a/& for a = l, 2,3. Gall V= y,a/d+= Ba/at-t-isa" V. We shall im@ne

We are rintpky wlving by suamive apprdmaticns an in-1 quation (deducible dimctly from (1) WE& l j l - H ~ + U and (8 with R=&@>,

This urn be radily verifid directfy fram ( I f just as (9) whae the first inte@ exleads aver ciU s w e and all times peater than the 6, alpwring h the m n d term, and k>rb

k e t a WDWP OIIMB, LQ, (84)

F%@ 2. The Dim equsrtien wrmits anaiher mlutien K+(2 1) if oat k n ~ d e r s tbat wava ~~ibttemd by the ptential can M w m d a in drne as ia Fig. 2 (a). This is intcpreted in the wand orda rw- (b) (c) by nortiag that there is new the pad- Mty Pcf of virtuadlpai: Probuction at 4 Ihe positron goin to 3 to b. aoi&tied. ?his o n be pirrvd as dmilar to o r $ l i ~ ~ y w t t e fb) a m t that the ektron ts mttclled bwkwwds in time horn J to 4 TL wavm sfatkred mm 3 to in (4 r rmat tht mibiitit 2 a arrivfng at 3 fram 2' and a n z k a n g the dmtron &m X. %%is view is proved equlvdent to hk h t y : electram tmveLing bekwards in time are rmmizad m etrons.

herdter, purely for relativistic convenience, that +,* in (3) is rephced by its d j o k t &,= cbn*fl.

Thus the I)& equation for a partick, mms m, in sn exteml field A = A ,y, is

m d m, (G determining the propgation of a free prticle bwoma

(iVa-m>K+(2, 1) = i6(2, l), f 12) the hdex 2 on 17% indiating dgerentiation with rmpct to the coordhtes sss which are reprewated as 2 in IE+(2,1) and W , 1).

The function 11=,(2,1) is d&ed in the a b n m of a field. If a ptentiilf A is acting a s h i k r function, ay K+(h",(2,1) m be d e h d , It differs from K+(2,1) by a %mt order comeetion given by the analoee of (9) namely

expmion of the intern1 equation

K+cA'(Z, 1) l])

wKch it a h satbges. We would now expet to choo*, for the spttciaf solu-

tion of (12), II+mXe whem Ke(2,I) w n a w for 12<it

and for t3>& is given by (3) where 4, and E, are the eigenfwctians and energy valum of a parGcle satis- f h BkacJs equation, and @,* is by $n.

The formuk arising from this choice, however, s&er from the drawback that they apply to the one electron theory of Dirac rather than to the hob theory of the witran. For example, consider its in Fig. 1(a) an electron after kkg za t te rd by a potential io a small region 3 of space t h e . The one elecbon ays (as does (3) with K+= KO) that the m t b r d ampEtude at another poitlt 2 will p r o c d toward mitive t h e s with both positive and nqatlve energh, that is with kth p i t i v e and negative rabs of c k w e of phase, PITo wave 19 mtLerd to times previous. to the time of mtkring. nrn are just the prowrtim of R0(2,3).

OR the other haad, according to the p i t r a n neptive e n e w stittes are not availabk to the electron after the mtteriw, Therefore the choice K+=Ko is unsa&facby. But there are other mlutions of (12). We shall choose the solutrion d e h h g Kc(2, 1) so that K+(Z, 1) for tt> 1% is 1h I(!%= u j (3) m podioe w g y stales o ~ l y . Now this new solu~on mmt =My (12) for ilU tima in orda that the rwrewnk&a be cornpieb. I t must *erefore dBer from the old wlution K@ by a mlution of the homogenaus Dirsc eqwGon. It is c k from &e deftnition t h t the difference K@-K+ is the sum of (3) over all neetive enerp;y statm, ars long as 12>11. But this gaerence m a t be a wiuGoa of the homogenmus Dhae equa~on for all t ims and m a t &erefore h rreprwnteci by the =me sum over nega~ve e n e w s k t a aka for Ca<ll. Since K B ~ O in Skt cam, it fobws that our new kernel, I), for In<lx i s the mrg&iw Ql the sunr (3) ouer wgolifx? m g y states. That is,

K+")(Z, 1) = - ~SK+(Z, 3)d(3)Kt(3, i)drr, (13) R+(2$ l ) = x p o ~ a. 4n(2)6n(1) Xexp(-iB,(tg-lll) for t2> 18

repreen~ng the amplitude to go from 1 to 3 as a free -Crvso 4i)nf2i(i;n(l) lf 7)

particle, get mttered there by the p & n t M (now the X~p(-iE~@a-h)) for It<lt. matrix A(3) ins-4 of U(3)) and coatlnue to 2 as free.

order mmction, italogom to is Ifih koim of K+ ~ a t i o m (13) and (14) wiif now give results quivdent to tfiose of the

X+m(Z, 1) - - s s K I ( 2 , 4 I A ( 4 ) 6 t m n halie a w ~ ,

That (141, for a m p l e , is the mrrHt mond order

M+(4,3)A(3)15+(3, l)dzdrs, (14) e ~ r e & o n for h&g at 2 an or&&& a t 1 according to the msitron thmq m y be wen ar, follows

and so on. In generat K+") ~ati$fies (Fig. 2). &sme as a speciat w m p b h t tz3El and ( l ) , ('5) that potential vani*la a rep t in ktcrml f z - h

&at E4 and tr both lie betwan tt and k and the succmive t e r n (131, (14) are the power =rim First s u p p 4> t s (F&. 2(b)). Then (since is> t l )

" T H E O R Y O F

the electron assumed originauy in a positive enerw state propagates in that state (by f(+(3, X)) to posihn 3 where it gets scatter& (A(3)). f t then prmeds to 4, which it must do as a positive energy ekdron. Thh is correctly descGbed by (l+ for fC+(4, 3) contains only positive energy compnents in its expansion, as tr>t,, After being scattered at 4 It then p r w d s on to 2, a e i n n e c m s ~ l y in a positive energy sCate, as It> tr.

In positron theory there is an additional contribution due to the possibiiity d virtual pair prduction (Fig. 2fc)). A pair could be created by the potential A(4) at 4, ihe electron of which is that fwnd later at 2. The positron (or rather, the hob) PEW& to 3 where it annihilates the electron which has arfived there from l.

This alternative is alrmdy included in (14) as; contributions far which l,< La, and its study will lead us to an inltrvretation of K+(4,3) for Ir<la. The fmtor K+(2, 4) de~ribes the electron (after the pair prduction at 4) proceeding from 4 to 2. Ckewiss: K+@, It) repmnts the electron prweding from 1 to 3. K+(4,3) must therefore represent the propagation of the positron or hole from 4 to 3. That it d m SO is clear. The fact that in hole theay the hok prweeds in the mnner CA and electron of negative energy is r e k t e d in the fact that K+(4,3) far ( P < ~ J is fmhus) the sum of only negative energy componenb. In hale theaq the real enera of these intermdiate s b m is, of coum, positive. Thb is true here too, since in the phasa ap(-iEn(lr-ls)) desmg K+(& 3) in (17), ER is negative but so is I r - l g . That is, the contributions vary with 11 as e:p(-i\E,/ (to-14)) as they would if the enera of the mtermediate state were t E,/. The fact that the entire sum is taken as negrttive in computing IC+(4,3) is reflected in the fact that in hale theary the amplitude has its sign reversed in accordance with the Pauli principle and the fact that the electron arriving at 2 has been exchanged with one in the %a.@ To this, and to higher orders, all procesw involving virtual p i r s are correctly dwritrted in this way.

The eqressions such as (14) can still be d e w h W as a pawge of the ekctron from 1 to 3 (K+(3, I)), scatkring at 3 by A@), proceeding to 4 (K+(4,3)), scatrering agin, A(4), arriving finally at 2. The scatterin@ may, bowwer, be toward both future and past times* an electron prapagakimg backwards in time king recag- &ed ats a positron.

This tkerefore sul~gests that negative energy corn- ponents created by scattering in a p ten tk l be consider& as waves propgating from the scattering mint toward the p t , and that such waves r q m n t the propgation of a positron annihilating the eltt~tron in tfie p ~ t e n t h i . ~

"t has often boen noted &at the one-electron theory s patently gives the m e matrix e h e n t s for lhi pr- as dam hofe theory. The pr&lern is one of iniepretadon a p d I y in a way that will alsa 've urrreet radts for other r&-, e.g., self-enmgy.

7 &e idea that p i t m m a n & repiewnted as elrrvoos with proper time r e v e d retative to tme em &S been k w d b the author and othms, particuisrly by StBckeikr~ E. c, C!

With ehi inteqretation real pair grduction is aho dmribed carretly (W Fig. 3). For example in (13) if t l< la<ta the equatbn gives the amplitude that if a t time $1 one electron is present at 1, then at time just one ctl~tron wilt be present (having been scattered a t 3) and it will be a t 2. On the other hand if IS is less than h, for examph, if t2= tI < t g , the same expression gives the amplitude that a pair, electron at 1, gositran a t 2 will annihihte at 3, and subvequently no particles wijl he prewnt. Likewis if 6% and I 1 exceed t a we have (minus) the amplitude for Snding a single pair, electron at. 2, psitxon at 1 crezrtad by A(3) from a vacuum, If ta>ta>ls, ($3) dacribes the wattering of a psitron, A'fl these amphttldes are rehtive to the ampiitude that a vwuum will remain a vacuum, which is taken as unity. (This will be discussed more fully later.)

The analsue of (2) can be easily worked our.@ Xt is,

where d8iVr is the volunte element of the c l a d 3- dirmensioml surf;tcse of a re&on of space time contaking

Fto. 3. Several dilferent prw- can be demibed by the =me form& de ndiag on the time relations of the variables 11, It, Thus P,]GA~(~, I)/* is the probattUity that: (a) An electron at 1 will be ~cattered at 2 (and 00 other psirs form in vacuum). (b Electron at 1 and pasitmu at 2 annihilate leaving noFmg. (3 A &n& pair at I md 1 is rreaM frrm rmm. (d) A at 2 io metered ta 1. (~+(41(2,1) is tbe sum of the e e k 2 sflttteh in the potential to all orders. P, is a normalizing mtanttt)

paint 2, and N(1) h N,(l)r, where N,(I) is the i ~ w e ~ d drawn unit nomaI to the s d a c e a t the paint X. That is, the wave function #(2) (m this case for a free particle) is determind at any point inside a four-dimen- sbnal region if its values on the surface of that re@on are sp&ed,

To intewret this, cornider the case that the 3-surfam comkts m n t k l l y of all space at =me time say t = O previous to h, and of aU spce a t the time T> t g . The q lhder connecting these to wmpIete the chsure of the s u d a e may be very distant from X* so that it gives no appreciable contribution (as R+(2, l ) deamws exponentially in spce-&L ddir~tbns), Hence, if yr= @, since the inward &awn norm& N will be B and -8,

tivistic calculations, om be removed irs follows. Instead of dehing a sk te by the wave function f ( x ) , which it h a at a given time tt=O, we define the state by the function $(l) of four varirrblm XI, t.1 Ilvhi& is a satution of the free particle e q u a t h for all 4 and is f(xr) for k=Q. The h f state is m e w i ~ defined by a functian g(2) over-all s p ~ e - t h e . Then aur sudace integrah can be p r fo rmd since JK+(3, l)@j[xl)@x~= f(3) and JQ(s;~)B@x&+(2,3) = #(3), There results

the i n t e p l now biryq aver-aU swe-time, The transition amplitude to wwnd order (from (14)) is

for the mticXe a r ~ * ~ a t 1 with am~litude {(l) is scatter4 (A(z)), prsg&m to 2, [~+\2 , I)), ihd is

where fr...@, 2'. positive energy (elmtron) scattered again (A(2)), and we then ask for the ampli- comwnenls in $(l) contribute to the first integral and tude that it in slate g(2), E g(2) is a negative energy only nwtive energy (wsitronf com~nen:ntf of @(X? state we are sol+@ a problem of annihilation of elec- the =con&, That is, the amplitude for finding a charge tron in 1(1), positron in g ( ~ ) , ek, a t 2 is deter&& both by the amplitude for Gnfmdinh: f;Ve have h e n emphasizing scattering problem, but an electron preriotls to the measurement and by the be motion in a fix& V; say in a amplitude for h d h g a positron after the measurement. hydrogen atom, can dealt ~f it is first

migbt be inteqreted as mmning that even in a view& a %atrering prablem can ask for the problem involving but one char@ the amplitude for amplitude, dra(X), that an efwtran with original free findkg the charge a t 2 is not determined when the only wave function was scattered K times in the potential. V thing known in the amplitude far finding an electron either famard or backward in time to arrive a~ t 1. Then (or a positron) at an earlier time. There may have been the gfter one more scattering no electron pment initially but a pair was created in the masurement (or atso by otber external &eieXds). The ampfitude for this contingency is spified by the amplitude for fxnding a po~itron in the future.

We clan a h obhin exprwions for transirion ampli- An equation for the teal am~htude tudes, like (S). For emmpk if at 1=0 we have an electron prmnt in a state with (positive enerw] wave fmction f(x), what is the amplitude for finding it at

T w i a ae (pgi~ve enew) wave function for a r ~ G n g at 1 either dirwtly or after any numhr of The amplitude for h&mg &e electron anywhere after m t t h n e is ob~sined by summing (24) over all K from l= 0 it; given by (19) with $(l) r q h d by f ( x ) , the 0 to CO ; second in&@ait vanwing. Rence, the transition element to h d it in slate g($ is, in anatqy to (S), just $(2)= 4 . ( 2 l - i b ( z , I)V(l)+(l)drx. (25)

s ine g* = #B. If a potenthi acts sommhere in the internal betwan

O and T; K+ i6 repked by K+cA). Thus the first order e g s t on tbe transition ampEtude is, from (131,

- iS@(xi)~x+(z. 314(3)K+(3* I)@j(xr)hlhr . (21)

Eqrwions such as this ean be simplifid and the ;dsurfoce htwbls, which are inconvenient for rela-

Viewed as a s M y state problem we w y wish, for emmpie, to find &at initial condition 4s (or better just the $1 which l& to a p ~ d i c motion of +. This is mast p m c t i d y done, of coumt by mlving the Dirac equation8

(iV-m>+(1)= v(l)#(l), (26)

deducd from (25) by bpmting on both sides by iVa- m, athg the h, and using (12). This illus-

trates fhe rehtion be twm tht? p o h b of view. For m n y problem the total potenhl A+ V may be

split conveniently into a h& one, V, and another, A, comiderd as a perturktioa, If K+") is de&& as in

(t6) with V for A, eqressions such as (23) are valid and u%ful with K+ replaced by and the functions f(t), g(2) reQl"ced by solutions for all space and time of the Dlrac Eq. (26) in the potential V (rather than free particle wave functions)

We wish next to consider the case that there are two (or more) distinct charges (in addition to pairs they may prduce III virtual states), In a succeding paper w e d k w s the interaction between such charges. Here we amurnr: that they do not interact. In this case each particle behrlves independendy of the other. lnJe can expect &at if we have tvvo particles a and b, the ampfitude that particle a goes from x l at Is, to xa at while b gaes from X$ at t* to xp at tq is the product

The s p b o l s a, 6 sinpliy indiate that the matrices apgearing in the K, apply to the Dirac four component spinors corraponding to particle a or b respectiveiy [the wave function now having 16 indices). In a. ptential X+, and K+b &come K+,tA) and K.+I . (~ ) where is defind a d cjzlculated as for a single particle. They commute?. Rereafter the a, b can be omittd; the space time varhble appearing in the kernels suffice to d e h e on what they operate.

The par~cles are idential however and satisfy the exclusion principle. The principle requires only that one calculate K(3,4; 1,2)- R(4,3; 1,2) to get the net ampIitude for arrival of charges a t 3,4. (It is normlisd wuming that when an intqral is yx?rform& over mints 3 and 4, for example, since the electrons represented are identical, one divida by 2.1 This expmsion is correct for p i t r o n s also (Fig. 4). For emmple the amplitude thi~t, an efectron and a psitron found Initially at x2 and X, (say I t s i r ) are later found at xa and (with t 2 = ig> 13) is given by the =me expression

The h t term repreen& the amplitude that the electron prwee& from 1 to 3 and the psitron from 4 to 2 [Fig, 4(c)), while the mcond term reprwnts the Intedering amplitude that the pair a t I, 4 annihilate and what is found a t 3, 2 is a pair newly creaee$ in the potential. The generaEzation to ~ v e r a l particles is clmr. There k an additional factor K+(A' for each particle, and antisymmetric combinatims are always taken,

No account need be taken of the excitusion principte in Lntemdiate states. As an example consider wain rqression (14) for I r > l ~ and suppw t4<ta SO that the situation reprant& (Fig. 2(c)) is that a pair is made a t 4 with the e i ~ t r o n prweeding to 2, and the psitrsn to 3 where it iuxnikiktes the electron arriving from 1. It may tT& obje-cted that if it happns that the eiwtron cmted a t 4 is in the sam state as the one coming from 1, then the process annot occur bemuse of the exclusion principle and we should not have includd it in our

P O S I T R O N S 755

FE. Q. Some problems involving two distinct charges (in ail&- tion ta virtual airs they may prduce): c,JK+(AA"(3. I)k;fAjf4, 2) --K,1Ajj4, 1 ) & f ~ l @ , is the prabab~iity ,that: (a) Etmtrans at I and 2 are -9attered to 3,4 (and no paws are farmed). (W Starting with an electran at 1 a single pair i eleetrans at 3 4 (c) A pair a t 1, 4 is found sian princiF1ef r&uirw that the amplitudes exchsnge of two eiclctrons be subtracted.

term (14). We shall see, hovvever, that considering the exclutusion principle also requires another change which reinstate the quantity.

For we are computing amplitudes relative to the amplitude that a vauurn at tl will still be a vacuum at E*. We are interested in the alkration in this amplitude due to the preEnce of an electron at 1. Now one process that can be visualized rls occurring in the vacuum is the creation of a p i r a t 4 follow& by a re-annihilation of the same pair at 3 (a proces which we shall call a closed loop path). But if a real efectron is present in a cerbin state 1, those pairs for which the electron was created in state 1 in the vacuum must now be excludd, W must therefore subtrset f r m our rejative amplitude the term correspndiag to this prxess, But this just reis- states the quantity which it was argued shouM not have been included in (14), the necessary minus sign coming autamaticaIly from the definition of K+. It is obviously simpler to disregard the exclusion principle completeb in the intermediate states.

AII the amplitudes are relative and their quares give the rehtive probabilities of the various phenomena, Abwlute prohbiiities result if: one nnultipiim each of the prolsabilities by P*, the true probability that if one has no partictes prewnt inithay there will be none finally. This quantity P, can be crzkutatd by normalizing the relative probabilities such that the sum of the prohbilities of a11 mutuaUy exclwive dtemaGvm is unity, (For example if one starts with a vxuum one can calculate the rektive probability that there rewins a

756 R . P . F E V N M A N

vacuum (unity), or one pair is created, or two p k s , etc. The sum is P,-l.) Put in this form the theory is cam- plete and there are no divergence problems. Reai proc- emes are completely indepndent of what gaes on in the vacuum.

When we cam, in the succeeding paper, to deal with interactions between charges, however, the situation is not so simph, There is the possibility that virtunl electrons in the vacum may interact efectrompetially with the real electrons. For that reaon procesw %cur- ing in the vacuum are analyzeil in the next section, in which an independent method of abtsining P, is discussed,

An alternative way of obtaining absolute amplitudes is to multiply aU amplituctes by C,, the vacuum to vacuum ampLitude, that. is, the absolute amplitude that there be no particles both initially and finally, We can wsume C,= l if no potential is present during the internal, and otherwise we compute it as follows. It dsem from unity because, for example, a pair could be created which eventually annihilates itself q a h . Such a path would appear as a c l o d loop on s space-time d i q a m , The sum of the amplitudes resulting from all such dngle clo*d loops we tall, L, To a first approximation L is

For a pair could be created sily a t 1, the electson and p i b o a could both go an to 2 and there annihilate. Tbe spur, Sp, is taken since one has to s m over aU paible spins for the pair, The factor 8 arises from the fact that the =me loop could be comidered as s h d n g at either potential, and the minus sign rml t s since the interactors are each - iA. The next order term would beB

etc. The sum of aH such terms givm &.la

in- is taken aver all ri 7 3 and r~ this hss no effect and we are left with (-- lIr from cbanhngf the sign of A. Thus the e+~ur wmh its negative h m with &R odd number ~f otentisl intemtors give sera ~by&catly ihis is &mu% faf each Lop the eloctmn a n go arcrund one m y or in the op&te direction and we must sdd these ampEtub. But reversing the motion of an electron makes it kkve lib a posiitive chr& thus chmging the 54- of erreh potential intembon, sa that the sum is m a if the number of m-tions is odd, This aeorm is due to W, H. Furry, ??hp. Rev. SX X25 (1937)-

A 3 4 r e h n for k in terms of K+.") is hard to obbin because nf 1he78etor (l/$ in the nth term, However, thc per- twhtian in L, dL due to a. $mail chwe in potential AA, is m y to express, Tfrc? (I/#) L caneded by the fact that AA can a p p r

h ddiition to these si_ngle loops wa have the posrsi- bilily that two independent p i r s may be cratted and each pair m y annihilate itself again. That is, there m y be famed in the vacuum two c l o d h p s , and the conbibution in amplitude from this alternative is just the prduct of the contribution f r m each of the laops considered singly. The total contribution from all such pairs of loops (it is still consistent to dkregard the exclusion principle for these virtwl stzLtm] is L2/2 for in Ls we count every pair of bp twice, The total vacuum-vaeuum amplitude is then

C,= 1 -L+L2/2-LB/6+ . - exp(-L), (30)

the succemive terms representing the amplitucle from zero, one, two, etc., hops. The fact that the contribution to c, of single imps h -L is a coweqnence of the Pauli principle. For exawler consider a. situation in which two pairs of prticfm are creatd. Then these wirs later destroy themselves m that we have two h o p , The elwtrons ~ u M , at a given t h e , be inter- ckanged fanning a kind of figure eight which is a single Emp. The fact that the interchange must change the sign of the contribution requires that the terms in C, apwar with alternate signs, (The exclusion principle k also rapnsibfe in in similar way for the fact that the amplitude for a pair creation is --K+ rather than +K+.) Syxnmebical statistks would l e d to

The quantity L has an S n i t e imaginav part (from L;"" ',higher orders are finite). We will &&cm this in connection with vacuum p h r h ~ o n in the succw&g paper. This h a no e@et on the noorrnalizstion c o ~ a t for the probability that a vacuum remin vacuum is given by

P,= IG,j2=eq(-Z*reat part of L),

from (30). This value agrea with the one calculatrxf directly by ~normaiizing probbitities, The real part of t appears to be p l t i v e as a conEquence of the Dirac equation and prowrtim of K+ m that P, is lem &an one. Bose sbtistics gives Ce=exp(+l;) and cons- quentIy a value of P, greakr than unity which & p p = meaninglm il the quantities are interpreted as we lzave done hem. Our choice of K+ apitpparently requires the exclusion principIe,

Charges o k y b g the Klein-hrdon equation a n be qually well tmted by the mthctds which arc? dk- c u s d here for tbe l3irac eftf~trons. How this is done is d k u d in more detail in the succding p p r . The real part of L corn= out nqative for this equation so that in this case Bose shtktics a p w r to be requird for consistency.' in any of the n poten(ials. 3% result after m m i n g over n by (131, (14) and using (15) is

The term K&, I) 8etuay inkmte*j to are.

The practicrrt evahatian of the hetrix ektemenes in wme problems is often simplged by working with mornentm and enerm variabim rather than s p c e and time. This is k c a w the functiion &(2, l ) is fairfy complicated but we shall find that its F o u ~ e r transform is very simple, nameIy (ifi*(P-m)-Z that

= B ' ~ ~ - P ~ x ~ = P P ~ * S - P ? " ~ ~ C , f i f l " P l r ~ C ,

i?i]*dp~dppdlbd@4, the urtegraf over aU P. true can be seen imme&iately from (121,

for the represenktion of the o p r a b r iV-nt in energ (p,) and momentum (pLx B) spzlce L@-m and the t m e f o m of 6(2, 1) is a constant, The reciprod matrix &-.1)-%n be interpreted as &+nt)w--&)-l for p-m*= (p-m)CpT"E) is a pure n m h r not involving 7 ntstrices. Eence d one wishes one can write

is not a mac oprator but a functbn giitisfyiw

where --a+= (a/axg,)(a/dx~~). The irmtqrals (31;) and (32) are not yet completely

d e h d for there acre poles in the in&@&& when p--ta2==0, w e a n define how these poles are to h evalated by the rule that m s's tm&ed lo hawe an z ' ~ j s i & s i d n-egrstipte imgirsa*y p&. That is m, is re- pbmd by nt-i6 and the Emit taken a9 M from above. TKi a n be seen by i w i n i n g that we calculate K+ by iategrating on $4 first. If we call: E=+(mz$-$ae +pf+paa)c then the integals involve P4 essenthlly as J' exp(-z'~~(tg- Ir)]dpr(p*z- &?)-l which has poles at p,= +E and pc= -E. The rwhcement of nr by m-z"B mans &at E has a sml l negative imaginary part; the fust pole is blow, the sewnrf above the real axis, Mow if h-tr>O the cxrrntour can k completed around the wmicircie Lebelow the real axis thus gi.ving a residue from the $4- +E pole, or - (?E)-%q(-iE(1g-t1f), If 1%-tl<O the upper semicuck mcrst br? us&, an8 p,= - E at the pole, m that the function varies in emh mse as requird by tlhe other definition (17).

Other dutions of (12) result from other p r e ~ r i p tions, For example if 9 4 in the fachr w-nr2)-1 is considered to have a positive i m ~ n a r y part R;. becomm rephe& by KO, uhe Birac one-electfon kernel, zero for fa< k. Explicitly the fmction isl"(x, t =zzt>

I+(r, t ) = - (4a)-V(sP)+ (rnI8~s)Rt@~(ms), (34) where a=+(@-X*)# for P>xa and S= -i(#-P)b for

"Iq(x I) is (2iLi)-t(Da(~ 8)-iD(x i)) wllrrrs 01 and D me thn: functtad dL.firrrxt by W. $at&, R@"[ Mod. Php. 13,203 (1441).

P O S I T R O N S 757

P<#, Bl@f is the fiXa&ek function and 8(@) is tfia D i m &tta function of sZ. I t khaves asymptotiaHy as =p(-im), decaying exponentially in spce-like directiansmES

By means of such transforms the matrix elements Iike (221, (23) are easily worked, out, A free particle wave function for an eelectron of montenrum $8 is esl eq(-ipl.x) where ut is a constant spinor wtisffiq the Dirac eqation pl@l =msl so that Pb=m2. The m t r k eiement (22) far going from a state #l, to a strtte of momentum #a, spinor zc~, is -4e(M(q)?~) where we have i m g i n d A expand& m a Founer i n t e ~ a l

and we select the component of momentum q = f t ~ - P ~ . The second order term (23) is the matrix element

between and un of

since the efectron of momentum pi may pick up q from the wtentil, a(q), propgate with momeatm jh+q (factor (ar+g-m)-" until it is m t & r d again by the potential, ~ @ s - P r - a ) , picbg up the remking momentum, #*-Pt-q, to bring the to?d to p%. Since all values of q are possible, one integrates over q.

The% =me matrices apply directly to positron prob lems, for If the time component of, say, is negadve the state represents a prsitron of four-momentum -#z, and we are desribing pair prcxluction if is an electron, i.e., has p i t i v e time component, etc.

T'he p b ~ b i l i t y of an event w h m mtrix element is (z-i@lst) is proprtionaf to the absolute square. This m y aIso be writ&n (@E@u~) (~%MuI) , where .@ is M with the operators written in op ite order a d exglicir appearance of i changed to - i(E B times the complex conjugate transpose of @M). For m a y problem we are not wnceraed about the spin of the hnal sate. Then we mn slum the probabifity over the two 4 . l ~ corresponding to the two spin directions, This i s not a complete set be- a u s e $2 has another eigenvalue, -m. To p r ~ t sum- m i q over J 1 sb tm we can i m r t the projection operator (2m)-a@1+ m) and so obtain @?n)-Y&a(p~+ nt)iwul) far the probabitity of transition from Pi, m, to PI with ~ b i t m r y spin, If the incident state is u n p l a r i d we can sum on its spim too, and obtain

for (.twice) the probbility that an electron of arbimt.y spin with momentum pi will make transition t o h . The mpresious are all valid for positrons when $'S with

a U the --ib is kept with m bere tao the furzc~en I+ a p p r d e s rare for inhi& p l i v e and nepdve timm. W etay be wfuI in gmeral aaaiyses in avaidiw wmpLica&ns fmm inffniteiy remate fiudaces.

negative energim are inserted, and the situatian interpreted in accordance with the timing rehtions d i m m d abve. (We have u d functions nar&lized ta (8%) .- l instad of the conventional f'LZ@u) = (%*U-) = l. On our scale fzZB%)=energy/m so the probgbilities must be corrected by the 8ppropriate factars,)

The author has mmy pmpk to thank for fruitful convemtians abut this sugect, paxticuhrly H. A. Bethe and F'. J. Dyson,

cr. Deduction fr@m Second QwtizaG~ioa

In this section we shall show the equivalence of this t ha ry with the hole t h m v of the p&tron "aording to the theary of w a n d quantization of the electron field m a given patential," the state of this field a t any time is repreanred by a wave function X mtisfying

.iax/ar=Nx, where N-.$ef(x)fu-(-iV-A)+Ar+mB)*(x)d% and %(X) h an operator annihihtmg an electron at pmition X, while **(X) is the correwonding creation operatoor. We conkemplate a situation in wfrich at 1-0 we have present -me electrons in states represented by ordinary spinor functions fg(x), fs(x), - . mum& nrthogonal, and some pasitrons, T b e are descr iw as holes in the negative energy m, the efectrons which would nomally fill the holm having wave functions pa(xj, @*(X), . . .. We ask, at time T what is the amplitude that we find electrons in sates gr(x), g*(x), . . . and holes at gt(x), (i*(x), . - .. If the initial and 6nal state vectors representing this situation are X, and xt respectively, we wish to calculate the matrix element

We awume that the potential A differs from zero only for times lwtween O and T so that a vtzcuum can be defined at these times. IT rqresents the virruum slate (that is, all negative energy sates filled, all positive @ner@a empty), the amplitude for hadry: a vacuum at time T, if we h d one at l=O, is

writing S for exp(- iATli"dl), Our problem is to evaluate R and show that i t is a simple factor times C*, and that the factor involves the K,") functions in the way dissussd in the previous swtians.

To do this we krst express X , in terns of X* The opemtor

creates an electron with wave funcrion +(X). L i t w i ~ @i~=J@*(x) X*(x)B)x annihilates ane with wave function #(X). Zence s a t e X. is X , = & * ~ Z * . . .PIP*. - while the ha1 state is CI*G"S** . XQIQI. . where F,, C,, P,, Q* are oprators deftnd Iike 9 in (391, but withj,, g,, p,, (i, repiwing (9; for fhe initkl state would reault fmm thc vacuum if we crfrated the elmeons in j a r jzj . - . arid annibitated t h m in p ~ , pp, . -. Hence we must 6nd

R;.. (X$ . .QsfQt*. . .G&$SfiLFf * PIP^. - (40) To simpiify this we shall have to m commutation refationk be-

twwa a @* operator and S, Tn this end consider e~[-i&iEdI')cg+ Xexp(-t-i&H&') and mpand this quantity in terms of %*(X), giving f*"(x)#(x,l)&%, (which defines +(X, l)). Now muftiply this wuation by mp(+i&Wd~') . .~(-i&Bril" and hind

Xexpl-i&tHdfl. AB is well known @(X, 6) saws tbe Dirac equetion, (digermtiate * (X , 6 ) with rapcct to l and use cummuta- tion relations of N and Jlf

Gotlsequentfy 9 0 , i ) must also sstisfy,the Rirac equation (diger- entiate (41) with repect to 1, use (42) and integrate by parts).

That is, if @(X, T ) is tbat saluaion of the Dirac egustion a t time T whirh is +(X) a t 1-0, and if we d d n e @*= JQ*(x)+(r)dk and @'*-f**(x)g(x, T)d% then Qr"*=Slf"*PI, or

The principle on which the proof will be based can now be illustrstad by a Simple example. S u p m we h v e just. one electron iniriaily and tinally and & for

We mi&t try putting P thrwgh the operator S using (m, SF*mFW*S, whmef in E"*=$**(x)f(x)dfx is the wave function at T arising f r ~ m j(x) a t 0. Then

whwe the mend e x p r h n has, k e n obtirineci by use of the definition (38) of C, and the generai mmmu@&n relragon

which is rr conwuence of the groperties of efr) (Xhe others are FGm -(;F and F*@= -&W*). Now x@*P* in the East term in (45) is tine cornpia exnjugate of PXB, Thus if f antained oniy p i l i v e enerw amwnena , F"@ would wrnish and we would have vdu& r to a factor times C, But F" as work& cut hem, does enntda negative energy mmpnents creatd in the poltentht A and the method must be skghtly modified.

Beliore ptt ing P1 through the o p a t o r we shall add to it another Wrator F" arising from a function Y(r f containing DlJr negcJive energy compnents and so c h m that the resulting f has d y puziliv~ ones. That is we want

S(FW*+ Pm.,"'*) FWr*S, (44)

where the "'"p~'' and "neg" serve W remindm of the sign of the energy mmpnents fontknect in the oiperators. TKis we can now use in the form

SFpOlt~Fm~*S-SFer*l(l- (47)

In our one electron problem this substitution replxes r by two terms

r = tx$f;FV,2Sx3 - (x$GSr"~~,"'*xo). The first of these reducm to

as abve, for P , k x o is ROW bern, while the a o n d is nro since Ehe c a t i o n operator F.,,'"* givt?s zero when acting on the vacuum saw m all negative enerdes are full, This is the centrat idca of the demonstration,

The problem presc?nted by [M) is this: Givea a functien fm.(x) a t time 0, to find the amount, f..,"', of neggtive energy component w k h must be added in order that the soiution ol Ttimrs equation at time T will have only H t i v e e n s w mmponenq jIrg(II. This is a bou&ry value pmhlern for Pvhich ttie kennel K+td, is d&ped. We know tbe padtive emere commnents initially, fpoh

and the negative ones h l l y (zero). The M t i v e ones &ally are therefore (wing (19))

~ * * ( x ~ ~ f x l d . . " r m r)*(r, r)d% (412 fpoll"(xl) - J K + ~ A ) ( ~ , t)dfpae~~tld"rr, (48f

where we have defined @(X, 1) by ~ ( x , jJBf~&)~(X) where C-T, f1==0. Sidarly, the negative ones in ik ly are

for emmpte G Wentzel Ei~tjechllg is d& Quarrlm- ~ d m f & k ( ~ r a n z ~ i u t k k e , Leipzig, 19431, C h a p

f r m a k e , a d tt = 0. The fPoe(ns) is

T H E O R Y OF P O S I T R O N S 759

s u b t w t d to keep in f,,"(xg) onfy those wava which return from the potential and not t h m arriving directly at IS from the K+(2, l) p r t of k',(r'(2, I), as 6 4 . We muld aim have written

Therefare h e me-electron problem, r =Jg*(x)f,,'(afd*x.Cu, gives by (48)

as expect4 in accordance with the r m n i n g of the previous sections (i.e., (20) with replacing K,),

The prmf is rmdify atended to the m m gmeml e x p r d o n R, (#.l), which can be a m l y d by induction. First one replwa Fa* by s relation such as (41) obGning two terns

R=(xo** . -Qp*Qzf* ..G&lFgpgSr*SI;(P*. . .PIPS* . ex@)

- ( X $ - .Qs*Qtu* - -GLGISP*~,,'"*FI*. " 'PLPP' * 'WO),

In the first term the order of Pt,,"* and C;r is then inter&angd, prducing an additional term Jg~*(x)fi~~r'(xjd"r t ima an exprm- sinn with one tess eletron in iniriaX and ftnal state. Nett it is =changed with C* prociuetng an addition -Jg~*(x)ftpo7"(~)$% times a similar term, etc. WnaHy on rmching the Q* with w&ch it anticommutes it can be &mpfy mved over to juxawktion with X@* where i t gives zero. The s e n d term is imihrly b d l e d by moving Ptm,,*" thrmgh anti commuting PI", etc, until i t r w h m Pt. Then ir is exchanged with PI to produce an additional simpler term with a factor 7"Jpl*jx)Jtn,,"'(x)dt or FJPi*(xs)K+tA){2, I)J~~~(xI)$'x~QBXO from (49), with taixi.&=O (the extra fi(xg) in (49j gives a r a as it is orthogonal to pg(x1)). This dmcriba in the expected manner the annlhiiation of the pair, electron fi, W t r o n pt. The P,,,'"* is moved in this way accerr gively through the P's untif it. gives when acting on X* '&us R is rrducd, with the q e c t e d facltlrs (and with attwoating $gm

requird by the a lus iou prindpfef, to dmpler terms mnlaining two bss operatm which may in turn he furher rerluced by u ing h* in a fimitar maner, etc. After all the F* are used the @*'S can be r e d u d in a similar manner. They are movd through the S in the opposite direction in such a manner as to p r d u ~ e a purely neg;ttive enerw aperator a t time 0, using relstions amlogow to (45) to (49). After all this is done we an? left *ply with the expected factor times Cr (amming the net cha~ge is the same In initial and final state.)

Xn this way we hsve written the Ilofution to the general problem of the mntion of electrons in given potentih. The &tor CV Is obtain& by normalization. However for photon fie& it L deair- able to h v e an emlicit form for C. in terms of the wtenentiafs. This is &ven by (3Cj) m d (29) a d it is readity demonsirat& that this also is corrmt mwding to mond quaotilation.

b. Aodyds sf the Vacam Rablem We &aft a lmlate C, from second quantizrltion by induction

consideidering a series of problems each wntaining 8 patentiaf die ttibutiou more nearly like the one we wish. sap^ we know C, for a prohhm &e the one we want and baving the =me ptentials far time t ktween Jame k and T, hut having pawnlid zero for times from Q to Ip. Cal! this C,(@, the wrmpnd ing Eliamiltonian Bto and the sum of contrihtions for all dngle loops, Ufe). Then for lo=IX" we have zero potential a t slt times, no pairs a n be praduced, I;(T)-O and CI(T)= I, For 1030 we have the mm- pie& prohiem, w that C,(@ is what is d e 6 d ar, Cp. in (38). h e r a l l y we faave,

since B& is identicat to the c o n a b t vamnm Bmiltonian Br for ;<h sad X@ is an eigenfuncgon of HT with an eigwvalue (enwgy of va-m) which we m fake as zero.

The value of C,(lo-dlo) a r k s from th-e Narnstonian N60-ar~ which diEers from Hga just by baving an extra ptentiol during the short interval AC Hence, to first order in h ie , we have

we therdare obtain for the derivative of CS the exprmien

which will be reduced to a simple factor times C,ife) by methods anartogous to those wed in rerlucing R. The operator % can be imagined to be split into ttrvo pieces Wm. and Jt,,, operating on pasitive and negative energy states rmpectively. The iy,,, on gives zero so we are left with t w ~ terns in the current density, *,m*@ABaeI and S,$SA?ir,,, The latter V.neg*BA*aeg is just

.the qee ta t ion value of BA taken over aH negative energy states (dnus 'Vma,flAwnO,* which gives zero acdng on X*). This is the effect of the vacuum ewectatian current of the e l ~ t r o n s in the sea which we should hsve subtract4 from our original Hamif- ronian in the c n s b m a ~ , way.

The mmaining term 0,,,*@A~,,,, or its quiva!ent *p,,*@AIV. can be considered as Jt*(x)f,,,(x) where fw(x) is written far the p i l i v e energy a m w e n t of the m r a t a r @A*(X). Mow this opmtor, \t+fx)fPo,(x), or more precisely just the **(X) part of it, a n be pushed through the =p(-i&erEdfl) in a manner exmtly analogous LO (47) when f is a function. (An altermtive derivation results from the congideratian that the operator *(X, l ) which satisfies the Dirac equarion also satis&- the linear integral equations which are quivaient to it,) That is, (581 can be written by Wh ((50,

I

where in the ftxslt term h== T, and in the m n d l p t e = . t t , The (A) in &cA) refers to that part of the potential A after The ht term va&hes for it involvm (from the Kc(A"2, l)) only WCive $ n e w components of W, wMch give zero operating into xe'. In the m e n d term onIy nqative comwnmts of **(X$) appear. U, then ifr"fx8) is interchanged in order with %(X$) it will give zero oprating an X@, and only the term,

will renrain, from the usual commutation rdarion of o* and *, m e factor of C,(&) i (52) times --&a is, according to (29)

(mfe~nce 101, just L(t~-&)-l;(la) i a a this ddigermce arises from the tlrcra p o t m m AA-A during the s h r t time interval AteB Hen= -dG,(rd)/dCo- +(&L(IoI/&DZC~(IO) so that integration from T 0 Jo-O ebhti&es (30).

Starting from the theory of the eleictromagnetic W d in secand q m b G o n , a c fduc~on of the quations fw quantum electro- d m m i a W&& a p w r in tae succetding paper may be w r k d out uru'ng very rjnikr principles. Tlre Paul-Wei&opf Ebwv of the RIein-hrdon qmt ion can a m e n t I y be a n a l y d in emn- tiallv the =me way as that U& here for Dirac electrow.

P H Y S I C A L R E V I E W V O L U M E 7 6 , N U M B E R 6 S E P T I B A B E R $ 5 , 1941)

R. P, F 13epat.lntd 4 Pkyska, C d Usimsicy, Iikft New Fwk

(Receivd May 9,1949)

Xa this paper two things are done. (1) It is sbwn that a con- sidwable rjmptilteation can be attained in writing down matrix ekments for mmplen p r w w s in electrodynamics. Further, a phydcai point of view is availabte which perraits thew to be wrirtwt down directly for any swil tc probtem. Being simply a rmhtement of mnventional elwtrodynamia, howevw, the matrix elements diverge fox comp1e-x process. (2) EIectr&ynamics is modified by altering the interaction of electrons at short distanw. All matrix efements are new finite, with the e x ~ t i a n of tb* relating ta problems of vacuum pofarizstion. The Latter are maEwted in a mnner suaested by Pauti and Bethe, which gives finite rerrults for these matrkes atso. The only eEccts ~nGr ive te the m d i b t i o n are changa in m s s and charge of the electron%, Such e k n p cauid not be dirmtly abservd. Phmomem directly obwvable, are insnsitive to the deltlils of the modification used (exucpt at extrme energies). For such phmomena, a lmit can be takm as the ran@ of the mdificatlon goes to zero. Tke rmults then agree with these of Schwlnger. A complete, unambiguous,

and p r m w h i y consistent, metbad is therefare availabie for the involving electrons and ph tom,

The siroplifiaaon in writing the e rcphons results from on ecmphmis sn the over-abl @ace-time view rmlting from a study of the 3101ution of the equations of dmaadynamics. The rela(ion of this to the mnre mnventional Narniltoniau point of view is dwmd. Pt would be very dacuf t to mske the modifLcatian which is p r o p a d if one insisted on having the quatinns in Eamiltoniarn farm.

The methods appiy as well to chrges obeying the Klein-&don equaGon, and to the variow m m n t h m ~ e s of nuclear farces. IUustrative amples are given. Althowh a mdifreation like that used in elecrrodynadcs a n make alt matrices finite for all of the meson &wries, for m e of the theories i t is no longer tme that a!! d i r e l y obwmbie phenomma am inansicive to t6e d e ~ h oE the nrMfifiCation d.

The actual ewluation of integrals a p p r i n g in the matrix ehrrrents m y be faciliktect, in the simpler cam, by m e t M s dwribed in the qpmdix.

T EEE paper should be considered as a direct can- tinuation of a profcerling one1 (If in which the

motion of elttctrons, neglecting interaction, was am- lyzed, by dealing directty with the so&aliion of the HamiltonLn differentbl equations. Here the =me kch- nique L applid to include interaetions and in that way to express in simpb t e r n the miofution of problem in quantum electrdynamics.

For most practical ca1culations in quantum electre- dynamics tht3 galutian is ordinarily in term of a. matrk element. The matrk is work& out as an expawion h pwers of &/h, the successive t e rm corresponding to the inclusion of an increwkg number of virtual quanta, I t a p p r s that a considerable simplification can be achieved in writing down these matrix elements for complex procmss, Furthermore, each term in the expansion can be written down and understood dkmtly from a physical point of view, simifar to the spce-the view in X. I t is the p u p s of this p p e r to de&h how this may I.re done, We shall aka diseusr; metfis of handling the divr?rgent intgrtrlrj which appar in these matrix eIemen&,

The shpYIfication in the formulae results minly fram the fact that prevbus methods unnecmrijty ~ p a r a k d into individual t e rm pro that were claefy related physirsally. For example, exchange of a quantum Between two elctrctrons there were t w te rm 4epndiq on whkh electron emitted and which absorbd the quantum. Vet, in the virtual stam considered, timing relsrtions are not significant. 0 b y the order of operators in the matrk must be maintained, We have Been (I),

in wbicb virtual p& are produced can be combined with others in which only

1 R. P. Feynman, Phys. Rev, 76, 749 ft9&), Lrmfer &l& 1. 1

positive eneqy ektrons are iavo1vd. Furaer, the effwts of longitudinal and tramverscl wava can k combined together. The separations previously m d e were on an unrehtiv&tic bash (reA~ted in the circum- stance that apprently momentum but not enerw Es commed in intermdkte sb ta ] , When the t e rm are combhd and shpW&, the rektivistfc invaxhnce of the result is =g-eddent.

We kgin by dimming the solution in space and time of the Schrijdmger equation for particles interacting insbntaneously. The results are inrmdiateky general- imble to d e h y d intctractions of rehtivistic ektrons and we represent in &at way the kws of quantum electrdmamim. We a n then fee how the matrix element for any proems a n be written down directly. In partkular, the rjeE-enew expression is written down.

So far, nothing has k n done other than a resbte- ment of c o n v c m ~ o ~ l electrodymmia in other km. The~fare, the WE-energy diverges. A md%cation2 in interaction k t w e n chaqes is next made, and it is shown that the seff-energy is made convergent and corrapnds to a canection to the ektron =ss. After the msf correction is mde, other real pramsses are finite and i w n s i ~ v e to the ""width" of the cut-off In the interaction,'

Unfortunately, the mdAcation p ropod is not completely sat isf~tory theoretialfiy (it ke& to some &&- culties of conwmation of energy). I t does, hwever, seem consist-ent and satisfactoq to d e h e the rnatrix

"or cr d i m d o n of this rnodifiation in elmial phy* gee R. P, Feyman# P b . Rev. 74 939 (IW), herurfter referred to Ild k

8 b brief summ of the m e a d # and results wit1 be found in R P. Feynmn, g Y s . Rev. 74, 1430 (%W, bnesftcr refwrd to as B.

69

element for all real pracems as the lintit of that computed here as the cut-off width goes to zero. A similar technique sugeted by Paul4 and by Bethe can k applied to problems of vacuum polarktion (resulting in a renormalization of char@) but agab a strict physical basis for the rules of convergence is not known.

After malrs and charge renormatization, the limit of zero cut-off width can be taken for all real processes, The results are then equivalent to those of Schwinger4 who does not make explicit use of the convergence factom. The method of Schwinpr is to identify the terms corresponding to corrections in mass and charge and, preGous to their evaluation, to remove them from the expreaions for real procmses. This has the advantage of showing that the resul~s can be strictly indepndent of particular cut-o@ methods, On the other hand, many of the properties of the integrab are a n a l y d using fomal properties of invariant propagation functions. But one of the properties is that the intqrah are inPlnite and it is not clear to what extent this invalidates the demonstrations, A practical advanlage of the presnt methrxl is that ambiguitks can be more easily rwlved; sinrply by direct calculation of the otherwise divergent integrals. Neverthelesl;, it is not at all clear that the convergence factors do not u p t the physical. consistency of the theory, AIthough in the limit the two methds agree, neither methd appcars to be thoroughly satisfactory theoretically. Nevertbetefs, it does appmr that we now have available a complete and definite method for the cajculation of physical processes ro any order in quantum electrodynamics.

Since we can write dawn the solution to any physical problem, we have a complete theory which could stand by i t ~ l f . I t will be theomtically incomplete, however, in two respects, First, although each term of increasing order in &/&c can be written down it would be desirabk to see some way of expressing things in finite form to ail orders in 4 f i c at once, Swond, although it will be physicay evident that the results obtained are equivd- lent to tbme obtained by conventional electrdynamics the mthemticstl proof of tbis is not includtid. Both of these limitations will be removed in a subsequent paper (see afm Byson".

BrieAy the genesis of this theory was this, The can- ventional electrdynamics was expressrtd in the La- grangian form of quantum mechanics described in the Reviews of Modem lPhysics? The motion of the fietd osciIktors could be integrated out (as described in Sec- tion 1.3 of that paper), the result being an expremion of the delayed interaction of the particles, Xext the m d i - fication of the del&-function interaction crould be made directly from the analogy to the chssical c a ~ . ~ This

was still not complete b e a u s the hgrangian method had been worM out in detail only for particles obeying the non-reiativistic Schradinger equation. I t was then modiified in xcordance with the requirements of the Dirac equation and. the phenomenon of pair creation. This was m&e easier by the reinterpretation of the theory of holes (I). FinaIly for practicaI caicuktions the expressions were developed in a power series in @/he, It was apaarent that each term in the series had a simple physical intergretation. Since the result was easier to understand tfian the Qrivation, it was thought Best to pubtish the results fiwt in this paper. Considerable time has been spent to make these first two pawrs as corn- piete and as physically plausible as posible without relying on the hgrangian m e a d , because it is not generally familiar. I t is r e a l i ~ d that such a description cannot carry the conviction of truth wl~lch would ac- company the derivation. On the other band, in the intermt of keeping simple things simple the derivation will appar in a separate paper.

The possible application of these methods to the various meson theories is d k u ~ e d briefly. The formulas corresponding to a charge particle of zero spin moving in accordance with the Klein Gordon equation are also given. h an Appendix a method is given for calculating the in tq ra l apparing in the matrix elemen& for the simpler pracesw.

The paint of view which is taken here of the interaction of charges digers from the more usual point of view of field theory. Furthermore, the familk S m i I - tonian form of quantum mmhanics must be compared to the over-ali space-time view used here, The first section is, therefore, devoted to n dbcussion of the relations of thew viewpoints.

Electrodynamics can Be boked upon in two equivalent and complementary ways. One is as the description of the behavior of a fieid (Maxwetl's equations). The other is as a description of a direct interaction at a distance (albeit deliayed in time) between chitrga (the solutions of Lienard and Wiechert). From the htter point of view light is considered as an interaction of the charges in the source with t h m in the absorber. This is an impractical point of view because many kinds of sources produce the same kind of &ects. The fi& point of view separates these aspects into two slrnphr prob lems, prduction of light, and absaption of light. OR the other hand, the field point of view is less practical when dealing with clwe collisions of particb (or their action on themseIves). For here the source and ahsarber - are ilot radily distinguishable, there is an intimate (I~~(i,s'22$;~3~,"$~$F~437~~~3; ~~~~~~+~~ exchange of quanta. The fields are so closely determined

Rev. 75, 486 (1949). by the motions of the particles that it is just as well not R. P. f i ynwn , Rev. M&. P~Y.;. 2% 357 (1941. The apl"lb- to *prate the question into two problems but to con-

sider the process as a direct interaction. Roughiy, the c d i n e Voi. U1,3 (226) 1949. fieid paint of view is mmc pnrctical for probjems invofv-

Q U A N T U M E L E C T R O D Y N A M I C S 17 1

h g real quanta, while the interaction view is best for the discussion of the virtual F a n & involved. We shall emphasize the interaction viewpoint in this paper, first because it is less familiar and &erefore requires more discussion, and second because the important -pest in the probiernsj with which we shall deal is the effect of virtual quanta,

The Hamilttonian method is not well adapted to represent the direct action at a distance between charges because that action is dekyed. The Hamiltonian method represents the future as developirtg out of the present, If the values of a complete set of quantitim are known now, their values can be computed a t the next instant in time. If particles interact through a delayed inter- actioil, however, one cannot predict the future by simply knowing the present motion of the particles. One would also have to know what the motions of the particles were in the past in view of the interaction this may have an the future motions, This is done in the Namiltonian electrodynamics, of course, by requirhg that one specify ksides the present motion of the particles, the values of a host of new variables (the cmrdinates of the held oscillators) to keep track of that aspect ol the past motions of the particles which determines their future bhavior. The use of the Hamil- tonian farces one to chow the field viewpaint rather than the interaction viewpoint;.

Xn many problems, for example, the close collisions of particles, we sre not interested in the precise tem-

sequence of events. I t is not of interest to be able to say how the situation would look at each instant of time during a collision and how it progresses from instant to instant. Such ideas are only useful for events taking a long time and for which we can readily obtain informatian during the intervening perid, For collisions it is much easier to treat the process as a whole.' The Mpcller interaction matrix for the the coltision of two electrons is not essentially more complicated than the nonrelativistic Rutherford formula, yet the mathemtical machinery used to obtain the former from quantum electrodynamics is vastly more complicated than Schrijdinger" equation with the 8/rl* interaction needed to obtain the latter, The dBerence is only that in the latter the action is instantaneow so &at the Hamiltonian method requires no extra variables, while in the formr relativistic case it is delayed and the Hamittonian method is very cumkrsom.

We shall be dixussing the solutions of equatiow rather than the time digerential equations from which they come, We shall discover that the solutions, because of the over-all space-time view that they prmil, are as easy to understand when interactions are delwed as when they are instanbneous.

As a further point, rebtivistic invarhce will be self- eviclent. The &miltonian form of the equations de- velops the future from the instanhneous presnt. But

"his is. tk view~oint of the thfiary af the S matrix of Heisen- tterg.

far difftzrent observers in relative motion the instanhneous p rwnt is dlgerent, and corresponds to a dserent 34imensional cut of spam-time, Thus the tempsral analyses of diaerent observers is different and their Hamiltonian equations are developing the process in different ways, These digerences are irrelevant, however, for the mlutian is the same in any space time frame. By foraking the Hamiltonlan method, the wedding of rehtivity and quantum mechanics can k accomplished mast naturally.

We UIustrate these points in the next wction by studying the solution of Schrtidinger's equation for nonrelativistic particles interacting by an instantaneous Coufomb potential (Eq. 2). When the salution is mrdi- hed to include the effects of delay it1 the interaction and the relativistic properties of the electrons we obtain an expression of the laws af qumtum. el~trodynamics (Eq. 4).

We study by the same methods as in I, the interaction of two particles using the same notation as X. We start by considering the non-relativistic ca* descrikd by the Schr6dinger equation (I, Eq. X). The wave function at a given time is a functbn $(X,, X*, 1) of the cwrdinates X, and X& of each particb. Thus call K(x,, xt,, t ; xe", xt,', 6') the amplitude that particle a at X,,' at time 1' wiH get to X, at .i whife particle b at X; at 1' gets to rl, at i . Xf the particbs are free and do not interact this is

where Kffe is the KO function for particle a consikred as free. In this case we can obviously d e h e a quantity like K", but for which the time 6 need not tze the same for partides a and 6 (likewise for 19; e.g.,

can be thought of as the anrpliitude that particle a gaes from x l at 11 to ~8 at fa and that particle b goes from X*

at 12 to ~4 at 16.

When the particles do i n t e r ~ t , one can only define the quantity K@, 4; l , 2) precisely if the interaction vanishes ktween tt and 12 and also between ta and I,. In a real physical system such is not the case. There is such an enormous advantage, however, to the conapt that we shalt continue to use it, imgining that we can neglwt the effect of interactions between 11 and t s and between ta and t 4 . For practical problem this means choosing such Iong time internals t o - l r and t d - 1 2 that the extra interactions near the end pints have small relative egects. As an example, in a mttering prcsblm it may wet1 be that the particle are so well sparat& initially and finally that the interaction at these times is nqligihle. Again energy values can be defined by the avmage rate of change of phase over such bng iime intewals that errors initially and finally can be neglected, Inasmuch as any physical proMem can be defined in terms of scnttering procresses we cto not lose much in

This turns out to be not quite right: for when this interaction is repraented by photons they must be of only positive energy, while the Fuurier transform of S(Ebb-r5~f contains frequencim of both signs. Xt should instead be replaced by S + f f p ~ - r L ~ ) where

This is to be averaged with ~ ~ a - ~ 6 + ( - l ~ e - ~ r s ) which arises when tS<k and corresponds to a emitting the quantum which b receives. Since

Fre, I, The fundamentaf. interaction Eq. (4). Exchange of one quantum betam two eler-trons. (2r]-t(4+(t-r)+ 8+(- f - r ) ) = &+(@-rg)*

a general theoretiml se-rise. by this approximation. Xf it is not made it is not easy to study interacting particles rehtivistically, for there is nothing signgcrmt in chms- ing 11=14 if X ~ + X ~ , as absolute sinulitaneity of events at a disbnce cannot be defined invariantly. I t is thlliy to avoid this approximation that the wmpIicated structure of the older quantum dectrodynamics has been built up, We wish to describe electrdynamics as a dehyed interaction between particles. X£ we can make the approxiwtion of assuming a meaning to Iif(3,4; 1,2) the rmulu of this interaction can be expressed very simply.

To sez? how this may be done, imagine first that. the interaction is simply that given by a Coulomb potenthi e2/r where r is the disbnee between the particlm. If: this be turned on only for a very short time &lo at time C, the R& order correction to K(3,4; 1,2) can be worked out emctly as was Eq. (9) of X by an obvious general- iation to two particles:

where ts=.tl=.lo. If ROW the ptentiat were on at all times (so that strictly K is not defined unless t4= ta and fE=l,) , the frrst-order effect is obtained by intepating on to, wGcb we can write as an integral over both tp and 1s if we include a ctelta-function 6(15-ta) to insure contribution only when ts==fe. Hence, the first-order effect of interaction is fcallmg to- t a x tar;) :

this means rIe-f4(tgp) is replac& by 6+(s&$) where ~ , ~ ~ = r ~ r ; " - r ~ ~ is the square of the relativistically invariant interval between points 5 and 5, Since in classical electrodynamics there is also an interaction through the vector potential, the eomg~lete interaction (see A, Eq. (1)) should be (1- (V, .V~)&+(S~$) , or in the relativistic case,

Hence we have for electrons obeying the Dirac equation,

where ra, and ybr are the Dirac mtrices applying to the spinor corrmpnding to pilrticies a and b, respectively (the factor BD@b king abrjorbed in the definition, x Eq. (171, of K+).

This is our fundamental equation for e!ectrodynamics, Et describes the effect of exchange of one quantuin (therefore first order in 8) between two electrons, Tt will serve as a prototyp enabling us to write dowir the corresi>otwting quantities involving the exchange of two or more quanta betwen two electrons or the interaction of an elestron with itself. It is a coilsequence of conventional eliectrodynamicf. Retativistic invariarlce is clear, Since one sums over p it contains the effects of both longitudinal and transvem waves in a retati- visticalily symmetrical way.

\Ve shall rtow intevret Eq. (4) in tl, manner ivhich wiil prmit us to write down the higher order terms. I t can be understood (see Fig, l ) as saying that the amplitude for ""a" to go from 1 to 3 and " P t a go from 2 to 4

X&(~ , , )K~ . (S , 1 ) ~ ~ , ( 6 , 2)didra, (2) is altered to hGt order because they c& exchange a quantum. Thus, "as' can go to S (ampkitude K+(5, l))

where dr .;. @zdl. We h o w , bowever, in ckaical ejectrodynadm, that ? It, and a like term for the eEect of a an b, I&s to a themy

the aujOmb dctes instantaneous^, which, in the c h i c 4 tintit, exhibits interaction fhroueh half-

but is delayed by a time taking the speed of light advanced and hall-rekrded potentiais. CluGmliy, this IS rqui-

valent to purely reardd aiitbin a closed box from which as uaity, This su$gesls simply replacing rs88(ts,l in no light my (e.g ste A, or J. A. Wheeler and K. P. Feynmsn (2) by something like r,-~606e-m) to the R w . M&. hys. ii, 157 (194511. Analagous t homn. ezist i;

uantum mechanics but it would l& us too far astray to d i m s delay m the efxect of b on a, %cm now.

Q U A N T U M E L E C T R O D Y N A M I C S 773

emit a qumtum (longitudinal, transvers, or scahr r,,) and then proceed to 3 (&(3,5)). Meantime '%" gms to 6 (R+(6,2)), abmrlts the quantum fyb,,) and proceeds to 4 (K+(4,6)). The quantum meanwhile proceeds from S to 6, w ~ c h it does with amplitude S,(SSB*), We must sum over all the passitxle quantum polarizations and positions and times of emision 5 , and of abmrption 6. Actually if Is>& it would be better to say that ""a" absorbs and ""bbmits but no attention need be paid to these matters, as all such aftemativa are auto&aticail_y contained in (4).

The correct terms of higher order in 8 or involving larger numbers of electrons (interacting with themscllves or in pairs) can be written down by the same kind of reamning, They will be nhstrated by mamptes as we preceed. In a succeeding paper they will, all be deduced from conventional quantum electrodynamics.

CalcuIation, from (41, of the transition element between positive energy free electron s ta te gives the M6ltr scattering of two electrons, when account is taken of the Pauli principle.

The exclusion principle for interacting charges is handled in exactly the same way as for non-interacting charges (X). For exampfe, for two charges it requires only that one calculate K(3,4; 1,2)-K(4,3; 1,2) to get the net amplitude for arrival of charges at 3 and 4. I t is disregarded in intermediate states. The interference eEects for scattering of electrons by positram discussed by Bhabk will be seen to reult directly in this formulation. The formulas are intexpreted to apply to poscitrons in the manner discus& in I.

AS our primaqy concern wiH be for prmases in which the quanta are virtual we shall not include bere the detailed analysis of processes involving real quanta in initial ar finat state, and shaU content ourselves by only stating the rules applying to them? The result of the analysis is, as expected, h a t they can be included by the same line ot reasoning as is used in discussing the virtuat processes, provided the qumtities are norma'lized in the usual manner to represent single quanta. For example, the amplitude that an eliectron in going from 1 ta 2 absorbs a quantum whose vector potential, suitably normalized, is c,, exp(- i k - x ) =Ce(%) is just the exprm sion (X, Eq. (13)) far scattering in a potential with A (3) replaced by C (3). Each quantum interacls only

once (either in emission or in abwrption), terms like ff, Eq, (14)) ocwr on& when there is more than one quantum invoived. The Bo?;et statistics of the quanta can, in all cases, be disregarded in intermediate states. The only effect of the statistics is to change the wight of initial or final sbtes, XI there are among quanta, in the inithl state, some pl which are identical then the weight of the state is (l/%!) of what it would be if these quanta were considered as &Berent (similarly for the final state).

Ir Althougb in the esprmsions stemming from (4) the quanta are virtual, this is not actually a theoreticat limitatton. One way to daduce the correct mtea far reat quanw from (4) is to note that in a c l o d system att quanta can be considered as virtual (i.e. they have a known w r c e and are eventuaity absorbed) so the; in such a system the present dmriptiorr is complete and equivalent to tbe conventional one. In particukr, the relation of the Einstein A and B coeftlcients can be r(edumd. A more practical direet deduction of the aprmsions for real quanta will be given in tke aubmuent paper. It might be noted that (Q) can be rewritten as dmribtng the d? on a, If"B(3, t ) - iJK+(3, 5) XA(5)K+(5,l)d?seof tbe potential A (5)-dJKt/4, 6)6+(sr&~~ X&(&, 2)dvr arnsmg from awe^$ equations - C3tAA,=4xJ, from B " c u f ~ e ~ t ' ~ ~ ( 6 ) = & ~ (4 617 g (6 2) roduc~d b pat- title b in p ing fmm 2 to d ~ k s il 2rt;e of the f.rt tK.1 I* satid= - o/atlscj")4na(2,1). (5)

3. SELF-ENERGY PROBLEM

Having a term representing the mutual interaction of a pair of charges, we must include similar terms to represent the interaction of a charge with itself, For undm mme circumstancm what appars to be two distinct electrons may, according to I, be viewed also as a single electron (namdy in case one electron was created in a pair with a positron destined to annihilate the other electron). Thus to the interaction between such electrons rnust comeswnd the possibility of the action of an efrrctron on itself,*

This intemtion is the hewt of the s d X energy prcib- lem. Consider to first order in 8 the action of an eelwtron on itself in an otheherwisc: force free redon. The amplitude K(2,1) for a singte particle to gt from 1 to 2 diEers from K+(2, l) to first order in 8 by a term

I t arises becaufa the electron instad of going from 1 directly to 2, m y go ((Fig, 2) first to 3, (K+(3, l)), emit it quantum (re), prxeed to 4, (K+(4,3)), rtbwrb it (y,), and finally arrke at 2 (K+(2,4)) . The quantum rnust go from 3 to 4 (&+(srsZ)).

This is related to the self-enerw of a free electran in the fonowing mzmner. S u p p e initklfy, time It, we h v e an electran in state j(1) which we imrrgine to be a positive enerfly solution of Dim% equation for a free particle, After a long time ts-lir the perturbation will alter

*These considerations make it a p w E unlikely h% tb contention of f. A Wbsier and R. P. Fqynman Rev M&. Phya. 17, 157 (tM), b a t electmm do not aet on & e w i v a , wi8 be a sucewfut m n q t in quantam efeetr0dynatnies.

774 R. P , F E Y N M A N

the wave function, which can then be took& upon as volume. If normaliad to volume V, the result would a superposition of free particle mlutions (actually it simpfy be proportionaI to T , This is expected, for i f the only contains f). The amplitude that g(2) is contained eEs t were quivalent to a change in enerw AE, the is m1cuXated as in (L, Eq, (2f)), The tlmgonaI element amplitude for arrival in f at f a is alter& by a factor (g=f) is therefore exp(-idE(t%-&)), or to first order by the BiBmence

- i f a Z : Hence, we have

J J~(~)BK~o(z , 1)fij(1)8rt@s. (71 AE=Z [(1271~+(4, 3 1 ~ ~ ~ ) e1p(ip.~a)6+(IaI)dr(1 (9)

The time intewal T= 12-ll (and the spatial volume V over which one integrates) must be taken very large, for the expraions are only approximate (rmaXogous to the situation for two interacting charges).la This is ha, fox example, we are dealing inwrretfy with quanta. emitted just before which would normaHy be rabsorbed at times aher t ~ .

If k'("(2, 1) from (6) is actually substituted into (7) the surface integrs~is can be perfomed as was done in obkining I, Eq. (22) rmulting in

Putting for fll) the plane wave u ex:xp([email protected];,) where p, is the enerp (p3 and momentum of the electron w=n2), and ~1 is a constant &-index symjbot, (8) becomes

the integrals exten&ng over the vofurne V and time intervai 2= Since K+($, 3) dqends only on the &Eerence of the =ordinates of' 4 and 5, the integmi on 4 $v= a result (except nmr the sudaces of the redon) i~dependent of 3, When integrated on 3, therefore, the mult is af order VT. The efftsst is promrtional to V, for the wave functions have been normalid to unit

Firi, 3. Xntewdan af m eiatroa with itself. Mommtum space, Eq. Uli).

Tbb is d i s f u d in rrfemam S in which it is inted out that cclnccpt of a mve function IOW mm if &e arc de~syd

pleE*Gam*

integrated over all space-time $74. This expression will be sirnp1ifit.d prexntly. In interpreting (9) we have Witty warned that the wave functions are nomalized so that (as%) = (iiy41r) = l, The equation may therefore be made independent of the normaliation by writing the left side as (M>(ar4%), or since (&/m)(Bu) and mhnt.=EbE, as Am(au) where Am is an equivalent change in mass of the electron, Xn this form invariance is obvious.

One can likewise obtain an expression for the energy shift for sn eIectron in a hydrogen atom. Simply replace K+ in (g), ,by Il+.Cv', the exact kernel for an electron in the potenttal, Ir=@8/r, of the atom, and j by a wave function (of space and time) for an atomic state. In general the &E which rauEts is not real. The imaginary p r t is negative and in exp(-ddET) produce an ex- ponenthlty decreiasing amplitude with tirne. This is because we are asking for the amplitude that an atom initsly with no photon in the field, witl diB appear after tirne T with no photon, If the atom is in a state which can radiate, this ampfltude must decay with time, The imaginary part of"^ when catcufated does indeed give the correct rate of mdiation from atomic states. It is zero for the ground state and for a free eledron.

In the mn-relativistic region the exprmion for bE cm be worked out as hhas been done by BetheeL"n the rehtivistic regian (pints 4 and 3 as close together as a Gompton wave-length) the lil;(n which should appear in (8) can be rqhced to first order in V by K+ plus K+(i",(2,1) given in I, Eq. (53). The problem is then very shilar to the radiationless scattering probjem

m d below.

The evaluation of (91, as are11 as all the other more complicated eqressions ariing in t h m problems, is very much simpl%& by w o r ~ n g in the mmentum and energy varkbles, rather than s p a end time. For this we shall need the Eourier Transfom of $(ss?) which is

which a n be obt;ii~ed from (32 and (5) or from I, Eq. (32) noting that 1+(2, 1 ) for &-0 b 6+(st;") from

B, A, Bcthe, Pbys, Rev, 72,339 (1947).

Q U A N T U M E L E C ' r H O D Y N A M I C S 775

FIG. 3, Ittirttarrvt correction to scattering, momentum space

I, Eq. (34). The F means ( K e R ) - I or more precisely the limit as M of (k-k+i&)-L Further @R means (2~)-~dkldFE2dkadkp. If we imagine that quanta are particles of zero mass, then we can make the genera1 rule that all poles are to be resolved by considering the Inasses of the particles and quanta to have inffnitesimal tregative imaginary parts,

Using these results we see that the xlf-enerm (9) is the matrix element between a and 16 of the matrix

where we have used the expr~sion (I, Eq. (31)) for the burier transform of K+. This form far the self-enerm is easier to work with than is (9)-

The equation can be understood by imagining (Fig. 3) that the electron of momentum p emits (7,) a quantum of momentum R, and makes its way now with momentum P- R to the next event (factor w- k-m)-i) which is to abfitrrb the quantum (another 7,). The amplitude of propagation of quanta is k2. (There is a f&tor b/*i for each virtual quantum). One integrates aver a11 quanta. The reawn an electron of momentum P propagates as l j ( p - m ) is that this operator is the reciprocal of the Dirac equation operator, and we are simply solving this equation, Likewise light goes as l/&, for this is the reciprocal D'AAiembertian operator of the wave equation of light. The first y, reprexnts the current which gneratm the vector gatential, while the second is the velocity operator by which this potential is multiplierl in the Birac equation tvhen an external field acts on an electron.

Using the same tine of rmoning, other probfems may be set up directly in momentum space. Fbr example, conider the scattering in a potential A = A,?, varying in space and time as a eq(-iq-z). An eliectron initiebtly in stare of momentum p~=pl,r, will be defiected to state $2 where p t = p ~ C q . The zero-der answer is simpty the matrix element of a between states l and 2. We next ask for the first order (in 8) radiative comec- tian due to virtual radiation of one quantum. There are several ways this can happn, First for the case illus-

l0 1 fbf FIG, 5. Comptan scattering, Erl, (151.

trated in Fig. &fa), find the ntktrix:

For in this case, firstf2 a quantum of momentum k i s emitted (r,), the electron then having momentum PS-- k, and hence propagating with factor @I- k-?)-', Next ~t is scattered by the potential (matrix a) receiving additional mamntum q, propagating on then (factor @- k-m)-q with the new mommtum until the quan- turn is reabsorbed (7,). The quantum propagate front emission to absorption (k+) and we integrate over all quanta (&K), and sum on polarization p. When this is integrated on R#, the result can be shown to be exactfy equal to the expressions (16) and (17) given in B for the same prceess, the various terms coming from resi- dues of the pies of the interand (12).

Or again if the quantum is both emitted and reabsorbed &fore the scattering takes place one finds (Fig. .Q<b)l

or i f both emission and absorption occur after the scattering, (Fig. 4(4)

These terms are discussed in detai! below. We have now achieved our simplification of the form

of writing matrix elements arising from v~rtual processes. Procesm in which a number of real quanh is given initially and finally offer no problem (assuming carret normalization). For example, consider the Gompton eflFect (Fig . S(a)) in which an electron in state p, abmrbs a quantum of momentum q,, plarizatian vector e t so that its interaction i s e~,y,=rz"~, and emits a screond quantum of momentum -q?, polariation ta arrive in final state of momentum p*. The matrix for

"S First, next, etc., here refer not to the order in true time hut to the suats ion of events alan the trajectory of the electron. Tbst is, more prwkky, to the or%r af qpearance of the mat+ in the exprmians.

this prKess is ez@l+ql-ns)-%el, The total matrix for the Compton effect is, then,

@ z C P ~ + q l - mj-'@z+@~(P~+(la- m)-1@,, (1 5)

the second &m arising b a u s e the ernhion of sz may also precede the abmqtion of el (Pig. Sfb)). One takm matrix elements of this between initial and find electron states @ 1 + ~ ~ = # 2 - ~ 2 ) , to obtain the Klein Nishina formula. Pair annihgation with emission of two quanta, etc., are given by the =me matrk, p i t r o n statw being t h w with negrttive time component of B, Wether quanta are abmrbed or emittd depends on whether the time compnent of q is positive or negative.

5, TEE GQ~XCiENGE OF PROCESSES WTE VIRTUAL QUANTA

Thew expressions are, as has been indicated, no more than a re-expresion of conventional qusntum electro- dynamia. As a consquence, many of them are mean- ingkm. For example, the &f-energy exprasim (9) or (l l) gives an infinite result when waluatd* The infinity ariscrs, apparendy, from the coincidence of the B-function singufarities in K+($, 3) and 4(s4a2f. Only at this point is it necasary to m&e a real dqarture from conventional electrodynamics, a departure other thain simply rewriting ezpressions in a simpler form.

We desire to make a modification of quantum eleetro- dmmics analogous to the modification of classical electrodynamics described in a previous article, A, There the &(slag) appearing in the action of interaction was rephced by where J(s) is a function of small width and great height,

The obviom comespondin@; mdif"ication in the quantum theary is ta =$ace the &+(ss) appsring the quantum mechanical interaction by a new function f+(s2). We can postulate that if the Fourier transform of the classical J(st22) is the integral over all R of F(k2) exp(-ik.xtz)d4k, then the Fourier trwform of f+.($\s") is the same inlegxa1 t&en over only positive fre- quenciies for &> tl and over only negative ones for I.r<l~ in analop to the relation of &(ss) to S($), The function l(.@) ==j(z-z) can be written* as

X cos(]R ~)dkrEKg(K. R),

where g(KeK) is h---" times the density of oscillators and may be a p r e ? r ~ d fox pwitive RI as (A, Q. (26))

where &"G(X)dX= l and G involves values of X large compared to m. This sixndy means that the ampfitude

*This reIalion is given incarrecdy in A, equiitian just preceding lb.

for prol~agation of quanta of mrnenlum k is

rather than W. That is, writing F+{#) -.- - ~ - ~ k C ( k ~ ) ,

E v q int-al over an intemediate quantum which prwiousty invofved a fczctor b k / R Z h now suppiied with a convergence factor C(#) where

The poles are defined by rephcing k2 by @+is in the limit 6-4. That is X2 may be assumed to have an infxni- teshal negative imaginary p a t ,

The function f+(s1z2) m y still b v e a discontinuity in value on the Ii@t cone, This is of no influence for the Dirm electron. For a particle satisfying the Rlein Gordon equation, however, the interaction involves gd ien t s of the potential which reinstates the d function if f has discontinuities. The condition that f is to have no discontinuity in value on the light cone impties k2C(ka) appraaches a r o as k2 approaches infinity, In terms of G(k) the condition is

%'his condition will also be U& in discusing the convergence of vvacuun? polarization integmIs,

The exprasion for the sezenergy matrix is now

which, since e(kz) faus off at temt as rapidly as 1/k2, converges. For practical pup- we sfxall suppose herafter that Cf k2) is simply -X2/( k2--. kg) implying that =me average (with weight G(X)dX) over values of X may be taken afterwards. Since in all procmes the quantum momentum will be cantabed in at teat one extra factor 01 the form Cp-R-.m)-' representing propagation of an electron while that quantum is in the held, vve can expect aU such integrals with their convergence hcrors to converge and that the result of all such pracessrts will now be finite and detinite {ex- cepting the p Itb c l o ~ d loop, discas& below, in which the inlegr%Is ars over the moments of the electrons rather than the quanta),

The integral of (IQ) with C(P) - X2(kZ-- h2)-koting and dropphg terms of order d h ,

is (see Appendix A)


When applied to a state of an electron of Illamenturn P satisfying pzl=mu, it gives for the change in mass (as h B, Eq. (9))

We can now cornpteb the discussion of the radiative corrections to sattering. In the integrails we include the convergence factor C(kZ), so that they converge for large k. Xntqral (12 1 is also not eonverpnt because of the well-known infra-red cabstrophy. For this reason we catculate (as disrzusd in B) the vatue of the integml meuming the photons ta have a smaff mass Xm;,<<m<X, The integral (12) becomes

which when integrated (see Appendix B) gives (e2/2r) times

where fq*)t = 2% sin@ and we have assumed the m;ctrix to operate between statetr of momentum #I and $z=#~+q and have neglected term of order &,,,/m, d X , and q*/kz. Here the onIy dqendence on the convergence factor is in the term fa, where

r=;1n(XJrn)-+9/4-2tn(~ltm,,>. (U)

Vile mwt now study the remaining terns (13) and (14). The integrar on k in (13) can be performed (aftw mdtiplimtion by C(R3) since it involves nothing but the i n t e p l (19) for the self-energy and the reult is. &low& to operate on the initial sb te SE, (so that Ptlkr=ml). Hence the factor folloGring af#I-mf-%iii be just Am, But, i f one ROW tries 10 expand l/(P1-m) =(P~+m)/(nP-tlf~) one obtains an infinite result, since pla=mz. This is, however, just what is expected physically. For the quantum a n be emitted and absorbed a t any time previous to the scattering. Such a proceps has the e8ect of a change in mass of the electron in the state 1. I t therefore changes the energy by dE and the amplitude to first order in AE by -iAB.! where 1 is the time it is acting, which is infinite, T b t is, the major effect of this term wuld be eanceted by the e @ ~ t of ckange of mass 6%.

The situation can be analyzed in the following manner, We suppofe that the electron approaching the scrtttering potential a has not been free for an infinite time, but at =me time far past suffered a scattering by a potential b. 11 we limit our discusion to the egects of Am and of the virtual radiation of one quantum between two such scatterings each of tlre eEfects will be finite, though larp, and their Blgerence is determinate. The propagation from b to a is represented by a matrix

in which one is to integrate possibly over P7depending on deails of the situation). (If the time is long between b and a, the energy is very nearly dekrmined m that 8'2 is very nearly m2.)

We shall compare the effect on the matrix (25) of the virtual quanb and af the chanp of mass Art&, The egect of a virtual quantum is

As we shall see in a moment, the other terms (131, whik that of a chan~e of mass can be written (14) give contributions which just cancel the ra term. The remaining terms give for smaI1 q,

and we are interested in the digerence (25)-(27). A

(24. simple and direct rnethod of mrsking this comprison is just to evatuate the integral on k in (26) and subtract from the resuit the exprmsion (27) where Am is givm

which shows the change in magnetic moment and the in (21). The remainder can be expressed as a muttiple Lamb shift ils intepreted in more detail in B.'* -r(f2) of the unperturbed amplitude (25);

"That the result given in B in Eq. (19) was in error W= re- - ~(191~)a(#'- r t d i y pointed out u, the author, h private communiration,

(281

y V. F. ~e iwkopf aad 5. B. Rench, as their ai~utatton, am- This has the same result (to this order) as replacing pleted simulaneously with the author's eady in LOIKJ, gave a diflerent result. French has finally shown that a l h u g h the ex- h e potential$ a and 6 in (25) (l -%rwt)a and pression far the radiationless ~at&d:ring B, m. (18) or (2.61 a&? - rs correct, it was rncrrrrectty joined onto Bethe's non-relativistrc Phyg. Rev. 75 1248 (19%) and N. E. Kroll and ,W, Fn hp4 result. He shows that the reiatiain Ln?k,,,-- l used by the Phys. Rev. 7 ~ ~ ~ 3 8 8 (1941)). The author feels unhepptly raponsble. author should have been in2k,n-S/5--.lnX,,,. This raufts in for the very considsnrble delay in the publiarran of French's adding a term --(l/@ to the I arithm in W Eq. (19) so thst the result o t c ~ o n e d by this error, Tllis foolnote is spprwriately result now agrees with that o f 7 R. Fren& ind V. F, Weidopf, numbered.

(1-&r(p"l))b. In the finit, then, as fie-* the net effect on the scattering is -3ra where r, the limit of I @ ~ ) as f i ' * q S (assuming the integrals have an infra- red cut-off), turns out to be just equal to that givm in (25). An equak term -&a arEses from virtual tmnsitions after the wttering (14) m that the entire 7a tern in (22) is crmeeled.

The reason that r is just the value of (12) wben q2==0 can also be seen without a direct calculation as follows: Let us call the vector rzT length m in the dimtion of p' so that if &'z=m(l+~}2 we have P'== ((+E)# and we bke t as very smalt, king of order Pi where 2" is the time between the scatterings b and a, Since W-m)-" - w+m)/(P'2-ntl) =,Cp+mf/2m2e, the quantity (25) is of order €4 or T. R e shall compute corrections to it only to its own order (s-9 in the limit e-4 , The tern (27) can be written approximatelyBP as

using the expression (19) for Ant, The net of the two egects b thmefore approgmatelyg6

a term n w of order l/c (since W-rn)-"@?~) Y (2m2e)-9 and therefore the one daired in the hmrt, Comparison to (28) gives for r the e ~ r e s i o n

The integrd can be imediatdy evaluated, since it is the same as the integral (12)) but with q= Q, for a repfacd by Pdm. The result is therefore r.@~/nt.) which when a-cting on the state ut is just 7,as$l%t= mwt. For the same r m m the term @t+nr)/2str in (28) is efiectivefy 1 and we are left with -r of (23).16

In mare complex problems stmting with a free efec- " f i e wrwian is not exact becawe the substitution of btn

b tbe. integral in (19) L vaLd only if opmteg on a state such &t ) can k replmed by m, The error, however, is of order a(p"-nr)-l@-nt)W-m)-ib which is o((L+e)&+nt)(p--ntf X ((X+ef#++#(2~+.3%*. But since mv,,we have pp@-m) -.-nu-m)=@-mlg so the net resuit ts ap rommatdy a@-m]b/4m%nd is not of order I/c but smaller, ga t f a t i ts eAmt (traps out in the limit.

We have used, to first order, the general expm&on (valid for any operators A, B)

tron the same type of term arises from the eBerts of a virtual embsion and abmvtion both previous t;o the other prmesm. They, therefore, simply lead to the same factor r so that the exprmion (23) may be used directly and these renormalization integrah need not be computed afresh for each problem.

PR this problem of the radiative corrwtions to scattering the net result is insensitive to the cut-off. This means, of course, that by a simple rearrangement of terms prwious to the integration we could have avoided the use of the convergence factom completdy (see for example Lfuuuisw7), The probkm was solved in the manner here in order to illustrate how the use of such convergence factors, wen when they are actually un- necasary, may facilitate analysis some~vhat by remov- ing the effort and ambiguities that may be involved in trying to rearrange the othmwl'w divergent terms.

The rephcement of 8+ by f+ given in (161, (17) is not determined by the analam with the cksical problem. In the clrossical limit only the real part of 4 (i.e., just &l is emy to interpret But by what should the imaginary part, l/(&%), of 6, be rqlaced? The choice we have miLde here (in defining, m we have, the Ioctation of the ppoles of (57)) is arbitra-ary and almost certainty incorrect, If the radiation resistance is calculated for an atom, as the imaginary part of (81, the result depends slightly an the function f+, On the other hand the light ra&ted at very large disknces from a fource is indepndent of f4, The total energy a h r k d by distant absorkn will not c h d with the energy loss of the m;ource. We are in a situation analogous to that in the classiral theoq i f the entire f function is made ta contain only retwded cont~butions (we A, App~dix). One desires instad the analogue of (I;),,$ of A. This prob'tem is being studied,

One can say therefare, that this attempt to find a consistent moditiation d quantum efectrocfynamics is incomplete (see aim the quation of closed loops, bejaw), Far it could turn out that any correct form of j+ which will guarantee energy consemtion may at the same time not be able to make the self-enera integral finite, The daire to make the methods of simplifying the calculation of quantum electrdynamic prwesses more wide@ available has prompted this publication befare an analysis of &e eorrecg form for f+ is complete. One might, try to take the psition that, since the enerm discrepancies discussed vanish in the limit h+w, the correct physics might be considered to be that obtained by letting A-+ccl after mass rrmormahtion. I have no proof of the matlrematicaf. consistenq of this procedure, but the presumption is very st-rong tbat it is sadsfac- tciry, (It is a k strong that a ~ t i s fac toq form for f+ can be found.)

In the awlpb of the raaative corrections to sattering one type of tern was not considmed. The potentiali


which we can assume to vary as e, exp(-iqex) creates a pair of electrons (see Fig, 61, rnormenta B,, -#b. iTlfis pair then mnnihilatm, emitting a quantum q=#b-f i , , which qwntum scatters the original electron from slate FXG. 5. V m u m piariaation ef- 1 to state 2, The mtrix eiment for this procem (and fect on mLtenngz (30)-

the othm vvhich can be oblirined by reanan&ng the or&r in time of the vadous events) is

This is b m u ~ the potential praducm the pair with amplitude proprtionai to a,y,, the electrons of momenta p, and -M,+@ proced from there to annihilate, prohcing a quantum (factor y,) which propagates (fa~tox TV($)> over to the other eiedron, by which it absorbed (matrix element of 7, between states 1 an& 2 of the ori&naI electron (%qy,%,)). AI1 momenta P, and spin slates of the virtual. electron are admitted, which mans the spur and the inte@ on d4p, are ealculatitd.

One can imagine that the ctosd loop path of the pitron-eiectron produces a current

which is the source of the quanta which a t on the ~ c o a d e l~ t ron . The qmndty

is then chmacteristic for this prablem of ~ h r i m t i o n of the vacuum.

One sees at once that J,, divergm badly. The modifi- fation of S to f afters the amplitude with which the cumnt j,, will aEe t the scattered e l~ t ron , but it can do nothing to prevent the divergence of the integral (32) and of its edects.

One way to avoid such difffmjties is apparent. From one p i n t of view we are consideriry~ dl routes by which a given elelron can get from one lugion of spa=-tim to another, i.e., from the source of electrons to the apparatus which measuretr thm. From this point of view the clmeif loop path I d i n g to (32) is unnatural. I t might be: assumed that the only wths of meaning are those which start Irorn the source arid work their way in a cantinuous path [pssibly containing m n y time reveals) to the d e t ~ t o r . Closed Ioops would be ex- eluded, We have alrady found that this may be done for electrom moving in a fixed potentid.

Such a suggestion must m e t -era1 qurrstions, howW ever. The d o 4 lwps are a conNquence of the w a l hole theory in electrodynmicts. Among o t k r things, they are require$ ta keep probability conmwed. The pmbbility that no p k is prduced by a potential is

not unity and its dwiation from unity arise from the imaginary part of J,,, Again, with closed Imps excludd, a pair of electrom once created cannot annihilate one another again, the xatiering of light by fight would be zero, etc. Atthough we are not exprimeatally sure of these phenomena, this d a s seem to indicate that the closed loops are necessaxy, To be sure, it is always passibk that these mlnatters of probabiliity con- sewstion, etc., will work themselves out as dmply i t 1

the case of interacting particIes as for thow in a fixed potenthi. Lacking such ii. demonstration the prmump- tion is that the diaculties of vacuum polarlzsltion axe not sia easily

An alternative prmedure discusrjed in B is to msume that the function K+(2,1) used above i s i n c a r r ~ t and is to be replaced by a modified function K+' having no shgularity on the light cone. The eEect of this is to provide a convergence fmtar C(p-m*) for every inte- grsI over electron momenta.lS This will multiply the inteeand of (32) by G(p-m2)C((P+ q)2-ntZ), since the integral was originally S@,-$a+~)d~p&'pb and both p, and f i b get wnvergence factom. The integrat now converges but the result is unsatisfa&~ry.~

One expca the current (31;) to be conserved, that is q,j,=O or qJ,,=O. Also one expects no current if a, is a gradient, or @,=g, times a constant. This leads to the condition J,g,.=O which is equivalent to q J,,==O dnce I,, Is symmetrical. But when the eqnpresion (32) is integatd with such canvergmce f ctors it does not satisfy this condition. I& altering the kernel from K to another, K', which does not satisfy the Dirac equation we have iost the gauge invariance, ib consequent current canservation and the general mnsistency of the thmry.

One can st?e this best by calculating J,d, directly from (32). The eqrasion within the spur Fcomes (P+q-rn)-lg(irCP-m)-Iy, which can be written as the difference of two terms : (P- m)-%# - &+ @- m)-LyL' Each of these terns would give the same result if the integration d4$ were 6thout a convergence fxtor, far

U Et would be very intcrmting to calculate the Lamb shiR wmrately enou h to be sure that the 20 memyclm -CM from vacuum p%xbtion are actually present. is W s techrii ue also makm self-energy and rdiationlm rcat-

tering integrals even without t k mdifieacion of 6, taJs for the diation (and the conseqjuent convergeace factar C(ltn) for the quant;l). Ziee B.

"Added tci the terms givcn k iaw (33) there is a tern 4{k"--2@'+.fqrjbr for C(kJ) 5 -Xt(&- X'))', which is not gaugg invariant, (In irddician tfte chsrge rmonnslization h -7/6 irddd to the iomithm.)

the first can be converted into the hecond by a shift of the origin of 8, namely lyP"=p+q. This does not reult in cancehtion in (32) however, for the convergenca: factor is aaltered by the substitution,

A method of ma-king (32) convergent without spiling the gauge invariance has been found by Bethe and by Pauli. The convergence factor for light can be lmked upan as the resuft of supwsition of the e@ects of quanta of mrious masses (same contributing nega- tiveiCy), Likewise if we take the factor G P - m 2 ) = - hz(bzl-- nz'-XZ)-l sa that (p-ns2)-Vw- nt2) - - p - m 2 } - - L ---m2-- X*)-+e are taking the dig=- enm of the resdt for electrons of mass ns and mass (Xz+m2n2)k. But we have taken this digerence for each propagation betwecm internetions with photons. They suggest instead that once created with a certain mass the ekectron should continue to propagate with this m a s through all t h potential illteractions until It closes its loop. That is if the quantity f32), integrated over some finite range of p, is ealied J,,(m22) and the corresponding quantity over the =me rmge of p, but with m repbced by (m2+Xz)k is J,,(d+X2) we should calculate

the function G(X) satisfying m ( X ) d X = I ernd &*C(h)XzdX=O, Then in the expression for JNFP the range of p inlegation can be extended to infinity as the integral now converges, The result of the integration using this method is the interal on dX over G@) of (see Appendix C)

with q2= b2 sina@. The gauge invariance is clear, since q,(q,q,- g*&,,) = O.

Oprrtting (as it always wiU) on a potential of zero divergence the (g,q,- &,,@)a, is simply -$a,, the B"1embertian of the ptentml, that is, the current producing the potmtkl. The term - $On(X2/nz2))(qsq. -g&,,) therefore gives a current proprtional to the current prduckg the potenthl. This would have the =me effect as a chnge in charge, so that we would have a Werence A(%) betwwn a2 and the exp~men- tally observe$ ckrge, &-t-A(l?z), analogourj to the difference between m and the o h w e d mass, This cbrge depnds Iogarithmically on the cut-off, d[e2)/d= - (28/3?r) Itl(X/m). After thh renormaliation of clzarge Is made, no egfects will be e d t i v e to the cut+E,

After this is done the finat term rem&ning in (331, contains the ~9134 effectsz1 of polruization of the vacuum,

A. Uehlin , Phya, Rev, 48, S5 (19351, R. &rbr, Phys. Rev. 48,4f) (1935f.

I t is zero for a free light quantum (qZ=Q). For smlt $ it behavm as (2/1S)q2 (adding -8 to the logitrithm in the Lamb effect). For qz> (2mj2 it is mmplex, the imaginary part representing the Ioss in amplitude required by the fact that the probicbility that no quanta are produe& by a potential able to produce p i r ~ ((@)b> 2m) decrea~es with time. {To make the necessary analytk continuation, imagine m to have a smdl negative imaginary part, so that ( I - q 2 / h t ) f becomes -i(qz/4m2- l)t as q2 g w from below to above 4ma, Then 8- lr/2+iu where sinhu= + (@/4m2- l]$ and - l/tanB- i tanhg= +i(@-4m2)1fqS)-b.)

Closed loops containing a number of quanta or potential interactions larger than two produce no troubte. Any Imp with an odd numkr of interactions gives zero (1, reference 5))- Four or moP-e potential interactio~ give integals which are wavergent even without a convergence factor as is well Emown, The situation is analogous to that for self-csnergy. Once the simple problem of a single closed loop is solved there are no further divergence diacutties for more complex p r o c e ~ . ~

In the usual form of quantum electrodynamics the tongitudinal, and transverse waves are given wparate treatment. Afternately the condition (M Jdx,)'lIr=O is carried along as a supykmenhry condition. In the prmnt form no such special camiderations are necessary far we are dealing with the dueions of the equation -mA,=4dr with a current j, which is conserved t3j Jax, = 0. That means at least a2(i),Q ,/a%#:,) = O and in fact our so1utioq dso satisfies aAddz,--O,

To show that this is the case we mnsider the aampii- tude for emiaion (real or virtual) of a photon and show that the divergence of this amplitude vanishe. The amplitude for emission for photons pohrized in the p direction invojvm matrix elements of 7,. Therefore what we have to show is that the corresponding matrix elements of q,y,=q vanish. For example, for it first order egect vve would require the mtr ix element of q betwm twa slilltes $S and Pa=bl+~. But since q=Ps-#f and (G&@*) = r n ( 9 ~ ~ 1 ) = (zi;e#z@l) the xnsztrix element vanish&, which groves the contention in this case. I t dso vanishm in m r e camplex situsttions (men- ti&y &came of relatlan (34), below) (for example, try putting ez=qz in the rnatrk (15) for the Compton EBect).

To prove this in general, suppse a,, i= l to N are a set af plane wave dtistwbing poentialis crrrrying mo- mentit q, (e.g., m m may be emisions or absoxptions of the same or &Berent quanta) and cornider a matrix for the Lxansitbn from a state of momentum t;o $p such B There are i q s mmpletely withntut externa.1 interactions. Far

exrtnrpk, a pair is am@ v~rtualty along with a, photon. Next they annihslate, absorbin this pbton, Su& loops are disrqardsd on the gmads that tghey do not intern! anything and are &meby completely unobmablc. Any rndirect eEi?cu &bey may b e via tha ezciusjm prin+te have a b d y k n hdudtd.


aw UN ffsptM-' (@1-%)-'4% w h ~ e p ~ = f i ~ t + ~ ~ (and in the XncirIentally, the virtual quanta interact through product, terms with larger i are written to the left). terms like 7,. . .r ,k2dbk. Real proceseces correspnd to The most general matrix element is simply a 1inar poles in the fomulae for virtual procmstzs, The pole combination of &m. Next consirier the matrix be- occurs when @ S O , but it look at first m though in the tween states p. and #M+Q in a situation in which not sum on all four vaiues of p, of r,. -7, we would have only are the a, acting but also another potentiai four Ends of polarization instad of two, Now it is c la r a expf -@-%)where a = q, This m;;iy act prwioustoalla,, that on& two p g ~ n d i d r to K are effective, in which case it givm U H ~ @ , + q- nc)-8u,(Po+ g-m)-" The usual elimination of bngitudinal and m l a r vir- which is equivaient to +a~n&,-E-q-m]-'ar since tual photons (fading to an instant i lne;~~ Coulomb + (Be+ g- m)-Iq is. equivatent to Cpo+ q- m)+ potential] can of c o w be performed here too (although X (P@-#- g-m) 8s a equivglent to m rrcting on the it is nat pzrticu:ularIy useful), A typkat &rm in a virtual initial state, Likewise if it acts after all the potentiab transition is 7,. 9 y r P d * R where the - represent it give q ( p ~ - n t ) - % ~ ~ @ ~ - m ) - k , which is equivalent some interve~irmg matrices. Let us choose for tbe valuepr to -aNn(p,-m)-%, since f t~ - f -g -m gives zero on the of p, the time 1, the? ciiretion ol vector part K, of k, final state. Or again it m y act between the pf&ntiaI and t m pqendicular directions 1, 2. We skI1 not ak and a&* for each K. This give change the exprwion for these two 1, 2 for these are

N-l N-1 reprewnted by transverse quanta. But we must find

Z: QN R[ (pl+q-nt>-fa,Cpk+~-m>-l (76- - *yt)-(yrr- - * yx). Now k=kdrg-krE, where k-t I-WI Xi"== (H;.K)l, and we have shorn a h m that k replacing

k-I the 7, givm ~ e r o . ~ IIence k t r K is equivalent to K 4 r t and

xq@r-mI-"ak If @$-m)-"a [yIrl. . (Yg* ' ((E(Z-ka"/ff2)(yt. - . r 3 , 1-1

However, so that on mulriptying by k2d*k=d4h(k,2-Ka)-1 the net effect is -(7,...yr)d4R/flCIP., The mans just scalar

&r+ Q- m)-"q(gr- m>-L )-,- (pk+p-n)-E, (34) wava, that is, potentials produced by charge density. The fact that l / R 2 does not contain kr mans tkat R4

tro that the sum break into the diBermce of tvvo sums, can be integrated first, resulting in an insantanmus the 5rst of which may be converted to the other by the interaction, and the d3W;/P is just tfie momenturn replacement of k by K-l . There remain onIy the term8 representation of the Coulomb potential, l/r. from the ends of the range of suxnmtion,

9. KLEfR \TOOWN EQUATEON N-l N- l

+a# a @Cltl-m)-ba,-~N If &e+~-m)-'@a,.. The methods may be readily extended to partic1a of -1 vat spin zero satisfying the Klein Gordan equ~ltion?

These cancel the two terns originally d i ~ u s d that ~ \ t - m 2 @ = i a ( A M ~ ) / ~ % ~ i ~ , ~ r f . / i ) n , - A s A r @ + ($5) the eatin: is zero. I-lence any w ~ v e emitted will care is rmuiret2 when on the c*rme satisfy dAJazs= 0, Likewise iongitu&mal wave (that prticie, Define x=krrt+RyK, and consider ik.. +r)+(x- k). is, wave for which A ,= aqildx, or a = q) annot be Exactly this term would arise if a system, acted on by potential X

have no dKt, for matrix ele, taming momentum -k, is disturbed b an added powriat k of momentum + k (the rwewd sign af rke momenta in the inter-

ments for em;i~ian and ~ h r p t i o n are si&lar. (we mediate factors in the seeond term X. . . k has no eRecr since we have =id kittie more than that a potenthl A,= dv/azR wiU tater i n t ~ m t e over ail R), Wmce as shown a b v e the result is ha no dect on a ~i~~~ eiectron since a tIransformation sera, but i nce (k. . -rf+(x. .k)=kig(rr- * .rd-lr""(rzs

we can still cancIude (rrr; . .?K) "R8K4(yr*. -yd. $'=@q(-i@):ctr)tt removes it. It is easy to see ia The equations d i s c u d in this m t i ~ n were deduce$ from the cmrdinate repremtation integrations by pab.) formuia(ion of the Kfein CIordon quauon given in reference J,

Thin has a uwful conwquence in that in Section 1 4 Tbe fvnrrian in this r f i o n has only one component and is not a spinor. An alternative format method of making the

computing probabilities for tmnsition for unplariad equatrions valid for spin zero and also for spin 1 is ( ~ a u m a ; b i ~ l light one can quard over au four by UM? of the Kemmer-Duffin matirccs F,, m e i s f ~ m ~ cornmu-

diretions rather than just the two special polarization @'On

~d&~+fl&,~,- arFflff+ &u#@~. vecam, Thus SuPPSe the matrix elmeat far some ~f W, interpret o to mean a& m t h r t h u Lt,r , far any a,,afl process for light polarimd in di r~ t ion e, is e,&f,, E the of the equations in momentum~prrce will wrnato &rmal17 identrcd

Xight wave vmtor we how from the argument fC" t k for the spin 1/2- with the exceprian of t h m tn which a denominator &-nr)-l h& been rat;ionalizr?d to (n3-n)W-m')--" above that g N ~ ~ 0 * For m ~ h r i ~ e d &ht BrOgresX since p no longer equal to a numhr, *-P. But p does equal

ing in the S directi~n we would ordharity calculate (p-p)@ so that (P-n)-1 may now be interpret& F (mp+d M,Z+M,S. But weaawejI sum M,2+M,2+lli( f

~t )a -P'#~(P .P-~I ) - lmmt . m@ impgm that %wuom .lu c* ordinate space will be valid of the function K $2 1) is sven ari

for p&, implies MP=@, ~infe qt= q, for flW2 quanta. (Z,t)== l;(iVz+m)-m-I(V$+ B2*)31+(2, 1) ;it6 v ~ L B ~ ~ / ~ s ~ -

This unpolar* tight is a r e ~ t i v i s t i a ~ ~ y &S ;S aU in vlrtue of the fact that the many fompnent wave invaknt EOncep t, afld permiB some shPr&atiom in functio~ 1(" (5 components for spin O 10 for in I) mtisfies

(6V-n)+=?@ which ir lormaliy idmtikl to the %irac W u a l i w computing cross e i o n s for such Eght. See W. Pauh, Rev. Mod. Fhya. 13, 203 f1W},

The important kernel is now I+@, 1) defined in (1, Fq. (32)). For a free particle, the wave function $42) satisfies +DJI-m2$=O, At a pint , 2, inside a space tine region it is given by

(as is readily shown by the usual method of demon- stratiqg Green" theorem) the inteeal being over an entire &surface boundary of the region (with nomal vector M,). Only the positive frequency companents of JI contribute from the surface preceding the time corresponding to 2, and only negative frequencies from the sudace future to 2. These can be interpreted as elmtrons and positrons in direct analogy to the Uirac case.

The right-hand side of (35) a n be considered as a source ofnew waves and a series of t e rm written down to represent matrix elements for processes of increasing order. There is only one new point here, the term in A&, by which two quanta can act at the same time. As an example, suppose three quanta or ptentials, a, expf-iq,.~], It, exp(-iqb-~)~ and G, exp(-iqc.12) are to act in that order on a particle of original momentum pep SO that. fia--flo+qa and Pb=fiD+qb; the final momentum being P,--@b+qe. The matrix element is the sum of three term w=p,p,) (illustrated in Fig, 7)

The first come when each potential acts through the perturbation ia(A,d;)/dx,+.iA,dJ./azP. These gradient aprators in momentum space mean respectively the momentum after and before the potential A, operatm. The second term comes from b, and a, act-tng at the same instant and arixs from the A,A, term in (a), Together b, and a, carry momentum qb,+q,, m that after B-a operates the momentum is #g+q,+qb or p&. The final term comes from c, and B., operating together in a similar manner. The term A d , thus permits a new type of process in which two quanta can be emitted (or absorbed, or one absorbed, one emitted) at the same time. Illrere is no a-c term for the order a, b, G we have assumed, In an actuaf problem there would be other terms like (36) but with alterations in the order in which the quanh a, b, e act. In these terms a-c would a P w r -

As a further example the self-enerp of a particle of momentum p, is

where the g,,= 4 comes from the A,A, tern and repre-

m t s the msibility of the simultanwus emission and absorption of the same virtual quantum. This integral without the C(k2) diverge quadratically and would not converge if Cf E ) = - XZ/(k2- F), Since the interaction occurs through the gradients of the ptential, we must use a stronger convergence factor, for exam@€ C<kf? -- X4(R?- X2)-*, or in general (27) with &XzC;(X)dX.= 0, In this ase the slf-energy converges but depends qmdratically on the cut-off X and is not necesmrily small compared to m. The radiative corrections to scattering after mass renormatization are insensitive to the cut-od just as for the Birac equation.

IVhen there are several particles one can obtain Bose statistics by the rule that if two processes lead to the same state but with two e l~ t rons exchanged, their ampritudres are to be added (rather than subtracted as for Fermi statistics). Xn this case equivalence to the second quantization treatment of Pauli and Weisskopf should be demonstrabke in a way very much like that given in I (appendix) for Bi ru electrons, The Base statistics mean that the sign of contributian of a closed Imp to the vacuum polarination is the ogpofite of what it is far the Femi case (see I). Xt is ftta=ft,+g)

giving,

the notation as in (33). The imaginary part for (@)t> 2% is again positive representing the loss in the probability of finding the final state to be a vacuum, associated with the possibiiities of psrir production, Fermi statisties wautd give a gain In prob;rbility (and also a char@ renormalization of oppsite sign to that expect&),

b. C. 0.

FIG, 7, Klein-Gordom particle in three plenkialq Eq. (36). The caupling to the etcsctromagnetic fteld 1s now, for exampke, h-a+ p..n and a new ps&bility arige$, (b), af simult&aeaus inter- aclian with two quanta a+. Thr: prctpagatian factor is now (P p- r ~ @ ) - ~ far a particle af momentum p,.

Q U A N T U M E L E G

it?. APPLECATXQR TO MESON T R B O m S

The theories which have been cfeveioped to dmribe mesons and the interaction of nucieons can be easily expressd in the languag us& kre. Calculations, to fowet order in the interactions can be made very emily for the various theories, but agreement with exprimental results is not obbined. Most likely alI of our present formuIatio~ls are quantitatively unsatisfactory. We shall content oumlva therefore with a brief summary of the methotis which can be used,

The nucleons are usualiy assum& to eiatisfy Dirac's equation so that the factor for propagation of a nuclean of momentum j5 is where M is the mass of the nuciwn (which implies that nucleons can tie created in pairs). The nucleon is then asumed to interact with mesons, the various theories digering in the form assumed for this interaction.

First, we consider the case of neutral mmns. The theo:ory closest to electrodynamics is the th-eory of vector mwns with vector coupling. Were the factor for emission or ahsolrfrtion of a mmon is gy, when this mefan is ""palarized" in the p, direction, The factor g, the "mesonic charge," replaces the electric charge e. The amplitude for propagation of a meson of momentum q in intemedhte s t a te is (q2- $]-"rather than q-* as it is for light) where p is the mass of the meson. The n e c e sary intqrals are made finite by convergence factors C(~%-B$) as in eiectroctynamics, For scalar msons with scalar coupling the only change is that one replaces the y, by 1 in emission and absorption. There is no loner a direction of polarization, p, to sum upon, f i r pseudoscalar mesom, pseudoscalar coupling repftrce 7, by ~ ~ = i y , y ~ ~ ~ ~ ~ . Far emmple, the self-enerw mtrix of a nucleon of momentum in this theory is

Other t y p s of maon theov result from the reptace- ment of y, by other expressions (for example by k ( ~ ~ ~ e - r p 7 ~ ) with a subwquent sum over all p and v for virtual mans) . Scalar mesons with vector coupling result from the replacement of r, by y" where q is the final mornentum of the nucleon minus its initial momentum, that is, it is the momentum of the m w n if absorbd, or the negative of the mornentum of a meson emitted. As is well known, this theory with neutral msans gives zero for all procam, as is proved by our discussion on Iongitudinal wavm In efectrodynamics. Pseudoscalar mesons with peud+vector coupiing carre- swn& to 7, king replaced by p-'~raq while vector mesons with tensor coupling comapond ta using (2fi)-q(rr~-517u). These extra gradients invoive the danger of producing higher divergencies for real proc- ees. For example, y& gives a logarithmially divergent interaction of neutron and electr~n?.~ Although t h w divergencies tan be held. by strong enough convergence

M. S b ~ & and W, HeitIer, Phya. Rev. 75, 1645 (1949).

factors, the resutts then are sensitive to the method used for convergence and the size of the cut*@ values of X, For IOW order prwesses p---"r,q is equivalent to the pseuctosalar interaction ZMF-"& beaus9 i f taken between free prticle wave functions of the nucleon of momenta fit and &=fi~+q, we have

since 75 anticommutes with p? and p3 ovrating on the state 2 equivalient to K as is f i x on the state l , This shows that the inleractiotl is unusua!)iy are& in tire non-relativistic limit (for example the expected value of 78 for a free nucleon is zero), but since ysZ== 1 is not small, peudoscalar theory gives a more important interaction in second order than it does in first. Thus the pseudomlar coupling comtsnt should be chosen to fit nuclear forces illcluding these imprtant second order prmesm.% The equivalence of psudoscaIar and psudo- vector coupling rvhich holds for low order processes therefore does not hold when the pseudoseatar theory is giving its most importa~tt effects. These theories will therefore give quite different rmults in the majority of prcrctical problems,

Xn alcuhting the mrrectia~ls to scattering of a nucleon by a neutral vector rneson field (7,) due to the effecls of virtual mmons, the situation is just as in eletrodynamics, in that the result: converges without need for a cut-off and depends only on gradients of the meson potential. With scaIar (I) or pseudoscalar (7s) neutral mmns the result. diverges loerithmically and so must be cut off. The part sensitive to the cut-off, hovirever, is diret!y proportional to the mcmn potentiai. I t may thereby be rmoved by a renormalization of mesonic char@ g, After this renormalization the results depend only on gradients of the meson potential and. are essentiaiiy iindepndent of cut-08. This is in addition to the mesonic chars rmorrnaIizatian coming from the prdudion of virtual nuclwn pairs by a meson, anaiogous to the vacuum wlarization in electro- dynamim. But here there is a lurtlrer difference from electrodynamics for scalar or pseudmalar mewas in that the polarizaf;ion also gives a term in the induced current proportional to the meson potential representing therefore an additional renormalization of the mass clf tlte mesoa wlxich usuatly de~xnds quadratimlly on the cut-off.

Next consihr charged mesons in the absnce of an electromagnetic field. One can introduce imtopic spin opratrtrs in an obvious way, (SpecifimIly repface the neutral y ~ , say, by r,yr and sum over i= X, 2 where

7++ T - ~ r p i(r+- 7-) and T+ change neutron to proton (T+ on proton=O) and T- chngm proton to neutron.) I t is just as easy for practical problems simply to keep track of whether the particle is a protan or a neutron on a &=gram drawn to help write down the

=H. AA, Wethe, Bull. Am. E"11)rs. Soc. 24, 3, 22 (Wahiqtan, 1949).

matrix ekment, This exciuda certain procBses. For example in the scattering of a negative meson from to qz by a neutron, the mesan 48 must be emittect first (in arder of operators, not time) for the neutron cannot absorb the negative meson ql until it beclomes a protan. That is,in comparison to the Klein Nishina formula (IS), only the analogue of s ~ o n d term (see Rg. ii(b)) would appear in the scattering of negative means by neutrons, and onIy the first term (Fig, 5(a)) in the neutron scattering of positive mmns,

The source of mesons of a $ven charge is not con- sented, for a neutron capable of emitting negative mesons may (on emitting one, wy) become a proton no longer able to do so. The proof that a perturbatim q gives zero, dirjcussed for longitudinal electromagnetic waves, fails. This has the consequencl? that vector mesons, if represented by the interaction v, would not satisfy the condition that the divergence of the poteen- tial is zero, The interaction is to be takenn as 7,- f-Zg,q in emission and as =y, in abmqtion if the real ensrssxon of mmns with a non-z@ro divergence of potential is to be avoided. (The correstion term p-%q,g gives zero in the neutral cm.) The asymmetfy in emission and ab- w ~ t i o n b only apparent, as this is clearly the same thing as subtracting from the original 7,. .y,, a term ~ - ~ q - .q, That k, if the tern --plu2q,q is omitted the raulting theory dexribe a combination of mesans of spin one and spin zero. The spin zero mesons, coupled by vector coupling 4, are removed by subtracting the term p-*q. . 4.

Tibe two extra gradients q. . . q make the problem of diverging intepals still more serious (for example the interaction ktween two protons correspnding to the exchange of t w charged vector mesons depnds q u d - ratically on the cut-off if catculateb in a straightforward way). One is tempted in this formulatkn to choose simply 7,. . e r , and accept the admixture of spin zero mesons, But it appears that this f a d s in the conventional formalism to negative energia for the spin zero component. This shows one of the advantages of the

The vector meson field pokntiats ortisfy

where s the source for such m w u h is the matrix element of betw&n states of neutron and protan, By taking the divergence

a/d+ of both sides, conclude that tttp,/ax, s. 4ari-?as, fdz, so that the giginat equation can be rewritten as

r j s ~ f i - c ; " ~ p - -.P~fr,+ci-"a/a.r(ds~/ax~)~. The right hand side gives in momentum representation 7, -~**~,p,r, the left yieids the (@--&-g and finally the interaction s , ~ an the hgran&n gives the 7, on absoiorption.

droceedlng in t h ~ way find generaliy that arueIes of spin one can be represented by a lou r -vm~r I(, jwhieE, for a free particle of momentum p satisfia q.rr-0). The promation of vixtuili prticles of momentum q from state v to C is reprmated by mllltiplicatian by the 4-4 mtrix (or tensclr) P,,=(b,,-#*q.&,) X ( m e 6rst+rder intmmtion (front the Proea equaann) w i t f i n rlectmmagnetie potendai o rrp(-ih.s) corresponds to nuttiplieation by the matrix I f , , = ( q z - a + q ~ - a ) 8 ~ ~ - q ~ p ~ - p 1 ~ ~ w b e p; and qg=qt+k are the mommta before and after the interaction. Finally, two tentiak a, B m y act simulmmaaty, with m&k - (a-bR,+bIraI.

method of second quantization of meson fielids over the present formulation. There such errors of sign are obvious while here we seem to be able to write seemingly innocent expressions which can give absurd results. Ptseudovectxlr mesans with pgudovector coupling correspond to using YJ(Y,-~-~~,Q) far absorption and for emission for bath charged and neutrnl mesons.

In the presencce of an electromagnetic field, whenever the nucleon it; a proton it interacts wi* the field in the w;di~ described far electrons. The meson interacts in the scalar or pseudoscalar c m as a particle obeying the Klein-Sordon equation, I t is important here to use the method of calculation of Bethe and Pauli, that is, a virtual meson i s assumed to have the same 'Lmass'Vur- ing all its intemctions with the etectromagnetic field. The result far mass p and for (fi2+X") are subtract& and the difference intqrated over the function G(X)dX. A separate convergence factor is not provided for each meson propagation between etectromagnetic interactions, a&erwise gauge invariance is not insured, When the coupling involves a gradient, such as rsq where q is the final minus the initial moanturn of the nucleon, the vector potential A must be subtracted from the momentum of the p t o n . That is, there is an additional coupling f r b A (plus when going from proton to neutron, minus for the reverse) reprwnting the new possibility of a simulti~neous emission (or abm~t ion) of meson and photon.

Emission of positive or absorption of negative virtual memns are represented in the %me term, thesign of the charge being determined by temporal relations a for electrons and positrons.

Calculations are very easily caded out in this way to lowest order in $ for the various theories for nucleon interaction, smttering of mesons by nucleons, meson production by nuclear cotiisions and by gamma-rays, nuclear magnetic momnts, neutron ekctron xattering, etc., However, no good apeement with experiment results, when these are available, is obtained, Probably aU of the formulations are incorrect. An uncertainty arises since the calculatians are only to fint order in g%, and are not valid if g v / L is large.

The author is particularty indebted to Professor H, A. Betk for his expknation of a method of obtaining finite and gauge invari~nt results for the problem of vmuum polarization, He ia also gratttful far Professor Bethe's criticisms of the manuz~pt , and for innumer- able discussions during the development of this work. He wishes to thank Profesmr J. Ashkin for his careful reading of the manuscript.

In this appn&x a method will be iltustrated by which the simpkr integrals appearing in problems in electrodynamics can he directly evaluated. The integmk arl%ng in more cornpiear prm- lead to rather eamplicated functions, but the study of the relatias of one intqrat to another and their exprminn in terms uZ simpler inlegrats m y be f d t i t a t d by the m e a d s given here.

Q U A N T U M E L E C T R O D Y N A M I C S 78.5

As a tpimi problem eonsider the integral (1 2) appearing in We Ben have to do inkmats of the farm tb first order radiatianlem mttering problem:

Jf l; ka; bL7ld%(V-L)*(A"-2pl.k- At)-" j~~(~8- k-m)-Ia&~ - k-m}-IrPk-"d4RC(kl), (la) X ( V - Z p z - R - A ~ ) - ~ , (98)

where we shall take C(RI) to be lypicaliy -A2(8"-Xyi and where by (1; R,; kOkJ we m a n that in the place of this symbol d" (2r)*dRldk&b&k4. we first rationJEze factors either ' 8 Or may s(;ind in diaTerenr In @- k-.r)-i- U- k+Itl){(P-R}*-rn*]-l obtaining, compkicatcrd problems &ere may be more factors f V- 2p6.K-d,i-*

or other powers of these factors (the (&--L)-"" may Fte considered

S"vP@1- k-f-mlailtt - k - t d ~ , k " d * k ~ ( @ l ss a special case of such a factor with #,=a, &,=L) and further factors R e k,k,R,. in the numerator. The potes in all the factors

X((p8-A]*-m"-"@2-k)~-m33~1 (24 am d e degnite by the assumptian that L, and the 4% have infinitesimal negative i rnag i~ ry parts.

m e matrix e ~ r d o n may be simpiifid, f t aPWrs to h to We shall do the integrals of sucmgve complexity by inducGen, do so aj te the integraeons are pertormed. Since. AB==ZA .B--BA W., start with he simplest convergent one, and show where A .B-A,B, is a, numhr mmmuting with all matricerr, Bnd, if R is any expreen , snd A a vector, since r,A-. --Ar,,+2A,, . f ; l % ~ ( ~ - ~ ) - s = (8iL)-1, (loaf

Expmeons betwmn two -ymBs can be thereby redue& by induction, Particularly useful are

where A, B, C are any three vector-matricm (i.e., linear combinations of the four 7's).

In order to crrllculate the integral in (2a) the inkgrat may be written as tbe sum of three t e r n (since k== ks rJ ,

-1: is far I; the (S; k,; k,k,) is rqtaced by 1, for JS by h,, and for 1 8 by k,k,.

More complex p r m s w of the Ernt order involve mare facton fike (@a- k)'- m Z , F and a eorraponding increase in the number of R's which m y appear in the numerator, as kOk,k,.- . . Higher order prmmses involving two or more virtual quanta involve simitar integrals but with factors possihiy involving k+# instead of just A, and the intqrsl eztending on k-gd4kC(Wk""d%'6flk"). T h q can k sinlpliged by metb& s ~ i o g m s t-o t hos U& on th h s t order integralri,

The factors @-k)*-m' may be writ&

whsxe h=&-Bf, dt=mI-P1: etc., and we can consider dmlinp with cases of greater generality in that the diEerent denomimtors need not have B e %me value of the msss m. In our specific problem (64, pp.-& so that Al..-@, but we desire to work with greater gmesality.

Now for the factor C(&}/@ we shall use -K(&- ha)-Wa. 'This am he written as

l"hufi we can q l a c e by (&-l;)* and ot the end integrate the result with respect to L from m0 to X% We can for many p r ~ t i w l p u w s consider As very I s s e retative t~ m* or p. When the ~ d d n a l integral anverges even without the convergence factor, it will be obvious since the & fntemtian wili then

wavergent to infutity, EX an infrered catastrophe exists in the b & g d one can imply m m e quantei have a smaE mass X,ta and ertend tfie integral on L from X#-,, to h', rather t h n frnm zero ta h'.

For this integral is J(2x)*dR&E(k4#- K - K-L)-a where the vector R, of magnitude K==(W;. K)b is R E , Rn, ka. The integral m kc s h w s third order polea a t k4= +(P+rl)b and k.5 -(hq+Ljb, fmwining, in accordance with our defmitions, that I, has B small negative ima&nary part only the first i s below the real axis. %c contour can be closed by an infinite semi-circle below tbto axis, without change of the value of the integral since the ~ontribution from the mi-circle vanish= in the b i t . Thus the contour can be sbmnk about the pole R4- +(k-"+L)f and the resulting R+ integral is - 2 r i times the residueai this pole, Writing kc= (ICVLL))-f-e and emandin8 (kr*-lc'"-L)-~~-*(e+2(Iri."s+L]~)-~ in powm of t, the residue, being the toeacitnt of the term cl, is seen to be 6(2(E+L)4)-6 SO O U ~ integd is

- (3i/32s)jeer 4r@dK(RI+ I;)-M- (318i)(1/3L)

estahkishing (10a). We akw have fkod"R(V-L)-a=O from the symmetry in the

k space. We write t h e results as

i ] (l; k,)dik(&-L)-ei. ( l ; O)&l;i, (Ifa) (8 1 where in the hrackeu (f; k d and ( 1 ; 0) corresponding entries are to be U&.

Substituting R= @--P in (1Xa),md calling L-pa=d shows that

By digerentiating both sides of (f2a) with respat to A, or with r a p t to p* there follows directly

i) (1, k R R )d'b(R"-2p.k-A)-' w j ' , U, Q .

.- -U; p,; p , p , - j~ , ,w+a)>w+&)+ . f f ~ a )

Further diBerentiations give dirertty succasive integrals including more k factors in the numerator and higher powers of (BP- 2p. k -d ) in the denominator.

The integrals so far only contain one factor in the denornilla.tor, To obtain results for two factors we make use of tbe identity

( s u ~ a r d by some work of Sehwinger's involving Gaussian integrals). This r q r e n t s the prduet of two recipro~lls as a para- metric integral over one a& wilI therefore permit integrals with two factors to be expressed in terms of one, For other powers of: a, b, we make use of all of tbe identiem, such as

deducible from f14a) by successjve dilferentiahns with r a p t to a or 6.

To perfom an integral, such aa

(8@J(1; ka)d%(kr-2pl.R-Af)*(iP1-2p~~h-A~-~, (l&)

write, using (IS*),

(kf-2@,. ~-A,]"(Y - 2 p z z ~ - d z ) - g = x 2 lcdx(k - ) -2&*k-&~,

where px=x#t't.f l-z)#~ and A,-xA~+(l -xj.Laz, (17s)

(note that d, is not equal ta m'-fir? so that the expraian (164 is (81)&tM~$(l;k,)d*kR()rZ-Zp~*k-rl~)-~ which may now be evaluated by (IZa) and is

where#,, A, are given in f17a). The integral in (18a) is etementary, being the integral of ratio of plymmials, the denominator of w n d degrw in x. The general expra&on aftbough rmdily ab- tained is a rather complicated camhination of rootsand lagarithms.

Other integrals can he obtained again by pramet& differentia- tion, For exampie difierentlation of (L&), (18a) with respect to ba or p*, gives

again lading to elementary integrals. As an erample, codder the case that the second factor is just

(@--L)* and in the first put #*=P, &#=it, Then f"l-rp, &-rh+(f -x)L. There results

(8i)l(1; k,; k,k,)d4k(V-L)*(l-2k"k-A)*

Integrals with three facrors can be reduced ta those involving tw by udng (I4a) again, They, therefore# l e d to integrab with two Emrametem (e.g., sec. application to radiative corrartion to scattering below).

The methods of calculztion given in this paper are deceptively simple when applied to the lower order p r m m . For p rocew of tncrea&ngly higher orders the wmplexity and difficulty in- creases rapidly, and these methads m n become impractical in their present form.

The self-energy inlegrat f 19) is

W that it requires that we find fusing the principle uf (8n)) the integral on L from 0 to XP of

since (P- k)t--mS= Rs- 2p.k, as p";.. me. This is af the form (164 with &$=l;, $#=Q, hsd?, PI=-$ so that (18a) gives, since

(X-X)~, Asm~Lp

or prforming the integmt on I;, ss in (81,

M m t n g now that h+>m* we nwfe t (X-s)Lpnf rehtive to xXs in the arwment of the l w t h m , which then h m e s (h*/ma]($(t -X)*). Then sjnee ,&i& ln(x(1 --a$+ 1 arid

~4(i-z)dxln(x(l-z)*)= --(1/4) find

sv that substitution into (19) (after the U-k-m)-' in (19) is replac& hy (p- k+m)(Re- 2p.k)'") gives

(1%- ~&/8rlrpt@+nrlt2 In(hg/=a1+2) -p(in(ht/='3 - #)l%

-. (@/8r)C8rn(tn(h2imZ)+1) -#(2 In(~~/nr")+5)7,

using (4a:ita) tr, remove the 7,'s. ThisagrecPs with m. (201 of tk text, and gives the self-eneru (21) whea ft is r ~ l g d by m.

The term (12) in the radiationlms mttering, after rationa2idng the matrix denominators and using Pf =##-r)i" requira the integrals (Ba), as we have dirscumd, This is an integrat with three denaminatare which we do in tw stagea. First h e fwtars (g-2pl.R) and ( l - 2 p ~ - k ) are cornbind by a Wrameter y;

(@-2pr -k j . -" l -2ps* k]-'i~f6 & ( @ - 2 h * h ) T from (14a) where - vP1-1- (1 -rPz. (21a)

We therefore need the integrah

whlch we witl then intelgrak with r m p t to y from O to 1. Next we do the integrals (224 imm&tely from (2&) A=@:

We naw turn to the integrals on L as rmuired in (h). The first term, (l), in {l; k,; R,R,) gives no trauhle for brge L, but if L is put qua I to zero there results z*p;* which leads to a diverging integral on z at, -0. This infra-rd catastrophe is anaiyed by using h,d for the lmer limit of the k integral. For the last term the u w r limit of L must be kept as X", Assuming X,,,"C(p#l<<X* the r integrals which remain are trivial, lis in the sell-energy case. One finds

The integrals on y give,


These in t e~a i s on y were performed as fotlovrs, Since &=#tf q where q is the momentum carried by the potential, it follows from p$--);2=mz that Q=-# so that sinct? @,=P~+g(l-y), pu~=~m2-g~yjf -y). The substitution 2y- l=ana/ tan@ where B is dehnetl hy 4mx sin2@= q" is useful forlr na wc*a/secV and @,-Sdy = j n ~ g sin2B)-Va where n goes from - e to 4-8,

Thew rrtsults are subtituted into the original swtexing formula (2a), giving (22). X t has been simpiified by frequent use of the fact that p, oprating on the initial s a t e is m, and likewise #I when it appears at the left is replacable by m. (Thus, to Smplify:

yltps~jb8~~- -~PIQ#O by = -2@2-qfa@3+ql = -2fm-g)a(m+g).

A tern, like qaq= -q1u+2(a.q)q is equivalent to just -@a ince -m-M has zero matrix element.) The renormalization

term requ~res the correqonding integrab for the special case 4-0,

C, Vacuum Palmizat-ion The expressions (32) and (32') for J,, in the vanrum polariza

ikon lxrobicm require the caIculation of the integral,

J.,(i if l= - 5 SSPEY.@- t 4 + n ) r p @ + 1 ~ + r ~ > ~ 4 ~

X (Q- &@lz- m?*)-"@+ f 1~17 -" , (32) Itbere ue have repiaad P by B- tq to simplify the cakculation mmewhat. We shall indicate the method of caieulstion by studying

where we assume X*>>& and have put some terms into the arbitrary eonsbnt C' which is indepndent of Xvhut in principfe could depend on 92) and which drops out in the inkegrat on G(X)dh. We have set 4m' sin%.

En a very similar way the integral with rtp" in the numerator can be worked out. It is, of coum, necemy ta dlEerentiak hiis m* also when calculating I' and I f i3he re mulls

with another unimprtant conshnt C'. The mmplek problem requires the furlher integral,

The value of the integral (31.a) times d differs fram (33a), of cuum, because the resuits an the right are not actually the inkgrals on the left, hut rather equal their actual value minus their value for nzP- ?n2+P.

Combining tbesa quanti.ties, as required by (32), dropping the constants 6, C" and evaluating the spur gives (33). The spun are maluated in the usual way, noting that the spur of any odd number of y matrices vanishes and Sp(AB)=S#(BA) for arbitrary A , B. The S#(I)=B anti we aim k v e

- - the integral, ts@f @%+mr)@s-~z~@a-t-~a)@~-mt)f

I ( 4 = ~ P , P P S ~ C P - iq)2-ma)-h((P+ t~)"f&~l-~. - (Pt.#s-mm~a)(pt-P4-"mrc) - (~z.~a-mrmd(P~.Pe-~@O

The factors in the denminator, p-@,g-mP+jqs and f P + p . g +(Pt.#~-mlmd(@~.lba-ms~z~~ (S@ - ~ t 1 ~ 4 - f q2 itre combined as usual by f8a"l for symmetry we where m,, m, are arbitrary four-vecbrs and constmts, sul>strtute .v-- g(1-f-rl), (1-2)- 4 U - 1)) and integrate s from I t is interesting that the terms of order X9nh* go out, so that --l ta-f-l: the charge renormalization depnds onky fogaritbmically an ha.

This is not true for Borne of the m w n rhm~.es. Electrdynamics ~ ( ~ a ' ) J-y : I P . P ~ ' P ~ - ~p ' 4- m".+i@l"d@iZ. C3Oa) h saviriowly unique in the miidnas of its divergeaa.

Btrt the integral on ) will not be found in our list for it is badly divergent, However, as d i s c u d in Section 7, Eq. (32') we do not L), More C@mpi@8: PrabXenrs wish I($+) but rather J6"[l(m*f -l(mD+ h))s(h}dhh We can calculate the difference I(="-I(mr+XP) by first cakulating the Matrix compla prohiems derivative l'(m9+L) of I with respect to m%t &+L and later =hnus that ad for lhe

L from eero to dE~erenlialinlJ (%l, three iilweatjons; higher order carrections to the MQIHhr Katter-

respect ta N10 find,

I ~ c ~ ~ ~ R =Jiff papde@w- ?.p g-+-~-t- ~ : p ~ - ~ d ~ , This still diverges, but we can cfigerentiste again to get

I " < ~ ~ + B -3c p . p ~ " a ~ - @ p . g - d- ~+ip2)-"drr (3 1 a) - - gil-lf:: ( t ~ l q ~ q ~ ~ t c p - i ( i , , ~ ~ f d n

(where D- $(g- f)@+~g+L), which now converges and Itas been evaluatlrd by (13a) with p-ifqq and A-&+l;.-fqz. How ta get I%@ may intggrate I" with respect to L as an indefinite integral and w may clmese any c m v e t f a ar&rwy cm&&. This is bemm a constant C in I-wilI m a n a term -CXa in I(mq-I(m2+XI) which vattisha since we will integrate the raults tima G(h]l?)dX and &PdXPG(X)dA=O. This means that the lwri thm ~ M n g on integnrting I, in (31aJ p rencs no probEem. We may take

~ ' ( r n ( + ~ f (%)-'E C t ~ ' g & ~ D - 2 + f 8.r bD3w-t- C&,

a aubsequmt integral on L and i i d y on pr-ts no new problems, Thm rwults

FIG. 8, The inkrxtlon betweem two electrons tn order One adds the contrribution af every figure ~nvoiving two vrrtual

+&,,C(A2+ma)h(h~m-2-f-f~-CfX2J, f32a) quanb, Appndix D.

ing, to the Campton mttering, and the interaction oi a neutron with an electromagnetic fieid,

For the Mllfer sattedng, consider two el~trons, one in state tgt of momentum p, and the other in state of momentum #S. t i t e r they are found in states us, and uh P,. This may hppen (first order in @/kG) because they exchange a quantum of momentum ~ ~ P i - f l a ~ P r - P ~ in Be manner of Q, (4) and Fig, l. The matrix element for this prmess is proportional to (translating (4) to Rlomentum space)

(sr~~~af(@rv,.dq-'. i37af

We shall diwuss corrections to @?a) to the next order in @/h. (There is also the possibility that it is the electron at 2 which 6naUy arrrva at 3, the electron a t 1 gdng to 4 through the exchange of quantum of momentum PI-Pn The ampfitude for t h ~ s process, (ricr,~tf(irar,ua)Ipa-P*}-~, must be subtracted from (3Sa) in accordance with the exclusion principle. A imiiar situation exrsts to each order so &at we need consider in detail only the corrections to (37a), ~er~erving to the last, the subtraction of the same terms with 3,4 exchangd.)

One rmwn that (37%) is modified is that two quanta may be exchanged, in the manner of Fig. 8%. The total matrix element for all exchanges of this type is

as is clear from the dgure and the genera1 rule that electrons of momm;nmnt contribute in antpEtude (,P--m)-"etw~en inter- acGons 7 , and that quanta, of momentum k mntribute @. In intepating on d% and summing aver p and P. we add all alkrnrna- liver, of the type of Fig. 8a. If the time of a b q l i o a , y,, of the quantum k by elweon 2 is tater thaa the absorrpttion, rp, of g- k, this comapnds to the v~rtual state @S+& k ing a psitron (m that (3W contejns over thirty terms Ctf the conventional methd of andyL).

In integrating over all the= alkmatives vve have considered all w i b I e distortions of Fig, 8a which prewrve the order of events along the trajectories. We have not included the p d b i l i ~ e s carrapnding tn Fig, 8b, bowwer. Their mntribudon is

as is r-dily 9wiFted by Iaheling the dkgram. Tht? contfibutions of all possibje ways that an event can =cur are to be added. This

FIG. 9. h d i a t ~ v e mrrection to the Compton e t k r i n g term (a) of Fig. 5. Appendix D.

means that one adds with equal weight the intepals corresponding to each lopologidly dishnct figure.

To this rame order there are a h Ihe p J b i E t i m of Fig, 8d which give

This integal on k will be m n to be precisely the inteuai (12) for the radiative corr~t ions to mttering, which we have worked out. The term may be combined with the renormalimtion terms resulting from the Blgerence of the eFtects of mass change and the terms, Rg. 8f and &g, Figures ge, &h, and 8i are similarly awlyzed,

FiwUy the term Fig. 8c b ctearly retailed to our vacuum poiarizalion problem, and when integated gives a term proportional to (@,yBsp)(~l~,~l) lr&+. Lf the charge is renormaliad the term lu(X/nt) in J,, in (33) is ontitttrd so t h e i s no remaining dependence on the cut-08,

?he only new in t emb we requtre are the convergent i n t e ~ a l s (38a) and (39a). They can be simplified by rationdizing the +- nominaha and combining them by (f4aI. For example (S&) m- volves the factors (E- Z$t.k)-'(kf+ 2P~,k)-~k*(q'+ P- Zq.k)-'. The first twa may be combined by {l4a) with a parameter z, and the mond pair by an exprwsion obtain& by digerentiation (15a) with resnect to b and callina the parameter y. There rcsults a factor ( k - ~ ~ ~ k ) ~ ( k ~ + ~ q ~ ~ ~ ~ ~ ~ k ~ - ~ SO that the intqrais nn d?5- now involve two factors and can be performed by the meshads given earlier in the appndix. The subquen t integrals on tbe parameters s and y are complicated and have not been worked out in detail.

Working with charged memns there is often a considerable reduction of the numhr of terns, For example, tor the interaction betwmn protons retrulting from the exchange of two merians only the term conesponding to Fig, 8b remains. Term 8a, for example, is imposrrible, for if the lrrst protsn emits a p s i t ~ v e m e n the second eannot abwrb it dircxtty for only neutrons can absorb pvsitive mesons.

As a m n d milmple, mm'der the radialive correction to the Campton scattering. As seen from Q, (1.5) and Fig. S this s-catter- ing is r ep rmted by two terms, so that we can cornider the car- rectians to each one separakly. Figure D shows the types of terms arising from corrections to the term of Fig. Sa, Calling k the momentum of tbe virtual qumtum, Fig, 9a gives an integral

convergent without cut-o8 and reducible by the methads outIine$ in tbb awendix.

The other terms are relaacivety easy to evaluate. Terms 6 a i d G

of Fig. 9 are &wIy relrrled Co radiative corrections (altfnough s o m w h t more dtffiatt to evaluate, for one of the slaka is not that of a free e l ~ l r o n , (nl+q)~Zm~). Terms e, j are renormdiza- tion terms, From term d must be subtracted explicidy the effect of mass Am, as analymd in &S. (25) and (27) leading to (28) with @'=fit+qt a=es &=ea. Terms p, h e v e zero since tbe vacuum polarimtian has zero eetect an free fight quanta, gts=D, qs*-=O. The total is insensitive ts the cut-08 X.

Tbe result shows an infra-red atastrophe, the largat part of the egect. W h cut-vg at X=,,, the effect proportional to h(m/h,,,) goes as

(&h) In(m/h,,.)(I -2@ ct&@l,

timct-s the uncolrected ampfitude, where @*-@~)g=h%n%. ?&is is h e %me as for the radktive correction to wattering for a deamtion P1-&. This is physically clear since t)re fong wave quanta are not decked by byhart-lived intermediate states. The infra-red eBects ansem from a final adjustment of the field from the asymptotic coulomb fieid characteristic of the electron of

F, Bloch and A. Norrfsieck, Phys. Rev, 52, 54 (1937).

Q U A N T U M E L E C T R O D V N A M f C S 780

momentum fit before B e mllision to tbat characte~scic of rur electron movinp in a new directian $2 after the cotlision.

The complete w r d o n for the coaection is a very compEa& expr&on involving tramendental i n t v l s .

As a tinal e m p l e we consider the i n b a r i o n of a neutron with an ekctromagnetic field in virtue of the fact that the neutron may emit a virtual ~ g a t i v e mmn, We cheose the emmple of p n d a - wakr m m s with pseudovector aupting. The change in amplitude due to an Jmtramagnetic field A=a exp([email protected]) determines the kcactteflng of a neutron by such a held, h the limit of small p it will vary aa qa-aq which reprmntci tlre int-eraaian of s par-

ng a magnetic moment, The irsl-order inkractian between an elmtron and a neutron is given by th heme calculalion by amidering the =change of a quantum between the eleetroron and the nuclmn. In this cage U, is p times the matrix element of v,, between the initial and Rnaf states of the electron, the states diEe~ag in momentum by q,

The interactiou m y m u r h u e the neutron of momentum #X emits a w t i v e mema k o d n g s proton w&ch proton inter- a c t ~ with the field rnd then rwbsorh the m w o (Fig. 10a). The matrix far tfiig pm- is (P~~f#t+q),

aternatively it may be the m m n which interacts with the lieid, We assume that it does this in the rtlennw of a d a r potential 8a&fying the Kfein &rdon @. (351, (Fig. IOb)

where we have put I=- k11-Q. m e change in sign arises km= the virtuel m w n i s nestive. Finally there are two terms wiging fmm the rdi part of the pudovector mapling (Figs. l&, 1Odf

U&ng convergence factors in Ibe m n e t d m m d in the stsction on meaon theeries a& integral can be evaluated and the resuib mmhind. E z p a n a in powers of q the timt k rm gives the W- netk momcmt of the neutron and is insensitive to the cut-o8t, the next gives the Iseattering mplitude of slow electrons on neutrons, and depends logaritbmially on the cut-off,

The qrEssions may be simplified and a m b i n d =mewhat hiore integrstion, %is makes the intepals a little mie r and also shows the nfation to the case of p s e u d m b r a q l i n g , For emmpie in (rila) the find yr& a n be written as ydk-#t-f-R.if sin@ when operating on the initial neutron state. Thk Is

Frc, 10. According to thn: meson tkory a neutron interacts with an electrornwetjc potential a by first emitting a virtual cchsgetl mem. Tbe figure rtlusmtes the case for a mudoscalar megon with m u d o v ~ t o r =upling. Appendix D.

@1-R-dd)yoS-2By, since y, anticommutes wtth fit and k, The first term cancels the @~-k-m-%nd gives a term which just canceis (@%l. Xn a like maner the lading faetor y6k in (4la) is written as -2g~s-yp@s- &-M], the secand term keading to n simpler term con@ning no tpll-K-Mf-g factor and combining with a slmilar one from (-1. One 9implifies the rnRs and in (42a) in an analogow way. There finalty r au lb terms like (4laf, (42aI but with m d o e a h r coupling 2dfy6 instad of ysk, no terms like (43a) or (&a] and a remainder, r p m ~ n t i n g the differenu? in eBwts af w u d o v ~ t o r and pseudwalar coupling, m e p s e u b l a r terms do not d ~ e n d seensitiveiy an the cut-all, but the diaerence term dewnds on it w t & c a i i y . The difference term affects the detron-neutron inktaction but not the =-tic moment of the neutPon.

Inteat ion of a protsn with an e ! ~ t m r n ~ e t j c pknendal can be slmjtarfy amtymd, There is an d k t of virtual m w n s on the e l e c t r a w e t i ~ p r v r t i e s of the proton even in the c m that the m m n s are netltmi. It is a n d q o u ~ to I-he d ia t ive wrrstions to the mtterlng of eIeetrons due to virtual photons. m e sum of the magnetic mamenb of nmtron and protan for charged m a n s is the m e as the proton m m n t dcuiatedl for the comaponding neutrd -R#. In fact it is mdily w n by camparing d i w r n s , that for arbitrary q, the sfattering matrix ta f i s l W& in Ilrc? &r@mgdk powliol far a proton m d i n g to neutral m a n &wry is qual, if the m m n s were charged, to the sum of the matrir for a neutron and the matrix for a proton. This is true, for any type or mixtures of mcson coupling, to all mdem in the coupling fnqls t ing the masg diarence of ncutron and pmwn).

richard p. feynman-quantum electrodynamics-westview press(1998)

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