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 ~ mMechanics, 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
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Editor's Foreword
Addison-Wesley's Frontiers in Physics series has, since 1961, made it possible for leading physicists to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics-without having to devote the time and energy required to prepare a formal review or monograph. Indeed, throughout its nearly forty-year existence, the series has emphasized informality in both style and content, as well as pedagogical clarity. Over time, it was expected that these 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 knowledge and reader interest dwindled. However, this has not proven to be the case for a number of the volumes in the series: Many works have remained in print on an on-demand basis, while others have such intrinsic value that the physics community has urged us to extend their life span. The Advanced Book Classics series has been designed to meet this demand. It will keep in print those volumes in 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;ynnran8sCaltech 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
EDITOR"
SFOREWQRD
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
PREFACE
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 1961
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 cWave 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 P o t e n ~ aProblems l 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
CONTENTS
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 h r i o u s ""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 ResoXution 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 electrod y n m i c s . The subject i s made to a p m a r 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 u d v e r s e a r e in a, box. Classically the box may be treated a s haviw natural modes d e ~ c r i b a b l ein terms of a distribution o-f: harmonic oscillatars with coupling between the oscillators and matter. The trarxsfUan to wanturn electrodynmics involves merely the assumption that the oscifladars a r e quantum m e c h a ~ e a linstead cif classieal, They then have energies (a +. X / 2 ) b , a = 0, 1 ..,, with zera-point e n e r w 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' t y m n t a increase by & l [emission or absorption). Waves in a box can be repressxrted a s plane e t a d i a g waves, spherical, waves, o r plane rmning wavtsa exp (iK * X$. One can say there i~ an instan-
t Revs.
Modern Phys ., 4, 87 (1932).
QUANTUM E LECTRODfxiNAMfCS
4
tanems Coulomb interaction e 2 / 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 i o n a r e M in Hamiltonian form, field o p r a t o r s , etc. Fermi's technique leads to an infinite self-energy term e 2 / r f i . It i s poasibie to eliminate this term in s u i a b l e caarr3inate systems but then the transverse waves contribute czn i d t n i ty (interpretaaon more obscure). This momaly 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 a s follows: 1, The amplitude that m atomic system will absorb a photon d u r i w the process of transition from one state to another 18 emctly the same a s the amplitude that the same transition will be made under the influence of a ptential equal to that af a classical e l e c t r o m w e t i c wave representing that photon, provided: (a) the classical wave i s normalized t a 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 a v a i l h i e p r cubic centimeter of a given po2arizatlon Is
Note this ia exactly the same a s the number of normal modes per cubic centirneter in classical theory. 3 , Photons obey Boee-Einstein atatistlcs, That is, the states of a csllesetion 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 r e 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 a s
I N T E R A C T I O N QF L I G H T W I T H M A T T E R A
ZZ
a e cos ( u t - f?;*x)
We want the nctrmalizatian of A to correspond to unit probability per eubic centinneter of findiw the photon. Therefore the average e n e r a density should be 60. Now
IFf = (l/e) (&A/ a t) =
(Ua/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 E r , The probability of transition p r second i s the same a s the probability of transition m d e r the Influence of a vector potential a8 expf-t-ifwt- K-X)] r e p r e s e n t i ~ gthe ernitted photon. Aecasding to the laws of quantum mechanics (Fermi's golden rule) Trans. prob./sec = 2n/A /f(potential)il2* (density of states) Density of states =
QUANTUM E LECTRQDUHAMICS
6
The matrix element U f i = /f@otential)f/2is 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 chwsin@ 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 i w a s i m c e 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
Pauli equation. i s
as follows. The
INTERACTION O F LIGHT WITH M A T T E R
The partial derivative with respect to time introducaes a term and this may be included with 4e-jx @. Therefore the substitutions ( @ X / B t)Y
leave the Paul1 equation mchawed. The vector potential A as defined for a photon enters the PauB H m i t tonian a s a perturbation pohntial for a transition from s t a k i to @talef. 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@ctae a timeindepndent prhrbation. U(x,y,z) beween initial and fha1 states whose anergies are, respectively, E ~ and ~ El. w 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 Meehantca; Non-Relativistic Theory," Addison-Wesley, R e A i q , Massachusetts, 1958, Sec, 40.
QUANTUM ELECTRQDYNAMICS
To determine U f i , write
Because of the rule that f i e potential acts only once, which Is the s m e a s 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 e f ' "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 = waveXewth, I[f +/A<< I, all terms of higher ordfits than the first in ao/h msby be neglected. To complete the dipole a p p r o ~ m a t i o n ,it i s also necessary to neglect the last term, This is e ~ i l done y 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 O F L I G H T W I T H M A T T E R
(efif/2mc)cr . (Kx 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
Uf*(a =
+I
(Kx p ) ) ~I d v o l = o
since the @tawsare orlthwoml. For the present, the dipole approximation i s to be used. Then
-
UZi =" -a---e Prr * e m
"f
Using o p r a t o r algebra, pfl /m = Rwff X ft , SO that Pft dS1 = a2[e2o'/(2n)'1(e
l
where Xfi = $r grading Ptf aver
* X @$ d vol.
x f f )' d a
The total probability i s obtained by inte-
din, thus
e2u4
Total prob./sec = j a 2 7 (e (2x1
lf
)' dS2
QUANTUM ELECTRODYNAMICS
10
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,
Total prob./sec =
e2 3 1 1 x ~2l 54 G k3
&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 O F 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-
iwt
(haccord with the m l e s of b c h r e 2, the a r p m e n t of the exponential ia minus and only t e r m s which a r e lin,@min the potential are included,) Using &is time dependence m d p r f m m i n g 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 u j k * Far e x m p l e , s e e 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 freency 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, it is necesrJary to integrate aver all fr-equencisa, Tbis ahsorption probability is
when T is large, the faotor a i n ? ~ ~ / z f ihas ) an appreciable value only for Ew near El Ek,ancl P(@,@,Q>) wf ll km substmtially eomtant over the small r a e in o which contributes to the integral s o &at i t may be t&en out af the integral. Similarly far ulk, ao that
-
' 1' Trans. prob. = 2 ~ ( % ) -lulr
where
$)dQ
Q U A N T U M ELECTROf)'YNAMPCS
12
This can also be written in terms of the incident iatensity (energy crossing a unit area in unit time) by noting that
U s i w the dipole approximdion, in which
the total probability of absorption (per second) i s
It is evident that there is a relation b t w e e n the probability of sgontaneous emission, with aceompaying atomic transition from etate 1 to atate k, Probability of spontaneous emission/sec
= 2n(1i)-'(2nc)-~ j
~ ulk2 ~
~
and the absorption of a photon MUI accompan9ng atomic transition from 8 t a k 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 a r e (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, $1, Trans. prob./sec = 2 n ( ~ ) *lulk nfw,~,(13)(2nc)-~ wk12 d o (4-4)
This equation may be indewreted a s follows. Since a(w,@,cp)is the probability thiit, a phdon state i s occupied, the r e m a n d e r of the h r m s af the rfght-hmd side must be the probhility per second that a jphoton in a& s t a b will be absorbed. Comparing Eq, (4-4) with. the rate of spontanems emission s h w &at ~
/
I N T E R A C T I O N QF LIGHT W I T H M A T T E R
R o b ./sec of absorptictn of a photon f r m a state (per photon in that stale)
prob. /see of spontaneous emission of a photon into that state
h what follows, i t will be shown that Eq. (4-4) is correct even when there is a possibility of more &an. one photon per: state provided nfo,8,$)is t&en a s 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 n w i s h 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~orptionwill be Mice that for one photon. TaXcjng the statistical weight times the square of the m p U 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 N o tons per initial photon state md one is absorbed, the f o l l ~ M n gsix 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 r m s i tion probability i s (1/3 1)(1/2 1) (6)' = 3 times that if them were one photon p r i d t i a t state, In general, the transition probability for n. photons per initial photon state is n times that for a s i w l e photon per photon state, s o Eq. (4-4) is correct if n(cr/,@,$yj) is t&en as the mean n u r n b r 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 i d u c e d part of the transition rate, while the 1 i s the spontaneous part of the transition rats, Since the gotentials used in computing the transition probability have been normalized to om photon Wr cubic centimeter and the trmsition probztbiiity depends on the: square of the amplimde of the potential, it 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 a s resultiw from the wrturbation by a potential normalized to the actual energy In the field, that is, to energy nEw if there a r e n photons, The correct transition pr&&ilily for e r n i ~ s i o ni s not obtained this waly, however, because it is proportional to n + 1. The err o r corresponds to omitting the spontaneous part of the t r a n ~ i t i o ngrabability. In the semiclassical theory of radiation, the spontmeous part of the emission probability i s arrived a t by general arguments, ixzctuding the fact that its inclusion leads to the observed Planck distribution formula. Einstein first deduced these re lationshipa by sem iclassicsl reasoning.
INTERACTION O F LIGHT WITH M A T T E R
%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 t N s matrix element to vanish, For example, if in hydrogen the initial and final shks a r e S stabs [spherically symmetrical), X+ = O and transitions b t w e e n 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, f o 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 r b i k l angular momentum of the initial sLak i s 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 $,?l o r P,."$. F o r a complex atam (more than one electron), the &miltonian i s
H = z ( 1 / 2 m ) P, a
- (e/c)A(x gl).
+ Coulomb terms
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 i s 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
Q U A N T U M 1ELE:CTRODYbJAMICs
16
in the initial state do not interact, then the produet (X* + x2)iEi (xI,xZ) can be formally regarded a s the wave bnction of a system (atom -t otsject) having possible values of J i -c- l, J1 , and Jz- l for total m ~ l a momentum. r Therefore the matrix element i s nonzero only if Jf , the final a.nwlar momentum, has one of the three values JI 1 o r Ji Hence the general selection rule A J = a 1, O, Parity, P w i t g is the prowrty of a wave function referring to i t s 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 i s
If the pwidy of
%cf
i s the s m e m &at of -$!l, it follows that
Hence tlze rule that p a ~ t ymust chawe in allowed transitions. For a oneelectron 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), s o hL = O transition8 can occur. The a- 0 transktions a r e always forbidden, however, since a photon always c a r r i e s 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, it bllows that e i a e r ?Pf -X) = Jir (X) o r Q (-X) = -S [X). If the state is degen;c {X) But then a complete solution would erate, i t is possible that ?fr (-X) be one of the linear combinations 4?(X)
+ %(-X)
even parity
-
odd parity
% (X) @(-X)
Forbzddelz Llirtcls, Forbidden. spectral lines mzty a p p a r in gases if they a r e 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 it
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 l a r i z e d in the x direction, &is ~GOIZLBB
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 r s t part of p,z ie seen to be equivalent, up to a constant, to xz, which behaves similarly to a wave function for
Q U A N T U M ELECTRODYMAMXCS
18
2, even pmity, The second p r l i s the operator $, which behaves like a wave function for even parity, Therefore the selecUm rules correspondi% to the Brst part a r e Been to be A J = 2, +l, O with na parity chawe, This type of radiation i s called electric quadrupole, The selection rules far the scseond part of p,z a r e A J = rt I*OS no parity clzawe, md the e o r r e a p n d i q radiation i s eaUed magnetic dipole. Hots that unless A J & 2, the two t y p s of radiation cmnot be d i a t i w i s h e d by the change in m ~ l momentum m or pwity. If A J = *l, 0, Lfiey eirn only bbe d i s t i n ~ i s h e dBy the polarizatia~of thr; raaation. Bo& t m a may occur simultmeously, producing interference. In the case of electric qudrupole radiation, it is impticit in Lhs rules that 1/2 112 and O X trmasitiona are forbidden (siren though &lmay ' be & l), since the required chawe of 2 for the vector m m l a 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 mw 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
lMultipole
Electric dipole
m p t i c EL~trie &wetic dipoh quadmpolt.: qwdrupole
Ekectric 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 a s
where 2' is the multipole order o r 1 i s the vector change in angular moment~,
XNTIERACTllOrJ O F 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 and the lowest possible muluet of the initial and final parities is ( - l ) l t tipole order i s J f J i , the transition probabilities for multipole typea contained witEtin the dashed vertical gnes in Table 5-1 a r e r o w h l equa1.t In and parity-unfavored transitions, where the parity product i s (-1) t -ji the lowest mullipob order 163 I Jf - Jy1 + L, this may not be true.
-
!r
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 e q u d the number goi% from l to k by emission. If n, photons of frequency w are present per cubic cena m e t e r , then polaabili.t;les of absorption are proportional to n, mci probaBility 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 s c a t b r e d by an atom into a new direction ( a d possibly e n e r e ) (see Fig, 6-11. This may be considered a s 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 r e 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 muItipole.
QUANTUM ELECTRODYNAMICS
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 m d
where r pins are neglected, In each i n b g r a l defining AXkreach. of the two vector potentials must app a r once and only once. Thus, in the f i r s t integral the term p 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 c r a s s - p r d u c t term 2A1&. The second in&gral will have no cont r i h t i o n from A . A , but will be efie sum of two terms. The f i r s 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 r e s u f t i w in these two terms can ba represented schematically a s shown in Fig, 6-2. The integral resulting from We first term vAlf now be developd in delai l.
-
Then the resulting integral i s
e
I N T E R A C T I O N O F LIGHT WXTB 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 a v e r q i n g 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 'reviously 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,$8 If l = k, the s c a t b r i n g is coherent, Further, note that if all the atoms are in, the ground state and l r k, f;hen the energy of the atom can only i n c r e a ~ eand the frequency of the Ught ~3 can only decrease, This gives r i s e to "Stokes lines The oppsaite e f h c t gives @%ti-Stokes Unes Suppose wg = (coherent s c a l k r i w ) but further Ewl i s very nearly equal to Ek -E,, where E, i s some possible energy level of the abm, Then one b r m in the sum over n b c o m e s extremely large and d o m i m b s the remainder. The result is called "msomnce s c a t k r i w . " .B' @ is plo-d agaimt o, then a t such vdctes of w the c r o s s section hais a s b w maximum (see Fig, 6-31.
."
.$'
FIG. 6-3
The @'indexps oaf refraction of a gas e m be obtained by our scattering forscatterilae;, ~ by consideriw the mula, ft can be obtained, a,@ for other t y of ~ lf&t s c a t b r a d in the forward direction. bJslf-Enem, h o t h e r phenomenon &at must be considered in q u m of m atom emitting a phobn and r e a b ~ o r b e l s c t s o d p a l c s i s t h pos~ibilft.Y ~ Ing the earne photon. This affects tfie diagonal element Akk. Its @Beatis 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 dto a v d d this difficulty of the infinite self-energy will be &scusaed later. The net result i s a very @mallsMft 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 movixrg uniformly t q e t h e r at velocity V; that is, xza experiments made entirely inside of a closed s p a c e ~ h i pmoviw mfornzly rrit velocity v (relative to the center of gravity of the matter in the universe, for e x m p l e ) can debrrrine e h i ~velocity. The p i n c i p l e has been verified exwrimentally Newton's laws satisfy this principle; for they a r e urrebmged when subject ta a Galllean transformation,
.
because they i nvolve only second derivatives , The e H qaations are changed, h w e v s r , 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 Maxwell equations a r e of the @%me form in any coordinate system; and, in particular, that the velocity of light i s the same in all coordinate s y s k m s . 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 s o 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 cos 8
X'
y' =
-X
i- y
sin 6
sin B + y cos 6
Successive transforrnatians vl and v, o r ul and u2 add in the sense that a single tran~farrnationva 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, a r e unchawed in form under a Lorentz transformation. An interestin%; consequence of the Lorentz transformation i s that clocks appear to run slower in moving systems; that is called time dilation.. In transforming from one coordinate system to another it i s convenient to use tensor analysis, To this end, a four-vector will be defined a s a set of four quantities that transforms in the same way a s x,y,z and ct. The subscript p will be used to desimate which of the four components is being considered; for e x m p l e ,
The fallowing wantities a r e f our-vectors:
a --8
W.,"-,..
W
-
a
+.,"-a
(C,) four- dimensional gradient
a ~ " y y ~ v t
.lxJ jyJ jzyP A,,
$8
A,* V
P,# Pyr P z r E
(jp ) current (and charge) density
(A,) vector (and scalar) potential
(p,) momentum and total energy f
p -
$ The energy E, here, is the total energy including the r e s t energy
me2.
S P E C I A L RELATXVfTU
25
An invariant i s a quantity that does not change under a brexrtz transformation. If a u and bp a r e 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 w c u r s 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 it 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,
= p e p = ~2
- pXZ - pY2 - p,'
= E'
-
= m'
(E = total energy, m = r e s t mass, mc2 = r e s t energy, p = momentum.) Thus,
It i s also inlteresting to note that the phase of a free particle wave function ~sxpf 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~ particle
sttltiona xy padicle
hboratarjr system
Center-of-mass system
QUANTUM ELECTRODYNAMICS
but
and
The equations of e l e c t r & m a m i e ~ B = V a r e easily written in temor notation,
X
A and E =
-- (i/c)(@A/@t)- V Q,
where use i a made of the fact that (p is the fourth component of the fourvector 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 r e zero; thus there are only six ~ndewndentcomponents ( 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 a r e wrltbn
m
0,
$3
C.)
g
g
a"
-2 21car
&
*t
k3
3
rr
l,
E
V
b a o
-l;;;
I Q
win
'2W
2x8
9.
EU
a<
S
9 car E9
S
cQ
-.-l
0
"3c5
"g .2
3 tn
d.$
83 4 9
*E:@
9:
=S
c ; .*Q ).aC . )m& Its @.a 3$ ."g @i L.3 CU 3 .,a$$ ,=5 % m k tI
.I?
*@.S
Q
$ 2 ".g -G
w*-;g;;:
*m ..It
It
z a
X
a&
II
- 4 k w g
SSm"@
mg 4 B g 3 2
s 2 g ga 3 $ z s % y
3.5:
;@@%g
G ;
--h g ; zo lI%Z s ;S
$a%
c
E Eg "S S@ 3, Q
lel
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 a r e rectangular). Thus,
Note that in these operations P, AI, actually forme a second-rank tensor, f a &ird-rank tensor, and then contraction on the index v yields a firat-r& tensor o r vector. The k p i s the propagation vector with components
V, (V, A,
and the condition k *k = 0 meam
Problem: Show that lthe LLOmntz codition
implies that k e = 0. m e n w o r M q in three dimensions it is customary to t a b the poiarization vector 63 such that; K * a = O and to let the @calmpotential 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 a e o r g . 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 still hold if
= 0,
the h r e n t z condition, Eq, (7-41, will
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
30
the VpVvX
-
QUANTUM ELECTRODYNAMICS
O,Vp X because the order of ditferentiations is immaterial.
The components of ordinary velocity do not transform in such a mamer that (tanbe 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 @ves the relation between proper time and local time to be
by dt2
The components of ordinary velacity a r e related as follows:
It i s evident that u p u p = 1, for
The four-momentum i s defined p, = mu, = m/(l- v2)'/ 2t mvx/(l
mv,/(l
-
mvy/(l
- v2)'/ ',
- v2)'i2
i s the total energy E, s o that in ordinary notaNote that pc = m/(1 tion 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 d e f i n ~ dby d/dt (momenhm) cannot forrn the components of a four-vector. But; the quantity f,
-
dp,/ds
does form a four-vector with the components
where FP i s the ordinary force. The fourth component i s f4 =
power
--
rate of chmge of energy -
4
This i s seen from the fact that m/ the o r d n a r y identity
3
i s tlre total energy and also from
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
QUANTUM ELECTRODYNAMICS
Problem: Show that the expressions just given for f and f4 a r e equivalent to f, = euP F p v
so that the relativistic a n a l o p e 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 = e(E
d/dt (mv/
-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
~+ mj2 ] t /~2
(8-6)
which satisfies (H - e@l2- (P- e k j 2 = m'. It is difficult to convert the Hamiltonian idea to a csvari m t o r 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
SPECIAL RELATIVITY
Note that by definition (ds/da lz = (dz p / d a ) ( d z P/daf
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 cFS i = O (variation 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 a s a useful reference far conversion to a stomary units, The following numerical values a r e useful: M p = 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 notati on
m
Mea~ng~
Customary notation
Value
Mass of electron
m
Energy
mc2
5% 0 '99 kev
Momenhnn
mc
1104 gauss ern
Frequency
me2/%
Wave number
mc/A
bwth (Compton waveliewtt.r)/Zr
li/mc
3.8625
f @-'%cm
Time e2
Fine-a truetwe constant (dimensionless)
evm
Classical radiue of the e b ~ t r o n
2/me2
Bahr radius
If the qumtum-mecbfctal operator -iV fs used f o r p , the operation d e k r mined by the square root i s undeaned. %us the reEativistie quantummechmieal Plmiltadan has not been obtained directly from the classical equation, Eq. (9- 1). However, it is g o e ~ i b l eto 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
Q U A N T U M ELECTRODYNAMICS
36
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 its 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 r s t term on the left gives the ordSnary S c h r ~ i n g e reqtlation. 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 j = -if% V 9
Then show (p,
- e s s * / a t ) - eqb3 ?fi*
- J1 V Jlr *) - BA* ?k*
1) is a four-vector
= charge density
= current density
and show FPjp = 0.
The Klein-Gordan equation leads to a result that seemed s o unreasonable at the time i t was first brozzght to light that i t was considered a valid basis for r e j ~ c t i n gthe equation, This result i s the possibiuty of negative energy s t a b s . To s e e that the Illein-Gordon equation predicts such e m r g y 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 e w a tion has the solution V = A exp (-ipp xp), where p p p p = m2.Then, since
there results
The apparent impossibflity of negative values of E led Dirae t a the development of a new relativistfe wzlve equation. The Dfrac equation proves ts be correct in predietiw the e n e r m levels of the hydragsn atom md i s the accepted d e ~ c r i p t i a nof 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 EQUATION
intent, Ms equation also leads to the existence of negative e n e r m levels, which by now have been satisfactorily interpreted. m o s e of the KleinGordan equation can also be interpreted.
Exercise: Show if 3 -- e x p f - i E t ) ~(x,y,z) is a solution of the KleinGordon 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 i n a e a t e s 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 equation. r With m analogous point of view, the Dirac equation can be d e v e l o ~ das the four-vector form of the Pauli e q a t i a n , h following such a procedure, the terms invalviw "spixltWill be included in the relativistic. equation, The idea of: spin was first i n t r d u c e d by Pauli, but it 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 it is often stated that only the Dirac equation produces a s 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, m d beemse the Kleixz-Gordon equ.at;ion was thought to be invalid, it is often stated that spin i s a relativistic requirement. This is incorrect, since the Klein-Gordon equation i s a v a l d 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
QUANTUM ELECTRODYNAMICS
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 sa 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 e p a t i a n s can be rewritten (only to arrive a t a particular conventional f o m ) by w i t i n g
-
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 9bhas been shown e q l i c i t l y , 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
since 1
0
0
crx =
0
X,
P
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 fourvector notation, the Dirac equation is
that is, show
Y : =l
= y y 2 = yz2 = -1
QUANTUM ELECTRODYNAMfCS
40
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 y p (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 ~ ~-}L2 'eyIIy, F,,
.Y = m'+
A W E B M OF THE y MATRICES In the p r e c e d i ~ glecture the Dirac equation, = m% yl,(iVI, - ehI1)%
was obtained, together with a slpt3eial represenLaLion for the y %,
RELATIVISTIC WAVE EQUATION
41
where each element in these four-by-four matrices i s anof;her two-by-two matrix, that isa,
The best way to d e f h e 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 r e
or, in a unified notation,
Note that with this definition of ucts
and the rule for forming a scalar prod-
Other new matrices may arise by forming produets of the matrices afready 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,
Q U A N T U M E L E C T R O D YltJtJlMleS
42
These are: the only new p r d u c 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 e q w l s o 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 =atrix. (It is maaemarclcally inbmtsting then that all four-b-four matrices can be e w r e s s e d 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
RELATIVISTIC W A V E EQUATION
For e x m p l e , the firet may be verified by vvritislg
and, movirmp; the sc;cond faetor to the front, by using the cammutation relationrahlips, Doiw Ghia with the first term, (bp?,) af the secoad factor produces
since y, commutes with itselt: and anticommutes with y,, performing &is o p r a t i o n on all terms, one obtains
&P = = t Y t I(-&,?,
+
+ b,y,[(a,yt
+ byr,[(atvt + bz?,[(a,v,
= -1391
+
a,?,
+
- a?,,
ayyy + a,y,)
+
h,and y, . By
2a,y,l
- ayYy - BzYz 1
+
- a,?, - ay?, - a,?,) Zayv,I - &,Y, - ayYy - a,yz) + 2a,yzl
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
28XYX1
QUANTUM E L E C T R O D Y N A M f G S
Suppose a n o ~ e representation r for the y @s ifs obt&ned which satisfies the s m e commutatio~relationships, Eq. (l0-3); 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-' ( ~ = l). 8 The ~ Birac ~ equation becomes
The r p and S commute, since n i s a differential operator plus a function of position, s o & i s equation may be written
Multiplying by the inverse matrix,
S i s called an equivawhere y ; = S-' y p S. The transformation y ; = S-' lence 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 m m n e r 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 a n d h e r representation for the y 'ss,md the Diraa equation i s in exactly the e r n e form a s the original, Eq. (10-l), md ie equivalent in all its 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 f o r m 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 equaF! 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 eqaivalenee 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 c o o r a n a h s ) . 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 r m s f o m a t i o n .
.
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 it in I f a i l t o n i a n form. The original term, Eq, (10-11, 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 follwutation relations: a X2 = a y Z= ( Y =~p2~ = 1 and all pairs anticommute. It will be noted that a,@ a r e Hernitian matrices in our special regresentation, ~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 ~ontfnuityeqmtiorr
Note: \Xr i s a four-component wave function. and
Xt should be noted that p mid a a r e Nerxnitim only la certain represenrlations . h particular, they a r e 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 l a b l e d 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 i e expressions for c h a g e 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 tXTZ, %V"3, ?ka of the waw function but the et$ term, Also the connponenb IYp2 here, All used in Schiff correspond, respectively, to -i&bf, -9b2p 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
REEATIVISTXC 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 a r e formally the same a s before. For e x m p l e ,
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 : of eigenvalues of a! a r e a l. Hence the eigexlvelocities of .k a r e ~ t speed light. This r e ~ u l its 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 r e 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 i s not completely acceptable, for k commutes with p; that ia, one should bs able to nsreaaure the two quantities simultansously.
QUANTUM ELECTRODYNAMICS
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 a s the analowe of Newton's equations. But, since there is no direct comection b e h e e n 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 a s true but their m e m i w i s not; yet completely understod, if at all:
where in
last relation a means the matrix;
so that o, = --tar, a y , etc. From analogy to classical physics, one mi&t e x p e t that the anmlar momentum clpemlor is now
From previous results for written
k and
P
(p
eh), the time derivative of L may be
RELATIVISTIC WAVE EQUATION
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 r s t krm;that eaneerved, even wl& central forces. is, the angular molnanhtm L is 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 a y terms, so that cr, = + l ( H a , a y - aXayH)= + ((Y,T,OI,(Y~ Q , @ y ( Y x ~ x+ QyTyaxay@x@y@ylfy). where
-
But
so that
This is seen to be the z component of -2or
and this is the first term of
X ar
. IFinal3,y &en,
iwith 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 o p r a t o r , where L represents orbimomenhm for spin 112. tag anwlar momentum and i-fi/2)o intrinsic a-Ear Thus total, ang~ularmomenhm i s conserved with central forces,
Pwblems: (1) h a stationary field
$3
= 0, BA/@t = 0, show tfiat
i s a coastant of the motion. Nob that this is a consequence of the anomalous gyroma@etie ratio of the electron. Xt also me cyclotron, frewency of the electron equals its rate of precession in iil mapetic field. (2) fn a stationary nrr2lgnetic field 4 = 0, @A/Bt = 0, and for a stationary s t a k , show that %fr. vIrz in
are the same as JlrZ in the PauXi eqwtion, Also, if EPHuliia the Mnetle energy. in the PauXf equation and EBirsc = W + m i~ the msrt plus kineWc errsrm in the?Erac ewtiont, show ~t
and explain the simplicity of &is relationship.
ltt will be assumed &at all ~ b n t i a l sare etatiomry and statimary states will be considered. This m&es the work simpbr but is not necessary, h &is case
T b t is, f ~ g i= (m
= a * (p- eA)@ +pm*
+ e@3
It wilE be recalled wiLh ?k written as Eq, (9-5)and wi& a,/3 a s given in b c t u r e 10, the previous equaaon may be writbn aa two eqiaaona (11)-4*),
RELATIVISTXG W A V E EQUATI[QN
where, a s befora, r = ( p - BA) a& (11-5) for ilib gives
V
=-
51
e$. Simplifying a d solving Eq,
2m, a e n JCrb -- (v/c)O,, For this remon It i s noted &at if W and V a r e are sometimes referred to a s the large and small components of and @, respectively. S&st;itution of qb from Eq, (11-6)into Eq. (11-4)gives
+,
and, if W and V a r e negiected in. eompari~onto 2m, the result I s
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 e r r o r may be expected from use of the Pauli equation.
Using the results of Leetrrre 11, given by Eqs, (11-6) md (11-7), the lowe n e r w 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)
QUANTUM ELEGTRQDYNAMICS
This 8ubstitut;ion also allowe easier interpretation sf Eq, (12-2). Rewriting Eq. (12-21,
[l + (o ~)~/(8rn~]l (W- V) l1 + (cr. g
Then applying Eq. (12-.4f and dividing by 1 + (cr*w]2/(8m2),there results
- (1/8rn~,(@.r1'~
(W-V)X = (1/2m)(cr*~)\
The techniques c>E o p r a l o r a k e b r a 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
Then, since rr = ( p
- BA) - (AB - BA)A
- ~fl3c), and since
-
there result8 fwiGh cr. r = A and (W V) = B in the foregoiqj,
f a i n ~ eV x E
-
8B/at = 0 here), s o Eq, (12-5) can be expanded as WX = VX +- (f/ltm)(p -- eA).(p (11
- sA)x - (c?/rztnr)(rr*B)~ 1-31
(21
-(1/8m')(~*p~~x (4
+ ( e 2 / 8 m 2 ) ~-+-2~0 . ( p -eAf x (51
EfX
(12-6)
(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 it has a p p a r d before.
RELATIVISTIC WAVE EQUATION
Term (2) can be interpreted a s the kinetic e n e r a . Term (31, the Paull spin effect, ilr~just a s i t appears in the P m I equation. Term (4) is a relativistic correction to the kinetic energy. The correction. 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 interpretation cornides the part of term (6) given by a * ( p x E). In an inversesquare 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* 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 a t 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 a s the cmfficient of term ( 3 ) , a d again of terms (5) and (G), i.e., (e/2m)(1/4m2), A classical a r g m e n 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 i h a 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). +
u/r3,
-
Problems: (1) Apply Eq. (12-6) Lo the Wdrogen atom and correct the energy levels to f i r s t order. The r e s u l b 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 approamation. (3) Pauli" e d i f i e d eqwl;ion can be applied to neutrons a d protons. It ilts obtained by a d a n g a term for anomalous moments to the Rirac equation, thus
Multiplyiq by P, this may be written in the more familiar "LHrzmiltoniarr" 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
f a r protans, and a similar expree%sianfor neutronw, but Mth e = 0. (4) Equation (12-7) cm be used to i n b r p r e t electran-neutron scattering in an atom. M a t of the s c a 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 i s e t a a warre which i n b r f e r e s with nuclear scattering. For slow n e a r a m , $hi@eEeet i s experimentally ohserved. lit is interpreted by term (5) of Eq. (12-6) [as m d i f i e d 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 m p l i t u d e for neutron-elecdron scattering, However, when the effect was first discovered, it w m @Xplained by the a s s m p t i o n 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 i a t given by term (5). ~ X Z O W t b t
In order to interpret cd(R) as a potential, the a v e r w e p-cltential 5 i s defined a s that potential which. acting over a sphere of radius e2/mc2,
would p r d u c e the same effect, = - 1.91 35 eB/22hlM, show that the resuiting V agrees with Using exprimental results within the slat;ed accuracy, i.e., 4400 -11100 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~vttnlentto 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 = y p a P i s invariant under a Lorentz transformstiaa,) Xt is neeesgarg to put the probhility density and current into a fourdimensional form. In the smclal repreaenttstion, the prab&iUty deneity m4 current are given by
SOLUTION O F T H E D I R A C 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, y y , N = y y y x = -N. If N = iyg = iy,yyy, y, , then N = -W, y,y,y, = - i y p The EolloMw p r o p r t y tabs the place of the Hiermitian property s o u s e h 1 in nonrelativistie quantum mechanics:
For a free particle, there a r e no potentials, s o q u a a o n becomes
85. = 0
and the Dirac
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 st;ljln&rd
h the
repmsenktion, Muktiplicactlon by P c b g e s tke sign of the3 third ing \k* from a column vector to
a& fau&h components, in addition bo c
a raw vector,
Q U A N T U M ELEGTEEODYNAMIGS
58
.?b is
13four-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 i s 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 produces each component by -Qp. s o that the result of this operation on
+
s o that Eq. (13-5) becomes
Thus the maumed solution will be satisfactory if &a = mu. To rslmplify paticte mmes in the xy plane, s o writiw, it will now be assumed &at LhaLL
Under these conditions, d = y, E
-
- y,p.
By components, Eq. (13-7) becomes
(E - m)ui (E - m)uz @,
- (p, - ipY)u4= O
- (B, + i%fua - i p , ) ~-~(E I- m)u~
@, + ip,)ut
-.
(E +- m)u4
=Q
=@ =O
in standard representation
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~tatesthat the p, must be &men a s 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
80
which also lsada to codition (13-10). A more elegant way of obtzliaiw exactly the s m e condition is to s t m t directly with Eq. (23-7). Then, by multiplylw %is i~3quationby $ gives
m e former is the same cmdition as obt&ned b f o r e , and the l a t b r $8 8 trivial solution (no wave knction) Evidently there we two Xinearly i n d e p ~ d e n solutiom t of the free-parlick Dirac equation. Thia is ao because bsubstihtion of the a a s m e d solution, E q . (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:
QUANTUM ELECTRODYNAMICS
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 ydetermine the wave h e l i o n . A ia h w n , for exmple, that in the coordlnab sgstern in which the p a r t i ~ bier staHomry acre me WO passibb spin orlentaexjistence of two solutiom to the eigentiane. Mathexnatioally swmw, value. equation #U = mu implies the exiabnce of an a p r a t m &At commutes with 6. n i s operator will have to be discovered. Qbserve h a t yti a~ticommutes with $; &at is, = -hI.Aleo observe that m y o p r a t a r [JFJ will anticommuter;Mt;h $ if W ep = O, besause
Ths combination yswof &ese two antieommutfng operators is an o p r a l o r that Is, which commuke? with
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 o p r a t o r iySJW a r e rt: l. The sigaflicanw of the choice W W = -1 ia a s fof lows: In the ~ y s t e n rin which =p, = O and pl = E. Then tXle particle ier at rsct, p,
-
S O L U T I O N O F T H E D I R A C EQUATXOPJ
62
m u s , W * W -W * W = -1 or W W = 1, This staWs that in the e m r a n a t e systei?.min which the particle is at re&, W is an o r a n a r y vector fit has zero fowth conzwnent) Mth unit Iene~;tX1, M e n the particle moves in. the xy plme, chmsts p to be y,, s o the omrator quation for iygv k c o m e s g
Using relationships derived in b c t u r e 10, this becomes, for particle, t
8
stationary
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 positive r (s = + X ) o r negative fs = -I). We
Exercise: Show that the first of the wave functions, Eq. (13-11), i s S = -il 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~farmationof the wave function as in Eq. (10-12). If the electron is initially at re& w i a i& spin up o r down in the z direction, then the spinor for an electron m ~ v with i ~a veloeity v in the spatial direction k ies
[For normafization, s e e 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 n a m d i z e d 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 somea i l i~b
flowever, 9*4!transforms sfmifarly to the fourth component of a fourvector (it is the fotutth component oaf four-vector current}, s o 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 O F 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 s k t e ,
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 o b h i m d f a 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 var l o u ~initkl and f h a l s a t e s , s o Table 13-1 has b e n worked out.
QUANTUM ELECTRODYNAMICS
64
T m L E 15-1. Matrh Elements for Particle Moviw in the xy Plane
2m
F2Fi
- Pi+Pz-
0
YX
~ P X
FzPt+ + Pa- F i
o
YY
2py
-fF2~1+"("
7%
0
1
Yt
2E
fP2-F I
0 t
+
Pt * Pz-
0
-Px*F2 + Pz+Ft 0
SQLUTXQEJ O F T H E D I R A C E Q U A T I O N
IkzrnzMng cases: To obtain the case &ere 71 is a p o a i t r o ~at rest, Lhe table gives mg~a~~u~) if oxre p ~ t sFi = 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 a yr e 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 ~ to d cornpub the matrix elemen& i s %limplyto write a e m sut in matrix form and carry out the o~ratiions.l[n W e way the data in Table 13-1 were obtained. M a r 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 ( a y p ~ ) But also n&e thiat { B ~ Y U) , = m(O[ypU)
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 , But p, ia just a number,
BO
Y,
=Z
it f~ilowtlthat
md aince flu = 2m,by normalization
(ilyp~) ZP, mrt-hermor.~?, the general relation
is obtarSned. norn tM~3It i s
P r o b l m : U&%
SBESIZ why fjhe possible
normalization
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'
SOLUTION Q F THE DIRAC EQUATION
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 a s having no mead*. But lit waa found that &ere 858 two h n d w e n tal objections to tba excluaian of negative e m r m stsws, Tbe first ie Nysical, &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 trasitions into mgative e n e r a s t a b s , Heme if they were exoluded thias would be a coneradiction. The a e c o d 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 enx direction with spinup in the z direction fs= +X), Then at t = 0, a constant pokntfal A=A,(A, = A y = 0) is turned on a d at t = @ it i s turned I' off. Find the probability t h t the parLiele is in a negative energy s k & a t t = T. Answer:
srw stab moving in the
fialiabiliw of beiw in negative enerm eta@ at t = T
= A~/(A' + m5 sin' [(m2+ A$'/'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 e n e r g e s if accepted a s such form a n o w r continuum from E = -m to -m. Between +m and -m &are are no availhle aner,eif;Ylevels (see Fig, 14-1). Birac proposed the idea that ail the negative e n e r w levels a r e normalfy fiflsd, Explanations for the apparent obscurity af sueb a sett of electrons in negative enerp;y. statea, if It exists, usually cozltain a psychological a s w c t and a r e 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 a k e a r e 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 a s a positron and behaves like an electran with a positive charge.
QUANTUM ELECTRODYNAMICS
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 occupy a given state, it would be impoaaibls to fill alI. the nagaX-ive e n e r m states, It i s in tMs way &at the Dirac a e o r y 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 l a ~ s i c a equa~on l of motion
.
reversing the dlrectit-ion of p r a w r time s amounts to the s m e m reversing the s i c of Iha charge s o that the electron moving b a c h a r d in time would look like a. positron moving f a w a r d 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 s m m i n g the mplitudes over all possible trajectories between xl,LI and xz,ta, aseuming t k t the f;rajectoriea atlways moved forwzurd in time. Thr~teetrajeetortazs might a p w a r i n one anerenslion aa shown in Fig. 14-2. But ~ t the h new pdnt of view, a pssaible trajectory mi&t b as shown in Fig. 14-3. Imtzginlng oneself an o b ~ e r v e rmovtw 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:
SOLUTION O F THE DIRAC EQUATION
X2
X1
FIG. 14-2
-
t
--e
tr-
t,
-
h% t,-
tz
only the initial electron present the irtitial electron still present but ~omewherfs, else an eleclrtm-posritrcm pair is formed the initial electron, and newly arrived electron positron a r e preerent the positran meets with the injtial electron, both of Wlem amihilating, leaving only the previously creabd electran only one electron presexll
T o handle this idea quantum m e c h d c a l l y two rules must be followed:
10
Q U A N T U M ELECTRODPNAMZGS
1, Xn calculating matrix rslementa far gositrom, the gmif;i~w of t b inireversed, That is, for aia electron move tial and final wave firnctiom must UIez mato a firwe state ing forward i a time from a past state trix elemsnt i s
But moving backward in time, the electron proceedsfrom the matrix element for a positron is
+,
to , Y
so
2. If the energy E i s positive, Ulen e - ' ~ i's~ wave function of an eleotron with energy p& = E. E E is negative, e m B Pis' ~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 b e i w scattered between the states 91 and were &seussed in, the last lecture. m e s e are: @z states of pocaitive energy, interpreted a s 9g electron Case I. Both 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 p s s i b f e . T h e ~ eare: Case ZfX. The positive energy, @2 negative energy, i n t e r p e w d as Jrl in ''ppa~t," %2 positron in. ""past." Both states a r e in the paat, and no%ing in the futwre. T U s 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
FIG. 15-1
IIX
Gase TV
QUANTUM ELECTRODYNAMICS
72
The EOW cases can be d i % r m m e d 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 s n e r m slates. h common lan@we,the arrows point toward pctsitive or negative time according ta whether 16 is positive or negative, that iia, w h e a e r the state represented is that of an e l e ~ t r o no r 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 e n e r w El to E2 and if the wrt;urbation potential ihs taken proportional to exp(--iwt), tfien this perhrbation b r i w e in a positive e n e r w w . To s e e 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, s o that the only contributing energies a r e 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 w a i n only if Ez= El + w o r -Epas, = -Ef,, + W , s o that Er., = w + E,,.,; that is, the perturbation c a r r i e s 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 , includng a perturbation potential, i s written
where V i s the perbrbation potential and He is the u n m r b r b e d Hmiltonian, 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 +mbe
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@ QUANTUM ELEGTRODYNAMfGS
73
Note that the kernel is defined to be Q for t2 tr. It can be shown t b t W;@ satisfies the e p a t i o n
The propagation kernel Kv(2,1) glving a similar mplituds, but in the presence of the p e r b b a t i o n potential V, must satisfy the equation
XL can be shown thrtt
Kv can be compukd from the series
]In ease the complete H m i l t o d a n H = H@ + V is f d e p n d e n t of:time, md all the e stationary s t a h s #, of the system a r e 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 p a t i e u l a r form for the H a i l t o ~ a n the , Dirac equation can be written
, the kernel is the aoluaon, to the Defining the propalyatim kernel a s K ~ then 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 e w e , Eq. (15-G).] lMultiplying &is equation by B, a simpler Eom results:
The equation for a free particle is obtained simply by letting calfiw the free-psrtiete k e r e l H;, ,
= O, then
QUANTUM ELECTRODYNAMICS
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 s e r i e s of Eq. (15-61, s o can be expanded m
Note that the kernel is now a four-by-four matrix, s o 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 eentimeter per seeand for the particle to be scattered anee at the paint f l f .Thue the interpretation of Eq. (15-11) is completely analogms to that of Eq. (15-6).
P~oblenz:Show that ltAas 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 ~ gwhich the particle reversed is no l o w e r true. its motion in time are excluded, Pn. the present erne The existence and interpretation of the negative ener&y eigenvaluea of the Birac equation allows the interpretation and inclu~fonof such p.aells. TaMw t4 2 t8 implies the elciaten~eof v i s a & pa;irs. The section from t4 t a tS repreeellf;~the motion of a positron (secs Fig. 15-21" h a time-slationary A d d , if the wave hnctiom @, a r e k n o w for all the states of the sysbm, then may be defimd by
rr
-
C
neg, energies
I-i~,(tz
- tr)l +,(xz)&
(xl)
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. (15-9) i s ~~A(2,1)=
C
m
exp[-iE,(t2-tlll$n(~g)$n(~t)
pos. energies
t
C
'
exp [ - i ~ , ( t l - t l ) ~ # ~ ( x ~ ) ? ~ t2 (x~)
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 i e 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"%;takescontribute, On the sUler hmd, Eq. (15-13)does not have s o 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 r e 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
Q U A N T U M ELEGlPR0DUWAMPC:S
'76
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 b e 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 it 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 e x p c t e d that a relativirstie ganc3raXization of this would be
FIG. 16-1
FIG. 16-2
m s turns out to b incorrect, however. It i s not sdficient, in the re1at;idsrtic ease, to h o w just the wave hnction ~t an e w l i e 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 at later t h e s e mis e q r e s s i o n c m be generalized to a t a k that it is meassary to h o w 9(xltx) on zt faw-d-innensfonat surilaee, 6~urrounid;iqthe point xz,tz (see Fig. 16-3):
where We 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 . 1in) 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 scatteriw and Lbe free-particle wave hncticsn g juet a f k r s c a t t e r i ~ and , that it elinninabs any computation of the moti on aist ai free particle, The amplitude for transitfo~,ta first order,
mily
be written
QITAMTUM E L E C T R O D Y N A M I C S
oigdMes integration over time a s well as space). The second-order Germ would bbe written -(1/2) Jjg(4)eA(4)K,(4,3)e8((3)f(3) d ~ drc s 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 i s 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 2and final wave funetism we plane waves: f(1) = u ~ ~ - ~ P I ' ~g(2) = u2e-@zex (four-component wave funetion)
Thus, foy Eq. (I&-$),the first-order mplitude for transitim from state f to s t a b 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 F o w i e r 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 ) - ' / Mx /(density ~ of final states) (16-6)
This i s a result from time-dependent ~ r t u r b a t i o na e o r y , the only new factor is a normalizing factor (EM)-"vvhieh takes account for the fact t;hat the wave functions a r e 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 o r (Gyt u) = 1 (so N =: 1 for that e w e ) , The matrix element M, a s salculabd in t;ttis nzamer, is relativistieally i n v a r i a ~ and t 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@ltyof etabe, Crosa boaon. For the efsetson s c a t t e r h g problem under consideration,
s o the transition probability i s
where t h density ~ of f i w l states has been obtained in the following mamer: Density of states = hut
=
% = l
+ m', s o dp2/dE2 = E2/p2 and Density of states = ~ ~ ~ $ n / ( z r ) ~
When the incoming plane wave i s normalized to one p & i c 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 w t u r e of the time integral. in Eq. (16-5)],and
( m w ~ t u d eof final momentm equal to mamitude of i d t i a l momentum follows from El = E2),
Therefare, vt = p i p t .
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
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 i s 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
-
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
Q U A N T U M ELECTRODYNAMICS
82
a& the sum over n becomes an integral over p. The up is the spinor cor-
respondiw to momrsntm p, poaitive o r negative e n e r w and spin up o r down, a s appropriate, Then the propagation kernel for a f r e e particle i s , far t2> tl, 1 spins
f o r E, = + (p2 + The factor 1/(2zl3 is the density of states per unit volume of momentum space per cubic centFetar. The factor 1/2Ep a r i s e s 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 r s t for the case of t2 tl. We first calculate u p ~p for p ~ s i t i v eenergy, and p in the xy plane and spin up. Under these conditions
Note that U,;, is the opposite order to that usually encountered s o that the product i s a msttrix, not a scalar. T h t is,
by Wle w u a l rules for mzzt;dx multiplieatioxz. But (p,
+ ip,)(-p,
and the matrix becornens
+ ip,)
= -pZ = -E'
+ m2
PROBLEMS. IN QUAPJTUM E LEGTRQDYNANLfGS
By the same process, the result in the spin down ease ia
W
upup =
Q 0
E+m -p,-ipy P x - 4 ~ ~ -E+m O
0
0
0 0
(spin dam)
Q
11, may be verified easily dhat the sum of these malrieea for spin up and spin down ia represented by
In Ule genr~ralcase when g is in any direetion, it i s elear h t t b only chwe is an additioml h r m -p, y, . So, in geukeral, N
N
up
+ ( ~ ~ u ~ b sdown p i n= : : ~ i ; " " ) " r - ~ * ~ + m = : ] p l + m
The sign of the energy was not used in ctbhinlrrg thia r~38ultSO 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 bcomes
-
The appearance of p in the form E p = (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
Q U A N T U M ELECTRODYNAMICS
84
where
h this form only one integral insbad of f m r m e d 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, s o t b t putting It/ in place of t in the formula, for I+(L,x) makes it good for all 1, mis i n b g r a l 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 ~ ~ ' ~ 'is( am 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 transformatf 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 diagonat 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 8 h 0 m that the asymptotic form of I,(t.@ f o r large s i s proportional to e-lmS. W e n one% region of accessibility i s limibd to thc3 inside of the light cone, large s implies t 2 >> / 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 nk 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 limibd to regions within the lig& cone, another region included in this a s p g t o t i c a g p r o ~ m a t i o nis 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~formtionto momentum representation will ie facilihted by use of the inkgral f o m u l a
lilow
be made, This
The ir b r m 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 i n k g r a l relation above, d,(t,x) becomes
But E:
= p2 + m 2 s o this i s
= where p i s now a four-vector s o that d$ = dp4 dp, dp2 dp3, and p p p p . 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@simalin e e t i v e 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 o m r a t o r i ( i p K;" m) to IE,(t,x) gives the propagation kernel (here x = ~2 - x i )
recalling that iJV o p r a t i n g on exp l-i(p From the identity
X)]
is the same a s multiplying by
Ifi,
the kernel
e m also be written
By the same process used for X,(t,x), t b transform of K,(2,1) in mornenturn 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 manner. 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.
PROBLEMS IN QUANTUM ELECTRODYNAMICS
87
Therefore the transform of &is equation can fie w r i t k n down immediately:
a s before. The fact that Eq, (17-1) for K(2,1) has mare &an one solution is reflected 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 m l e &at @electsthe particular form we wmt i s &at m h considered as. b v i n g an infinibsimal rtemtive imagiwry m r t ,
-
Eighteenth Lecture
Since the p r o p m t i o n 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 yuseful 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 ~ ~Fourier l t r a n ~ f o m s . . To convert the potential, define
Then the inverse transform is
The function a(q) is interpreted as the amplitude that the ptential. conhim 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 s m c e part of the mamerttunn. The delta EuaeLion 6(q4) arise8 from the time demndenee of & ( X ) .
Q U A N T U M E LEGTRODYNAM1C8
88
Utrfx Elfamentta, An adivankge of momenttlxkt represenbtioxl i s the aimplieity of computing m a t r k 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 i n e dsnailogclusly 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 d e p n d on
lexp tip xl) exp (-iql = (2n)'b4(p
71,this
part of the inbgrrbl i s
xl) exp (-ipl
- qg - PI)
xl) d r l
(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 i n k m a l 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 f u e u o n [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 a s a free particle crf momanhm Xli. A t point 3, it is scattered by a photon of m o 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-
QUANTUM ELECTRODYNAMICS
90
tun &)l, Fimlly, it 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 $l and gi2 a r e io debrmined by d2 = -pjl --&.The law of conservatim of eneven, ergy requires p12= m', p22 = m'; but, since the intermediate state is a virtual stak, it is not necessary that (pl + di12 = m2. Since the operator X/(& + dl - m) may be resolved as (fit + dl + m)/((Pt + dl)2 -mZ], the imporl the degree to which the hnce of a virtual state is iaversely g r a p o ~ i o n at~ consemaaon law is violabd, m e rasults @ven in Eq8. (58-3') &ad (18-6) may h summrized by the following list of handy ruleat for computing the matrix element M = ( 5 2 ~:~ 1 ) 1.. An eketron in a virGual st_ab of momenl-um 6 contrib-uttls the amplitude, i/(jp5 m) to PS. 2, A wtential containing the momentum q contribuks the amplihde -ie$(q) to N. 3. All indeterminate momenta qi a r e 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 ~ em 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 ewl i t h tt p h ~ t i a only l once (Born appro&mtfort) Is often, odffeiently aceumte A f b r the matrix element is debrxnined, the probabilit;y of trawltioa per second is given by
-
-
.
P = 2n/(n N) J MX ~ (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, m o s e pdentiala a r e a k a applicable to the relativistic theory if the m l r i x elements a r e eompubd a s described in k e k r r e 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 o w 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 ~ in Eq. (19-11,giving centimeter by dropping the ( 2 ~ ) - ' /factor 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 s k t e s . This becomes part of the normalization factor RN, which conk i m a similstr factor for each electron in the initial and final @&Le%.
QUANTUM ELECTRODYNAMICS
92
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 t o the p r a ~ ~ t i vector. on Heam e * e = -1, and e q = 0.
MDUTION FROM A T O m The transition. prokbility 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 ~ mf o m u k far the Gomipt~n-effectc r a s s 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 A l p = alp exp (-iq! X) and for the outgoing photon take Az = exp (-iq2 * X). The light is polarized perp n d i c u h r to the direction af propgation (see Fig. 19-1). Thus,
et ql = O S
eg q g = O
91 41 = 9t2 = 0
and
g
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 m s t ,
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 e n e r m 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 t y m s 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 y w s of polsllrization:
QUANTUM ELECTRODYNAMICS
(4"
62 = Y z
(B') 4%= y, cos B
- y, sin @
Conservation of energy of xnornenbxn d i e h t e s that either the angle of the recoil electron @ or the angle a t which the s c a t b r e d photon comes off @ completely determines the remaining quantities. If the electron direction i s unrimportant, its 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 i w Eqs, f 19-31, (19-41, and (19-6a, c, d). This can be written ~ ( W-I02)
- cos 8)
This i s the well-hown formuh for the Campton shift in wavelength (or fieqaeney)
.
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 e m r g y inrterval) can h obdained. k m of t o h l energy E and total linear moment= g disintegrahs into a twoparticle fiml state,
Density of states = ( 2 n 1 - ~
dGI
- Ej(P0Pl)
(B-1)
where EI = enerlS;V of pdlrticle 1; E2 = energy of particle 2; pg = momenturn, of particle 1; dQt = solid angle, into ~ i c mrtiele h 1 comes oul;; m1 = mass of m r t i c l e 1; m2 = mass of gartiole 2; a& El + E2 = E, p1 + p2 = p. Another useful formula is in t e r m s of the f i m l energy of m r t i c b 1and its @l). It i8 azfmul;h $1 [instead of Density of states = ( 2 ~ ) - ' (E1E2/ /p/)dE1 d&
I N T E R A G T f Q N O F 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 ~ dQl E ~/(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 m s= 03:
Density of states = (2n)-'E2 lpzl daz pt2
PI
(D-6)
The Gonrpton e f k c t has a Wo-particle final state: takiw particle I to be photon 2 and mrticle 2 to be electron 2, from Eq, (D-l), Dens3ity af states = ( 2 v 3c~ $2
W?
day
Calculation of /M/'. Using the Compton relation Eq. (19-7) to eliminate B, this bcomes
Density of states = ( 2 ~( ) ~ ~ ~~dh2,,/mw w ~ ' 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
b a d i n g from right to left the factors in the matrix element a r e interprebd as follows: (a) The initial electron enhra wit&litude ul; (b) the electron i s first scattered by a potential (i.e., absorbs a photon); (c) b v i n g 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 @Lateug
.
Exercise: Write down the mtrix; element for h e secand process 5, The t o b l matrix element is lthe e r n of thte;se two. Ratf onalize these matrix elements and, u s i w 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~dto be
- d2- m)]$2ul) = -i4aeZ( GZS U ~ )
-i4%eZ{&$1[~/(61
The complete m a t r k element is the sum of these, s o that the eross section bcomes
The p r o b l a now is actually to compute the matrix elements for R and S, First R will be emsidered, Using the identity
the matrices may be removed from the denominator of R giviw
The demomimtor i s seen to be 2mwl from the following relations:
The matrix elements for the various spin and polarizatlorr eombimtion8 can be calculated straightforwardly from &is paint, But c e r h i n prelimimry manipulations wiZX reduce the h b a r involved. Using the identity
it i s seen t b t
But pl has only a time component &nd e1 only a space c o m p ~ m n st o pl * el = 0, Wealling that 61ul = mu$, it i s seen &at
and thig is the matrix element of the ffrst term of R. Xt i s also f i e nemtive of the matrix element of the last term of R, so R may be replaced by the
equivalexlC
QUANTUM ELECTRODYNAMICS
98
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 bansposing 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 = 2ez eel -. Pi$z. Thus, with qz and &at ql (el *qi = 01 and
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
- y,
-
COS 8 )
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 i w e 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 ~ } ' / ~ ( ; ~+(s)u~) R = -(l --
C08
-
@)iF1p2+ sin 6 Pz+ Fi
The result@for the other combinations of polarization and spin a r e obtained in the same m a m e r 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 68
~
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 ~ ef o m u l a s a r e not adeqmLe for circular polarization, m ~ is,t (iy, + y y ) , it i s seen thst because of the phasif gil were, for example, 1/a
INTERACTION O F PARTICLES W I T H LIGHT
l01
all the calculations must be ing represented by the imaginary part of carried out before squaring the xnatrh elements in order to get the proper intederence. Finally the cross section Tor scatterine;: w i l o r e s c r i b e d plane polarization af the incomim and outgoiw photons i s
This is the mein-Nfshim formula for polarized light, For unpolarized light this c r o s s section must be averaged over all polartzalions. It is noted t b t diagram eases s u c h 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 eorrespond to emission and reabsorlption of a third photon by the electron, such a s in Fig. 20-2,)
Twenty-first Leetgre
as f?;latn-Hishim Formula. In the "Thompson limit,'" << m. Tnea the electron picks up very little energy in recoil, and wi .:oz. This can be seen from the relation Dlscussiclg d
wi
h this limit, the Mlein-Nishim formula, e v e s
QUANTUM E L E C T R O D Y N A M I C S
I02
which i s the Rayleigh-Thompson s c a t t e r i w eroras section, Note that w i s still very large compared to the eigenvalcres of an aloxur, in accordance with our origiml assumptions f a r 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 r a d i a b s to give the s c a t b r e d radiation
& = - - 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 i s then (times per unit solid angle and p e r unit incident inknslty)
The customary fii 's sand c % s a y be mplaced in Eq. (2 2-11 a s follows
(o i s an a r e a o r lenglth squared):
m2= (rne/Ii)' = l e n e h aquared
Avemgw 0 ~ 8 rFolari~ations,It is often desired to have the s c a t k r i ~ c r o s s section for a beam r e g a d l e s s 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 ~beam f & has polar-ization of t p A. The probbilitiea (or c r o s s 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 ineoming 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 if type A) f 112 @robbility if t;ym B), This ia the situation for unpolarized incomiw beam, and gives
cr (averaged over = ( 2 / 2 X M t + GB') + (112XE314".t BW) pdarizations)
INTERACTION O F PARTICLES WITH LIGHT
103
E, on the other b n d , the polarization of the outgoing beam is measured (still with an u w o b r i z e d incorniw b a r n ) , its dependence on frequency mid ~ c a t t e raiw~l e is given by the ratio Probability of polarization type A' Probability of plarization t m e B'
(1/2)[AA8+ BA')
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 ~ c a t t e r e dthrough 90°.P
Tok1 %kcat@riw Cross Sscti~n.l3 the c r o s s section (averaged over polarizations) given in Eq. (21-3) is i n k g r a h d over the solid angle
the total c r o s s section for s c i a t k r i x 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 1to + 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,
QUANTUM ELECTRODYNAMICS
104
Thus CompLon scattering is a negligible effect, a t high frec;fueneie~,where pair produeaon b c o m e s 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 s e a t k r i n g 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 s h o r n 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= -(m~mentum of ton di i~ reversed, and, since particle 2 i s positron, positron), So write $1
= IE-y,
- p,.
Y)
where the emrgies E- and E, of the electron and pasitran a r e both posltive 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), s o the matrix element for this ineraction i s
1'1.16second possibility, fnbistinguishble from the f i r s 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 2wz) a
I + &12jZ
x (density af sta&s)
In a system where the electron is a t r e s t and the positron i s moving. The density of final s k t e s i s
Since p a ~ i c l e2 is a positron,
$2
gives gii'
Then
This reduces to
pf*
=$it+
Pfi
= -g$,
so the conservation law, Eq. (21-41,
QUANTUM E LECTRQDYNAMICS
106
Taking the velocity of the wsftron a 8 (p+j/E+ the Cross s e ~ a o ni s @
=(2~)
dQ1/12E- * 2 /p+/ 4(2nj3 m(E+ + m)] x
I&&$+ ~ , 1 2
From a comparison of the d h g r a m s , it is clear t k t the matrix elements f o r 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 w l . T k n the cro@ssection ie
in analom with the mein-Nishina formufa.
TXON FROM REST
-
The formula for positron-electron amihilation derived in betrxre 2 l div e q e s as the positron velocity approaches zero (cr l/v; this i~ true for other c r o s s 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~llthat the p m c e d i w cm@@ section ws for a n 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 e q u d 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 i s necessary to sum over solid angle (IdQ = 4n) and average over polarfzations (sin2 B = %/g), giving
INTERAGTXON OF P A R T I C L E S WITH LIGHT f f a c h 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 u s l w matrix elements f o r an electron and positron a t rest. S h w that only the singlet state (spins antiparallel) can disinbgrale inb two photn>ns. The triplet state d i s i a b g m h s 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 disintegrab inlo &see pholons (spins must bt? parallel), The following procedure is s u a e s b d : (L) s e t 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 f o 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) o b a i n the b f a l disintegration rate by i n k g r a t h g over photon spectrum and mgle; and (9) compare with Q r 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 g4 f o r 6 in the m a t r k elements for the Compton effect gives m = 0.
When an electron p m s e s through the Coulomb field of a nucleus it i s deflecMd, A s s o e k b d with this deflection i s an acceleration which, wcording to the classical theory, resutLs in radiation. Acceding ta qumtum efectrodymmics, them i s a certain probability t b t the incident electron will m a b a tmnsition t a a different electron s b b with a, photon emit%d, wkile in the field of t h nueleus, h a r a e t i o n with the field of t k 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 w k l e tmve l i w along in a vacuum. Figure 22- 1 shows the process and defims a w l e 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 bremsstrah;luag proeeaa can occur: (a) the electron interacts with the Coulomb field m d s u b q u e n t l y emits a photon, o r (b) the electron first emits a photon and then i n b r a e t e vvfth the: Cougomb field. The diagrams for these pracTA, Ore and S, L. Pawell, Phys. Rev., 76, 1696 (19.49).
essels a r e s h o r n 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 it was s b w n that the Fourier t r m s f a r m 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 W I T H L I G H T Since the nueleus i s to be considered infinlkly heavy,
Notice tfnat there i s a, spectrum of phoans; t b t is, the photon energy i s not d e b m i n e d (as i t was in the Compton effect, for emmple). Letting "S1Z = (G2MU$).
where the f i r s t term comes from Fig, 22-2a and the second term from Fig. 22-2b, The explaimtion of tkm factors in the f i r s t term, for example, is, reading from right to left, that an electron initially in state u l i s scattered by the Coulomb potential acquiring an additional momentum 5& , the electron moves a s a free w r t i c l e with momentum 6%+ @ until it emits a photon of polarization gd, We t h ~ nask: 1s the eletctron in sbtf? u29 F o r the Coulomb pokntial
(see Momenturn b p r e s e n b t i o n , Lecture 18) in a coordinate system in. which the nueleus does mt move, [ F o r pohntial other than Coulomb, use agprop r h t e v(Q), the Fourier transform of the space depndenee of the pohntiafi,] 1Rationalkinf;f the denomimlor of the m a t r k , f
The outgoing photon can h polarized in either of t w directions, and the inearning and ouQofng electron each have two possible spin sbhs. The varis 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 i v i n gthe m e i n - M i s ~ n aeross section in Eczeture 20, Nothiw new is involved, so we omit the delafls. M k r (l) summing over phoWn polariz a t i o n ~ ,(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$(4E1E2 -GJ2+ 2w2) 2w2 (p; sinZe2+ (E2 - P2 f208 @2)(Ei- P1 f2cls01)
sin2et) (22-5)
An apgratximale e q r e s s i o n with a simple inkrpretation in b r m s of the Coulomb elastic s e a t t e r i ~cross section can be obbined when the gboan energy is small (small compared da resst mass of a l e c l r o ~but Zarge compared to electron binbiw energies). WriMng tlnc; matrix (22-3) fn b r m s of 4 insbad of @
using the relationskipa gOPlz = -$z$ + 2~3* p2 , $14 =: -661 in the numerator, since it i s small, this becomes
4
+ Ze
PI,and neglecting
where use is made of the fact thatJhe ma_trix element of M between states u2 and u l is to km ealeulakd and uz$z= dtui = mule writbn The cross section f a r photon emission can then
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 seatteriw from momentum pi Lo pz,
XMTERACTXQN O F PARTXCLES W I T H L I G H T
115
Problem: Calculate the amplitude for emission of two low-energy photom by the for~g-ctingmeth&, 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 e x t m photon.
It. is eas;Cly s h o r n that a s f w l e 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 s o 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 bremss t m h l u ~ there , a r e 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 n k r a c b with the field of the nucleus; o r (b) t k p b t o n creates a pair and the p o ~ i t r o ninteracts with the field of the nucleus. The diagrams for these altfveisl are shown in Fig. 22-3. The arlrows in the diagram indicate that
FIG. 22-3
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 d h g r a m s 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 r s t s c a t k r e d by the Coulomb potential and then by the phown; in case the order of the events is reversed. The differernce b t w e e n pair prduction. and bremastrafilung, when the direction sf tim is taken into account, is {X) is a positron state (melectron traveliw h c k w a r d 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 w e d for thia process if 6%i s replaced by --$, and 4 by -4,
$l i s the positron momenlum and
m)
Q U A N T U M ELlEGTROTZcYNAMICS
122
The $+ is t b n the positron momentum and i s the momentum of the absorbed photon, The density of f i ~ skks l is different, of course, since the particles in the fiml 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- f o r P2 Pi
-P+
-6- for O2
-8, for 6-
E+
E- for -E, for E, -W for W
F i m r e 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 first arrives 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 o b k h e d from t k t for " p l g r i ~ e d ?eefeetrens ~ by summiw prokbilltles over fiml 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 flml spin states do not Isrterfere lad these i s equal prokbility 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 W I T H LLCWT
one needs @
--
1 2 spins I
C
C
I(u2~uill~
spins 2
mean8 the sum over final spin states for only one s i m of the
where 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 a c e o r k n e e with t b u s w l rule f o r m a l r h multiplieatioa, the fallowing i s true: ( G 2 ~ u (I U ) I ~ u 2=) 2 r n ( G 2 ~ ~ u 2 ) all Y
where A and B a r e any operators o r matrices, the 2m factor on the right a r i s e s 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 o k 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 posispins 1 tive eigenvalue s k h s . So consider
C
(;2~u,)(;l@f+m)~uz) = (:2~(dt+m)~u2)2m
all U,
But rV
ui(Iljf+m) = Q 8w
= ui(2m)
for negative eigenmlue slaks f o r positive eigenvalue states
s o the p m c e d i q sum also equals
C
G 2 ~ u l ) a m ( GBuz) f
spins 1
CaneeXliw the 2m factors, this gives
C
spins 1
(Gz~u!)(';l Buz) = (G~AMI+ m)Buz)
(pJi -t- m) is a l h d a proleetion o p e r a b r for obviaus reasons. Similarly it f"alX0~8t h t
QUANTUM ELECTRODYNAMICS
spins 2
all
U,
where X is again any matrtx. Remembering the normalization &u2 = 2m. it 18 seen that the last sum i s just the tmce o r spur of the matr"ur. (;plj2+ m)X. -t- m ia immakrfal, Nob that the o d e r of X and Fimlfiy, when m e w m t s
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 e n c o m b r e d matrices may km verffied easily:
It i s also true that the spur of the product of any add n u m b r of daggerad o p e r a b r s 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 h c h i q u e . The c r o s s secUon f o r pohrized e l e e t r o n ~was previously fomd to b
Therefore, since Eq, (23-31,
7, = y,,
the c r o s s section for anpolarized electrons is, by
The spur can bt3 evaluabd irnmedia&ty from Eq. (23-5) with m2 = m4 Anotht?r way is: Since ytdI = 2Ei ~ % y tit,i s Been t h t
-
-g4 = p14 = yt.
5;
O and
, Spur of this matrix i s Usirzg ra few af the formulas l i s k d p r e & ~ u ~ 1 ythe seen to be
- p1
But p%* p2 = E
a
h, p% * p2 =
4~~-+ 4m2+ Also rn2 = E'
- ,'p
cos 8, and Et = .E2,SO thi8 i s
cos B
so that finally the c r o s s section becomes
@mpok
= 1/2 ( Z ~ @ ' / Q[ ~ )8 + ~4p2 ~ (cos
@
- 111
where v2 = p Z / ~ Z .This is the same c r o s s 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 f o r a Codornb pobntlal it3
where Q i s the mamenbm t m m f e r r e d to the nucleus o r p% - p2
- q.
Q U A N T U M ELECTRODYNAMICS
1l6
Clearly V(Q) gets large a s Q gets small. The minimum value of Q oec u r s when all three moxrrenk a r e I l n d up (F&, 28-11: Pa
Pr
g
FIG,23-1
F o r very high energies E
m,
so t h t in this ease
-
-
O a s Er 00 . This ahows clearly why the From this i t is seen that Qdn c r o s s sections for pair production and brc3mhsstrahXuw go up with e m r m . From the integral emression for V(Q) it is seen t b t the main eontributlon do the inkgrai comes when R m 1/Q. So as Q becomes small the imp o d n t range of R gets I a e e . It i s in this way that screening af the Coufor a conhmgiakd proclamb field Ldcorszes effective. The value of ess can be estimabd from the farewinli?; formula, The atomic radius i s given roughly by a o ~ - ' / ', where a. i s the Bohr radius. Thus if
or, w h t i s the same,
then s e r e e n h g effect wtll be i m p o a n t , and vice versa for the uppsi* inequalities, E from this estimate s e r e e n i w would appear to b imporknl, one should use the screened Coulomb podential. It gives the result
where F(Q) is the a b m i e structure factor given by
and nm) i s the electron density as a funcUon of R,
XNTERACTXON O F 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 c r o s s section for ernksion czf a low-energy pll~tQacan IM approximakd as
where cfo i s the ecat%rfng cross sectlan (neglecting arnlssian). Now consider a n 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 p a h n t h l is present and assume t b t they do not interact with one another (see Fig, 24-2). The amplitude f o r electron a moving from 1 3, wMle electron b moves from 2 4 i s given the symbol 33;(3,4; U ) .If it 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 r s t by any measurement in aceo&nce with the Pauli principle. This diff e r s from the first caae by the i n t e r c h q e of w r t i c l e s k t w e s n 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 a r i s e s 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 b t w e e n 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, o r the previous, or any other similar occurrence is @ven by the a b ~ o l u t es 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 vacu r n ; because of the possiMllty of pair p r o d ~ e ~ o int ,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 e q a l 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 s q w r e s of amplitudes (that is, oxnittiw the Pv b e t o r ) a r e actwfliy rehtive probabilities for various occurrences 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 t o b f amplitu& can always be &rlved from one by inkrehanging p r o w r spaw vari a b 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,
QUANTUM E L E C T R O D Y N A M I C S
120
The nonrehtivisac Born appro&mation to the amplitude for a n interaetion 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 tBand t 6 , 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 conceivable 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 t g 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
exp(-iwX) dw/tr
&,(X) =
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 t 5 5 tg and t 5 =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
-
a relativistic invariant, the potential is &fining t2 r2 as e26, (gS,2). Another term h i c h must be included i s the magnetic int e r a d o n , 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 exp r e s s this in the equivalent form -(Be), * ( P c u ) ~ and , in & i notstt;ion ~ The~c:P '8 the retarded Coulomb potential. is proporZ;iomf to P come from the use of the relativistic b r n e l . Thus Lhe complete potentfal 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 f o r a photon to go from 5 6 (or from 6 5 depending on timing) i s b , ( s s , t ) Equation (24-2) can b taken a s another a?&bment of t b fundamen&l laws of qumtum electr&ynamics. (2) Show t h t
-
-
Thus, irz momentum space,
From the r e s u l b of the fasrt lecture, it is evident t b t the laws: of electrodymmies could b s h b d a s follows: (1) The amplitude to emit (or absorb) a p h b n ie e y p , and (2) the amplitude for a phtobn to go from 1 to 2 is 6 , (91,$), where
2
in momentum representation. It is interesting to note that 6 , (si, ) 18 the same as I, (sI8$), the quantity appearing in the derivation of the propagation kernel of a free particle, with m, t b particle mass, s e t e q w l to zero, A more direct comeetion with the MaweXl equations ean km seen by writing the wave equation. 0 klI, = 47r JP in momentum representation,
We now consider the eomeeti.on witfa the "rules" of quantum electrodynamics @ven in the second lecture. The amplitude f o r 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 1Lo 5, emits a phobn of pohrizaaon 6 and direction K, t b a 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 potarization 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 absorbiw 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 n b g r a b over photon direction, sum over possible photon palarizations, and inbgrate over t 5 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
C lexp [iK. (rs- r8)]exp [-iK(ts - te)J
= 4?re2
P
QUANTUM ELECTRODYNAMICS
f 24
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 evldent by using the Faurier t r a n s h r m
exp ( -iKlt/ ) =
i 1 2 i ~ / ( w~ K'+ i ~ ) exp ] (-iwt) d w / 2 r
so that the foregoing egwtioa bcomera
and cornwring this with the result of the last problem of Lect;ure 24 estabZisbs that the rules given in Leeturs 2 a r e consistent with relativistic electrodywmiics develop4 in the last lecture.
The theory wit1 now be used to ob&in the electron-electron s e a t t e r i w e r a s s section. The diagrams far the two idistlngulshablo proeesse s a r e s h a m in Fig. 25-2.
XNTERACTXON O F 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
and electron s h t e 3 Since electron s t a e 1, i s a plane wave of momentum i s a plane wave of momentum $$, it is clear that in m_omentum representation the spinor part of the f i r s t bracket wJll become (u$Ypus) and the spinor part of the second bracket will become (u4ypuz) Integration over rg and 76 produces thtt consermtion laws given at the bottom of tbe diagrams. Dropp 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 r s t term comes from diagram R, the second from diagram S , and the summation over g Is implied. h the eenter-of-mass s y s k m , the prob~bility of transition per second i s
(see h n s i t y of F i m l g a b s , L e c h r e 19). The nzethd of L e c b r e 23 can be used to average over initial spin states and sum over_final spin states. F o r ?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):
QUANTUM ELECTRODYNAMICS
where x = E ' / ~ ~This . i s called M"ci11edllerscattering (see Fig. 25-3). Problems: (I) Calculate positron-el e c t r scatteriw by the pre cediq m e t h d . (2) Y i d the cross section for a, p rneejon to produce a back-on electron, Assume t b 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 e m e c b d 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 c a electmdymmics, l longltudim1 waves can always be elimimbd in favor of tranwveree m v e s a& an i n s m h n e a u s Coulamb interaction. This is tha approach uaed by Fermi (see Lecture I), and it will now be demonstrrz*d that the sum over four polarizatiam is also equivalient to drmsveree w8ves but plus an fnsbnbneous Goulamb 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 r e 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 = i s zero in general (from the argument f a r gaugr;. hvariance). t Thus y, with the result y~ can be replaced by
-
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 first &mm represents a Coulomb ineraelfon white the second term conkins the i n b r -
action tbough transverse waves,
f
Ln our special ease, it is easy to eee directly, for emmple,
Discussion and Interpretation of Various "Correction"Ter Twenty -sixth Lectzlre In many pracesaes the b e b v i o r of electrons in the q u m t ~ m -electrodymmfc 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 f o r 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
128
representation, to
where q E {Q, Q) and Q i s the momentum e x c h w e d by the two electrons. I t follows that Also, since 4 =: &
-
From this identity it 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 first term, it C- be seen that it is the momexxtzlm repreaenktfon of a pure, i n s k n m e o u s 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 dfrectisn of Q , F o r 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 hcomes
but
It is recalled that u
E
, where
M,
is the large parfi and ub the small
part and that in the nonrelativistic approximation
QUANTUM ELECTRODYNAMICS
130
Also, since
it follows that (taken bt;ween positive energy h3%&s)
In free space trix is
];I
= p,
s o the x component, for example, of the f o r e w i w ma-
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 first t e r n s in each of the brackets represend currents due to motion of the electron trmsverse do Q and the second k r m s represent the transverse components of the n n w e t i c 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 d e s c r i b d 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 z e r s 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% t b 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 a w e , s o the correction is still small. It to redo the a d y s i s ~ v e r t h e l e s s .
ifs
necessary
It would a;ppesar t b t , since the electron md positron a r e distinguishable, e diagram, leaving as the the Paul i principle w u l d not require tfi i n b r e only one Fig. 26-2.
FIG. 26-2 But it is still p s s i b l e by the same p h e n a m e n a l ~ i w lreasoniing to conceive 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 existing 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 tmnsf o m h k g r a l , i s the same a s -(qX V ) * (qX V) exp (-iQ * X ) , w h r e V is the grad operatar. This device enabtes an integration by p a d s , whichgreatly simplifies t h process and the result, Thus, sigcr?:the trmsform of 1[a2 i s l/r, the coupling i s -(vt V) * (CQ V )(X/r), wMeh i s the classical energy f o r inbraetin& m q n e t i c dipoles,
Q U A N T U M ELECTRODVNAMXCS
V"j3
FIG, 26-3 From the p o h t 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 diff e 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 b a r n 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 arrangement. 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 ~ o b l e m :Chc-tck the mean life 1/8 1 0 % ~for two-phobn amihibtfon usiw the c r o s s seedion already computed and u s i w h,ydrogen wave functions with the reduced =ss for psftronium. F i e r e 26-2 contrlbuks the Coulomb v t e n t l a 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 f a r b t h s h t e s ) . Thus this amounts to a fine-structure sepa5 and 'S slates which can Ltc? shown to be 4.8 10"' W, ration of the ' 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 &mariisiw 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 s p l i t t i q due to the ('new amfhilatfon force*'' In order to calculate the term arising from Fig, 243-3, one needs to eompub
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 r e s t in the positroniurn), s o the result is just a n u r n h r , This means that t a k i w 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 lo5 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 i w 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).
134
QUANTUM ELECTRODYNAMICS
having a value of 2.0337 l@ Mc. The experimental value for the positronium fine atructum is 2,035& QA00Mc, 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 its experimenbf observation 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 h e a d i q of self-action of the electron, to be treated in more dehf2. later,
TWO-PHmON =GUNGE: POSITRONS
BETWEEN E UCTRONS A m / OR
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 e x c h w e d . Although
it k s not been necessary tw, eansider such high-order processes to secure agreement with e q e r i m e n t , it may b c a r n e necessary a s experimenal res u l t ~improve. The amplitude8 f o r the proceases may be written down easily but their calauhtion i s dgficult. The amplfiude for case EE in space-time represenbtion is, f o r 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 b t w e e n the two photons, It ia for this reason that the integral over gI a r i s e s in the e q r e s s i o n for the amplitude.
Twenty- seventh tectgtiure 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 i s 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,
136
QUANTUM ELECTRODYNAMICS
nondiagoml elements in the perturbation m a t r k will all be zr?ro. A diagonttl element i s one in which the resulting wave functions of a particle remain in the same eie;enstab, F o r time-idependent perturbations, it 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 can l 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 i s represented by the inbgral on the right side of Eq. (27-1). The represendation is not yet m equality, since c e r b i n normaliziq factors a r e different in the two expressions.
To obbin the correct equation proceed a s follows: First, it is clear that the probability of the occurrence depends only on the interval in space and time b t w e e n 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 it should be omitted in Eq, (21-2).
Third, the normalization of wave funetlone i s different for the two app r o a c h @ ~F, o r 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 e s h b l i s h an equality, eqressictn (27-3) must tctl3 divided by the normalizing inbgral of Eq, (27-4). The r e s u 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, s o 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 h k e n a s a ehmge in the mass of t b electron, from
Using this expression, and t r a u z s f o r m i ~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 i s
138
QUANTUM ELECTRODYNAMICS
(l/k2)c(k2) instead of just {l/k2), where c(k2) i s s o 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 ~ i w i f i e a n t l yfrom 6, only for small intervals. This i s clear from the fact that if the high-frequency components are removed from the Fourier e q a n e i o n of a function, only the sharl-range details a r e moetifiedl. In the present case the size of the interval over wKeh the function i s m d i f i e d can Be described roughtg a s foUows: Consider a large number, )lZ, and suppose that s o 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 a 2 i s sort of a "mean width" of f, . If a 2<< 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 , laws will be r a this measure is negligible. Thus, depending on A ~the found undfected over any practical dislanee,
Choosing h2>> m2, a practical (and general) representation of c(k2) i s
and the simple form i s 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 s o 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 o r 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 righthand side the (1;0) i s I o r 0, respeelively. 410 grove the first of these, n o k
?R, P, Feynman, Phys, Rev,, 76, 769 (1949); included in this volume. N o b that i n 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 i s 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 , t b r e foXfows direetly
Further differentiations give directly s u ~ c e s s i v eintegrals i n c l u d i ~more k factors in the numerator and higher powers of (k2 - 2p * k - A ) in the denamimtor .
SELF-ENERGY mTEaML WTH A% EXTERNAL PQTENTZAL Last time it was found that the self-energy of the electron is equivalent
"CORRECTION"
TERM3
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 i w Eq. (28-3) with the substitutions - X) for h , giving
xp f o r p and L ( l
The integral over L i s elernenbry and gives
4
f
I = -2(32n2 i)""'
dx (l;xp,) In [(l - x)A? + m2x2/m2x2]
When h2 >> m2, it is legitimate to neglect m2x 2 in the numerator [it i s true that when x m I, (X - x)h2 i s l z o t much larger than m"', but the Lnbrval over which this IS true i s so small, for h2 a m2, Chat the e r r o 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, i s 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 d e p e d s on h and hence camot be determined theoretically. One can imagine an experimenkl mass and a theoretical mass which am r e l a h d 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 s b b d , A. t h e 0 0 u s i w mtt, and
e 2 h c self -action
is equivalent to
a theory using m e p , 8 1 % ~ e21fic self -action minas 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 meXpi s exactly correct. When the electron i s not free, e 2 h c self-action is not q u f h equaX to the Am term and there i s 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 s e a t k r i n g 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 f a r this process, excluding the possibiXfty of selfaction, a r e a s follows: lfor Coulomb potential Q~) Pobntial: d(9) = yt ( B ~ z ~ / &(Q) 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~ertingthe 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,
QUANTUM ELECTRaDYNAM.ICS
144
clear physical interpretation, discussed la%r). Temporarily the k2 ander d4k 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 >> h Z m i n . 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 u s i w 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 2n
fn--
m
X min
4
+@tan@+=
where r = In ( A/m) + 9/4 - 2 in (m/Amja) and 4m2 sinZB = 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 approximated by
The
(46 - $4) can be writ&n out
But qp is the gradient operator so this can be written, in coordinate represerxlatian,
[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 r s 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, s o that
doss not depend on the cutoff N o b that this result [and (28-9)md (28-%@)l h, and hence h em now be &ken to be. infinity, f
I t has been shown that w b n a paXlf.i~lei s scattered by a ysobntial, the primary effect is that of g, and that for diagram I (Fig, 28-2) a carreetion term a r i s e s 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 f k e t of diagrams II and E1 (Fig. 28-2),
when considered along with the effect of the mass correction, i s 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 b e i w developed must conlain the e m e r i m e n h l 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, it is seen to behave like a potential. with zero momentum and Involves 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 s u b t m c k d from the results obhined from d k g r a m s X, IX, and EX (Fig, 28-2) done, For diagmnn IL the amplitude would appear to be
"CORRECTION"
TERMS
and for diagram IfYFig, 29-X),
But the part of the amplitude for diagram TX. (Fig 28-2) eonbined in the parentheses is just Awnul, s o that XI and 11' ssem to cancel, A similar result m d El" This is a n e r r o r , however, arising from applied for d i a p a m s the fact t k t both of these amplitudes a r e i d i n i k , owing to the factor 4 - m in the denomimtor. Hence their difference i s i n d e b m i m t e , But by suht r a c t i w them p r o p r l y 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~storywill have alwaya fnvolved a s e r i e s of scatterings, a s will its future, These scatterings will h conaidered as oce u r r i w a t long but finite time intervals, Xt will be sdficient to calculate the effect of seE-action and the mass: correction k t w e e n any t w of these scatb 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 e g a r d i q a, eerrrectf on, q u a l to that calculated for om of the i n b r v a l s b t w e e n scatterings, a s being assochtc3d with the pobntial a t eaeh s c a t e r i n g ( n u r n b r of intervals e q u d s number of s c a t t e r i q s ) . Then, considering a s i w l e s e a t t e r i w event a s here, this correction to the pobntial represents all the effects of diagrams II[, In, B*,and 111" F o r an electron which is not quite free, pa .:rn2 exactly, but insbad
QUANTUM ELECTRODYNAMICS
by the uncertainty principle, and T is the interval between scatterings. gince T is h r g e , 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 potent i a l ~at a and R (any two s c a t k r i n g ~ ) ,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 perturbat-ions I s
up to brms of order
E
With the perturbations of self -action and mass cor-
rection, 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 w u c b 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"
TERMS
s o that t b foregoing m a t r h becomes
The f i r s t and last t e r m s a r e identical, up to terms of order E , hence may be cancefed. The integral in the second term h8 already b e n . done essentially in eomputiq diagmxn I (Fig. 28-21, except bere replaces 6, &, and g&, s o that Ilf = - pfi = Q in this eaae and gives the result
It i s To this order in E the $% in the numerator may be replaced by PJOfs. = mu, also noted that since
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 correetion 2erm due to diagrams 11, fII, E ' , and 111' i s 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 " v a b l e m of qtrmtum eleetrodymmics, It i s a tsiczal problem which could as well a r i s e in nanrelativistic quantum mechanics if, for example, one c b s e same n o m e m 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 i s e h an "energy correction'9or the f r e e e l e e t r o ~amlogous ta the -ss correction hvotved here, Then in
computing the anrrplibde for a s c a 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 if 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 ising from self -=lion and rnaas correction i s +8bn6+-
e2
4 tan 28
28
sk + 7-8nm Mid - dd) -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 e r o s s 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 c r o s s section diverges 1ol;~arithicallya s h ,in .+ 0, and it i s this divergence wMch was formerly referred do a s the "i d r a r e d c a t a ~ t r o p h e . ~ ~ This result, however, a r i s e s from the physical fact t k t it 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 met3ssarily accompanied by radiation, h the theory of brehmsstraMuw, it was s h a m t k t the e r o s s 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 c r o s s section i s
- ~ ) ~ / 4 r nThus ~ . the probability of emitting any photon where sin2 B = between k = Q and k = K&, is
which diverges logarithmically, Therefore, tbe dilemma of the diverging s e a t b r i n g cross section actually a r i s e s from a s k i q an improper question: What i s the c b n e e of s c a 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 3 of scattering ~ t the h emission of no photon of energy g r e a b r t h F o r 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 s e a t k r i n g and emittlng no plboton, the chance of emitting one photon. of e n e r m below Q , and the c b n e e of two and more photons below K, (but these t e r m s a r e of order e4 and higher and heme a r e 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 w r i l i w C h n c e of scatbring and emitting no ptrotoo of energy > K&, + (terms inde-
of s r d e r e 4 ) terms independent
This does not depelld on hmin and hence resolves the " i d r a r e d 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 wavelen@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).
QUANTUM E L E C T R Q D U M A M f C S
152
af free electrons, The best way tn handle the problem i s to calculate everything in t e r m s 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["mW
]EX? DIFFICULTY
h s t e a d 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 l e s s 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 t o be s u b t m c k d Is
, neglect both K and the f i r s t two k2 in this We assume ko a pi o r p ~ and integral. Then using d = 2p, - $', the integral i s approximately
Then
This i s the brrn to b subtracted from expression (30-1). Usiw sin2 8 = q2/4m2, for small q, Eq. (30-4)becomes
Subtracting this f m m Eq. (30-l), also with q small, gives
The last k r m is [ln (M[/2b)+ (131/24)1.
EFFECT 024 4 9 ATOMIC ELEGTROPiT Consider the hydrogen atom with a potential V = e2/r and a wave function (R) exp (-iEo t) = qo(xI,) Take the wave function to be normalized in the conventional =mere The effect of tlre self-energy of the electram i s to s h g t the energy level by an amount
The f i r s t integml fst written down from Fig. 30-1. The second is the freeparticle effect a s noted in previous lectures. The kernel i s not well
FIG. 30-1
enough determined to make exact calculation of this integral p o s ~ l b l e ,An approximate ctt.lculation c m be made with the farm
- similar sum over negative energies for The photon propilt;l;ation kernel can b @ w a d e da s
t 2 < t,
6 , (S,,$) = 4% $exp [-ik(ts
- t,) + ik(xZ- xl)j d'k/2k(2~)-~ %
= 4n J'exp[+ik(tz -tl)
+ ik(xz-~$1d3k/2k(2n)" tz
t1
I f s i q these expressions, Eq. (30-6) becomes
=
C +fbp
(-iK * R)lon (E, + K
+D
dSk/4nk -
--a
- ~ ~ 1(aP - l exp (iK R)],@ *
exp (-iK .R)]@,(JE, I + U + E,)-~
[a exp (iK R)lno dSk/4nk
- (A m term)
(30-7)
This form implies the use of +* instead of and a4= 1, = a. Another approach to the motion of an electron in a hy$mgen a b m i s the following. Consider the electron a s a f r e e 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 o r d e r 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 r e a b o r b e d before? two interactions b t w e e n 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 cornpubd in Eq. (30-5) for small q @lus anomalous moment correction), This is
in momentum space, To tmnsfarm to ordimry space, use
P
s2
(4a2
- Q')
$
M
(aZ/at2
- v2)V
Thus the correction i s
This cormetion i s of greatest inzporbnce for the s state, since with a Goulomb potential V ~ = V 4nze26m), and only in the s states is different from 0 a t 1R = 0. The c b f e e of fq is dekrmined by the inequalities m ko >> R y d b r g . A = 137 Ryd. With such a h,the effect of photons of satisfactory value i s 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
m(R)
'VGQRRECTJON"
TERMS
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 r n i h r terms in the A m correction for a, free particle, Thus for only one situation must the effect be eompuhd. Xn all cases, s i ~ c ek 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 i s 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 i s added. Thus the sum i s 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 by e d a similar quantity in the Am term, (2) Longitudinal positive energy states (ap Q k/k) : As an exercise the reader may show
--
QUANTUM ELEC
X@ k/k)
(ik * R) lno
=:
(En - Eg)/k f e wf
and the contribution of these bmms summed over positiv
- (E, - ~
~ ) ~exp / k(ik~ R 1),, exp (-ik * R),o (E, + k
Writing H = p2/2m (V commutes with the exponent), this
t V, a d thus is also eaneele This brm is f n d e ~ n d e n of tfon. (3) Tramverse p0sitive energy s h t e s : Since ko is la size of the atom, the dipole approximation can km used. t in the sum of Eq, (30-7) &comes
w r iting (E,
+ k - E@)-' = l/k - (E,
- Ed/(En + k -
the term in l[k can be split off from the rest of the ink independent of V and tfrtzs canceled by the Am correcti averaging over direetiom,
in the nonrelativistic approximation. Thus the inbgral
U s h g the relation
I.Cf. H, Betbe, Ph.ys, Rev., 72, 339 (1@47)*
>> E,
and the fact that energy s b t e s is
- E@,one part of the sum over transverse positive
This cancels with the In
-E-
of Eq, (30-77, leaving the final correction as
m o m a l o ~ smoment correction
This sum has b e n carried out n u m e r i ~ a l l yto be compared with the observed Lamb shift,
GL~IED-1;WP PROCESSES, VACUUM PQLmIZATTON Another process which is still of f i r s t order in e Z has not been consid ered in the s c a t k ~ n by g a potential. h a b a d of the potential scattering the particle directly, it can do s o by f i r s 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 o r d e r in time c h g e d slightly, The amplitude for these processes i s i4m2
C
( ~ Z Y ~ ~ I ) $ J U spin states
rrn Y~ 1
1
(l
where u is the spinor pax% of the closed-loop wave function. The f i r s t parenthesis is the ampffbde for the electron to be scattered by the photon; 1/q2 i s the photon propagation factor; and the second parenthesis i s the amplitude for the closed-loop process which produces the photon. The expression i s i n h g r a h d over p &cause the amplitude for a positron of mom 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 c a r e of the processes of dlagram I and two s k t e s take c a r e of the proce s s e s of diagram XI. No projection opemtors a r e required, s o the method of spurs may be used directly to give
a form which eorrlains b t h X and E (so aa usual i t i s not necessary to make separate diagqams flor pmeesrses whose only difference is the order in time).
ELECTRODYNAMIC5
QUANTUM
158
This integral also diverges, but a phobn convergence factar, as used in the previous lectures, Is of no v d u e b e c a u s e now the integral i s over p, the momentum of the positron. 1x2 the intermediate step, The method which. has h e n . used to circumvent the divergence
[email protected] 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 + ~
~
"+ 1 / 1/91 3
~
~
(3 f -3)
?See R. P. Feynman, Phys. Rev,, 76, 769 (1949); included in this volume,
"CORRECTION"
where
TERMS
= 4m2 sin2 8, which, for small q, becomes
Notice that (GzYpul) = (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 d e s c r i b d , Thus the correction can be interpreted a s a small reduction in the effect of all potentials, and one can introduce an experimental charge eeXpand a theoretical c k r g e eth related by
where B @ ) = -(e2/31h) ln ( ~ / m ) ' , in a manner analogous to the mass correction d e s e r i b d 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 (v2V). This carreetion i s r e s p ~ n s i b l efor 21 Mc in the Lamb shift and the {ln fnn/2(E,- E@)] + (11/24)) term in (50-7" i s replaced by (ln (m/2 (E, Eo)j + (11/24) - (1/5)) . The 115 term i s due to the ""palirization of the vacuume?*
-
One possible process for the scattering of light, and an indiatinguisbble a l b r m t i v e , i s indicated by the diagrams in Pig. 32-22. The second diagram differs from the f i r s t only in the direction of the arrows of the electron lines, Reversing such a direction i s equivalent to c b n g i n g 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 s h o r n in Fig. 31-3 can take place. The amplitude for the process is
FIG, 31-3
plus five similar brms r e s u l t f q from permutiw the o d e r of phatans, This integml appears b d i v e ~ elogarfthmlcaly, But when all six a l b r m t l v e s 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 d u e n c e of a potential w%scalculated, The potential c m create and amfhilate pairs (a closed-low process) htweerz times ti and t2, The amplitude for the ereatlon and ihilation of one pair is (to f i r s t nonvanishing or-der)
The amplitude f a r the creation and annihilation for two pairs is a factor L f o r 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 i s a s s b m 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
P A W L 1 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 proce s s , whereas the first process is a two-loop p m c e ~ s s, o 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
,,-,,,= /c,I
P
= 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? s e r i e s must a l t e r a t e in sign in order t k t this prokbility b not grealter than unity, We have, therefore, two arguments a s to wfi?Jthe e q r e s s i o n 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 i n t e v r e t the Dfrac equation as we do unl e s s the electrow o k y Fermi-Dirac s k t t s t i e s , 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 f a r 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 =Pv to 1 pair Probgbility of vacuum to 2 pairs
PV
23
a 1 p&rs I ~ l ~ a i r I '
C IK~
a i pairs ~
IKI pair/'
QUANTUM E L E C T R B D Y N A M I C S
164
Now, the sum of these probabilities i s the prohbility of a vaeuum becoming m y tMng and tMs must bit? unity. Thus 1 = Pv [ l -F @rob, of 1pair) + @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 i s P,/K, @,a)/ The probability that the electron goes from a to b and one pair is produced is P, [ ~ + ( b , ai2 )I K ( ~ pair) i2, 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,
y the Dirac kernel that However, it is a p r o ~ d of
and an inconsiskncy reaults. The Inconsistency can bt; elirnimtad by assurning that elsctrona obey Fermi-Dirae s b t i s t i c s md a r e not independent, Und e r tbse c ; i r e u m ~ W c e sthe 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. F o r t k k e r m l sf the Klefnaorbon equation, i t turns out that the sign of the ineqmli+ in Eq, (31-IQ) is reversed, Therefore, f a r a spin-zero pa&fcle neither F e m i - D i r a c statistics nor independent particles a r e possfbfe. If the wave f"urmctiions a r e h k e n 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 ) erthances the chance t h t another i s created in the same s h t e , So the meinGordon q u a t i o n requf reh~Base statistiee , t i q to t r y to s h a r p n , these a r s m e n t s to show t b t t b [~,@,a)l' db and 1 is quantitatively exactly compensabd f o r 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,
123
R CALCULUS t0.
SUBIXHARY OF N U M R m C a 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
is the ddesity of final states p r unit enerw where is the square of the matrix range at enerm E and 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~iclet;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 ] , isZkN,=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 electronsit 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 nomaIization (@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 debnition 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 frrcter (2x1- here serves no usefut purpage so the cmventian will be abandoned.In thiogeetion d4khas itrruhalmeaning,
-fusing
&&&1dRlaa,.
*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.
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 c,r 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 nidentity, 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?@) is then obtained The correct numerical value of 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 -proximately 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~trunit volume and time, additional fators of i for mch virtual electron propapation, and --i for each peential or photon internetion, are necemry, Then, for enwu perturbation problems iRZ for the the enerw shift is the e w t e d value of E unperturbd starte in question dividd by the normalization consrant N , belonging to each particle comprising the unprturbed state. The author has profitcsd from diseussio~s with M, Pahkin and L, Brown,
-
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PMVSfCAL REVIEW
VOLUME
76,
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~retrttianof the solutions of the Dlrac equation. It is 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 ysuch , a wave carrebponds to a positran appr-hin$ the potential and annihilating the electron. A particle moving forward in d a e (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
NlfWBER 6
SEPTEMBER
15.
1949
in time (po~trona 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~arionis 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 t h a r y of bletj is proved in arm ~pacia;.
1, INTRQDUCBON as a h a l e rather than breakhg it up into its pieces. & the first of ~t of papers dealing with the I t is as though a bambardier wnglow over a road solution of problems in quantum electrdyna&. 'ud"n'Y three an' it '8 when two of The main prinGpb isto 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"". over-aa spm'-tim' Point of v'ew 'ads to cont;he nrotia,sn ejectrons and porjitrons in given pwr We consider the interactions siderable simplifiation in m n y problems. One can take patentiais, into account a t the same time process@ which ordiof &eae particles, that is, quantum elEtrdmamics, of chargm in a fixed polential is usually n a ~ l ywould have to be considered separately. f i r The of second quantization of the example, when cansidering the scattering of an electron tr
THIS
~~
tude for each space-the path availab1e.l In view of the fact that in cltassical physics positrons could be viewed as electrons procding along world lines toward the pSt (reference 7, the was lnade removes in the relativistic case, tbe restriction that the paths must proceed always in one direction in time. It was dkc o v e d that the results could be even more easily understoob frorn a more familiar physical viewpoint, that of sfattered waves, This viewpoint is the one used 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 Schrifidingerquation, 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 ~ not s to be conistent unless the electronsobey 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
(where we write 1 fox xi, tl and 2 for X*, la) in t h i case 1)=x(bm(xa>@$(xz) exp(- iE,(ba-ti)), (3) L
for le>lI.TiVe shall find it convenient for Is<&r to define =Q (2) ia not valid tr). then readily shown that in general K can be defined by tht solution of
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 SCHRI)DTNGBR% EQUATEOR visualize. We desire eventurtlly to d k u s quantum We begin by a brief discussian of the rehtion of the electrodpamics from this point of view. non-rektivistic wave eqmtion to its solution. The ideas To gain a greater familiarity with the K function and will then be extended to rehtivistic particles, satisfying the point of view it suggests, we consider a simple Dirac's quation, and Plslaitfy in the succading paper to perturbation problem. Imagine we have a partick in hteracthg rehtivistic particles, that is, quantum a weak potential U(x, 11, a function of psition and ekctrodynamics. time. We wish to cafcufate K@, 1) if U ditfers from The Schriidinger equation zero only for 1 between 9 and to, We shall expand X in i n c ~ e s h gpowers of U : X(2, 1 ) s Ro(2,l)+K(lf(2, 1)+K(2)f2, l)+ * - .. (6) dwribes the change in the wave function J, in an infinitminzaf t h e At as due to tbe aperation of an To zero order in U, IC is that for a, free prtkfe, Ko(2, l),' I), first conoprator exp(-iHdl). One can ask also, if $(xi, 11) is To study the first order correction K""f(Z, tlze wave function at x, at time tl, what is the wave sider the case that U digers from zero only for the between s m e t b e t a infinitesimal time interval function a t t h e tz>&7 I t can always be written as and la;+dta(lg
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 since from i 8 to tl the particle is free. For the short ~ e r a E t y . 1If II is a cnnst;mt operator having eigen- intemaf &a we solve (I) as values B,,, eigedunctions 4, M, that $(X, Ir) can be ex- $(ss fa+ bla) =exp(- iEhla)J"(~, fa) pand& C, C n + n ( ~ ) , then $(X, 2%) =ern(-.iE,(tz- h)) = (1-iH,&aiUd13rt(x,IS), XC,rb,(x). Since C,= J@,*(xjarl)Jr(xl, E1)C%72gP one fiads 4 For 8, non-relativistle free particle, where S$,= enp(ip.xf, 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-rampiesof the general relabon af $pipin and stortislia d d a d by W,h&,Phys. Rev. 58, ?E6 (1Mj.
E, .IpP/2nr, (31 &v@, as is weiI b w n Ke(z, 1)
-1e 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
for #*>t~,and KB=O fw k
exp(#dn(xl-
XI)*(~~-~I)-~)
where we put 8-;Be+U, No being the Hamiltonian of a free prdcle, Thus $(X, l i ~ i b t , )digers from what it would be if the potential were zero (namety (1-iHobta)$(x, l a ) ) by the extra piece
T
1:
-
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 $(X.,
t.)=
S K . ( X ~ . 1%;X,, laCbl.)$&~,t.+d.ldh%,
s~ha (01
OR=,@
19) fbi
CI(1DER.
E@(@#
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~fitudefar a free prtide rarting at point 3 the change in the wave function at 2 brought about by the potenthi is (suhtitutc (7) into (8) and f8) into cm*, (Eq, (9)). In (b) is ilfustratd the secmd order r w a s (Eq !10)),, the waves mitered I 3 are %atfeteda a i n at P H ~ W the equation for $(X*,In)): ever In Dirae one-electron theory K44 3) wenld rqrmat elcc-
2
~$(2)
iSK0(2,
$g:r:p; ~ & a ~ ~ ~ & ~ ~ "
troni both of poGiive and of nmlive inergkm proceding from
3) (1(3)Ko(3,2) ~ ( l ) @ ~ , @ ~ & ~3K,(4,3), , to 4 %isE%. is2,mrdied by chmdng 8 different ~atteringkernd
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 ccswe11 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 tzS11. 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 al t 2 from l can be considered as the sum of the amplitudes for variaus alternative routes. It 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}). It goes f r ~ m1 to 5 ( K @ @I)), , gets s c a t t e d 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
This urn be radily verifid directfy fram ( I f just as (9)
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 ( t h e s e} A, A , isi A,. The four tnat~ces@a,B tan be comidered m trmforming: 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 satisfyr p 7 * 3 - . ~ p ~226, f i, = where &M= -f- l, 4, Note that ab+ba= 2ct-b and that &=G@@, a pure number. The sym'boI a/8z, will m a n a/a6 for p ~ 1 . 4and ~ a/az, -a/ay, a/& for a = l, 2,3. Gall V= y,a/d+= Ba/at-t-isa" V. We shall im@ne
and S,,=
= @ . ais
-
-
We are rintpky wlving by suamive apprdmaticns an in-1 quation (deducible dimctly from (1) WE& l j l - H ~ + U and with R=&@>,
(8
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
expmion of the intern1 equation
l])
K+cA'(Z,1)
wKch it a h satbges. We would now expet to choo*, for the spttciaf solution of (12), II+mXe whem Ke(2,I) w n a w for 12
& is given by (3) where 4, and E, are the eigenfwctians and energy valum of a parGcle satisf h BkacJsequation, and @*, is by $n. The formuk arising from this choice, however, s&er from the drawback that they apply to the one electron keta W D W P OIIMB, LQ, (84) theory of Dirac rather than to the hob theory of the F%@ 2. The D i m equsrtien wrmits anaiher mlutien K+(2 1) witran. For example, consider its in Fig. 1(a) an if oat k n ~ d e r tbat s wava ~~ibttemd by the ptential can electron after kkg z a t t e r d by a potential io a small M w m d a in drne as ia Fig. 2 (a). This is intcpreted in the wand ays (b) (c) by nortiag that there is new the pad- region 3 of space t h e . The one elecbon orda rwM t y Pcf of virtuadlpai: Probuction at 4 Ihe positron goin to 3 (as does (3) with K+= KO)that the m t b r d ampEtude to b. aoi&tied. ?his o n be pirrvd as dmilar to o r $ l i ~ ~ yat another poitlt 2 will p r o c d toward mitive t h e s w t t e fb) a m t that the ektron ts mttclled bwkwwds in time horn J to 4 wavm sfatkred mm 3 to in (4 r rmat with both positive and nqatlve energh, that is with tht mibiitit 2 a arrivfngat 3 fram 2' and a n z k a n g k t h p i t i v e and negative rabs of c k w e of phase, PITo the dmtron &mX. %%is view is proved equlvdent to h k h t y : electram tmveLing bekwards in time are rmmizad m etrons. wave 19 mtLerd to times previous. to the time of mtkring. n r nare just the prowrtim of R0(2,3). OR the other haad, according to the p i t r a n herdter, purely for relativistic convenience, that +,* neptive e n e w stittes are not availabk to the electron in (3) is rephced by its d j o k t &= , cbn*fl. Thus the I)& equation for a partick, mms m,in sn after the mtteriw, Therefore the choice K+=Ko is unsa&facby. But there are other mlutions of (12). exteml field A =A ,y, is 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 . this new solu~onmmt =My (12) for md (G determining the propgation of a free stales o ~ l yNow ilU tima in orda that the rwrewnk&a be cornpieb. prticle bwoma I t must *erefore dBer from the old wlution K@by a (iVa-m>K+(2, 1)= i6(2, l), f12) mlution of the homogenaus Dirsc eqwGon. It is c k the hdex 2 on 17% indiating dgerentiation with rmpct from &e deftnition t h t the difference K@-K+ is the to the coordhtes sss which are reprewated as 2 in 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 IE+(2,1) and W , 1). The function 11=,(2,1) is d&ed in the a b n m of a homogenmus Dhae equa~onfor all t i m s and m a t field. If a ptentiilf A is acting a s h i k r function, ay &erefore h rreprwnteci by the =me sum over nega~ve O S k t cam, K+(h",(2,1)m be d e h d , It differs from K+(2,1) by a e n e w s k t a aka for Ca
TL
m,
K+")(Z, 1) = -
~SK+(Z,
3)d(3)Kt(3, i)drr, (13) R+(2$l ) = x p o ~a. 4n(2)6n(1)
repreen~ngthe amplitude to go from 1 to 3 as a free particle, get mttered there by the p & n t M (now the matrix A(3) ins-4 of U(3))and coatlnue to 2 as free. is order mmction, italogom to
X+m(Z, 1)
--
ssKI(2,4IA(4) M+(4,3)A(3)15+(3, l)dzdrs, (14)
and so on. In generat K+") ~ati$fies ( l )
,
('5)
and the succmive t e r n (131, (14) are the power =rim
for t2> 18 Xexp(-iB,(tg-lll) 4i)nf2i(i;n(l) X~p(-iE~@a-h)) for It
-Crvso
lf 7)
Ifih koim of K+ ~atiom (13) and (14) wiif now give results quivdent to tfiose of the 6 t m n halie a w ~ , That (141, for a m p l e , is the mrrHt mond order or&&& a t 1 e ~ r e & o nfor h & g at 2 an according to the msitron t h m q m y be wen ar, follows (Fig. 2). & s m e as a speciat w m p b h t tz3El and that potential vani*la a r e p t in ktcrml f z - h &at E4 and tr both lie betwan tt and k First s u p p 4> t s (F&. 2(b)). Then (since is> t l )
"THEORY OF
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 yin 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
&
With ehi inteqretation real pair grduction is aho dmribed carretly (W Fig. 3). For example in (13) if the equatbn gives the amplitude that if a t time $1 one electron is present at 1, then at time just one ctl~tronwilt 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= tIta>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, tl
where d8iVr is the volunte element of the c l a d 3dirmensioml surf;tcseof a re&on of space time contaking
Fto. 3. Several dilferent prwcan be demibed by the =me form& de ndiag on the time relations of the variables 11, It, Thus I)/* is the probattUity that: (a) An electron at 1 will be ~catteredat 2 (and 00 other psirs form in vacuum). (b Electron at 1 and pasitmu at 2 annihilate leaving noFmg. 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)
P,]GA~(~,
(3
paint 2, and N(1) h N,(l)r, where N,(I) is the i ~ w e ~ dtivistic calculations, om be removed irs follows. Instead drawn unit nomaI to the s d a c e a t the paint X. That of dehing a s k t e by the wave function f ( x ) , which it is, the wave function #(2) (m this case for a free par- h a at a given time tt=O, we define the state by the ticle) is determind at any point inside a four-dimen- function $(l) of four varirrblm XI, t.1 Ilvhi& is a satution sbnal region if its values on the surface of that re@on of the free particle e q u a t h for all 4 and is f(xr) for are sp&ed, k=Q. The h f state is m e w i ~defined by a functian To intewret this, cornider the case that the 3-surfam g(2) over-all s p ~ e - t h e Then . aur sudace integrah can comkts m n t k l l y of all space at =me time say t = O be p r f o r m d since JK+(3, l)@j[xl)@x~= f(3) and previous to h, and of aU s p c e a t the time T> t g . The JQ(s;~)B@x&+(2,3) =#(3), There results q l h d e r 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 i n t e p l now biryq aver-aU swe-time, The transithe inward &awn norm& N will be B and -8, tion amplitude to wwnd order (from (14)) is
where fr...@, 2'. positive energy (elmtron) comwnenls in $(l) contribute to the first integral and only nwtive energy (wsitronf com~nen:ntf of @(X? the =con&, That is, the amplitude for finding a charge a t 2 is deter&& both by the amplitude for Gnfmdinh: an electron preriotls to the measurement and by the amplitude for h d h g a positron after the measurement. migbt be inteqreted as mmning that even in a problem involving but one char@ the amplitude for findkg the charge a t 2 is not determined when the only thing known in the amplitude far finding an electron (or a positron) at an earlier time. There may have been 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~itronin the future. We clan a h obhin exprwions for transirion amplitudes, 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 The amplitude for h&mg &e electron anywhere after l= 0 it; given by (19) with $(l) r q h d by f ( x ) , the second in&@ait vanwing. Rence, the transition element to h d it in slate g($ is, in anatqy to (S), just
for the mticXe a r ~ * a~t 1 with am~litude{(l) is scatter4 (A(z)), prsg&m to 2, [ ~ + \ 2 , I)), ihd is scattered again (A(2)), and we then ask for the amplitude that it in slate g(2), E g(2) is a negative energy state we are sol+@ a problem of annihilation of electron in 1(1), positron in g ( ~ )ek, , f;Ve have h e n emphasizing scattering problem, but be motion in a fix& V; say in a hydrogen atom, can dealt ~f it is first view& a %atrering prablem can ask for the amplitude, dra(X), that an efwtran with original free wave function was scattered K times in the potential. V either famard or backward in time to arrive a ~ 1. t Then gfter one more scattering the
An equation for the teal am~htude
for a r ~ G n gat 1 either dirwtly or after any numhr of m t t h n e is ob~sinedby summing (24) over all K from 0 to CO ; $(2)= 4 . ( 2 l - i b ( z ,
I)V(l)+(l)drx.
Viewed as a s M y state problem we w y wish, for emmpie, to find &at initial condition 4s (or better just c of This is the $1 which l& to a p ~ d i motion mast p m c t i d y done, of coumt by mlving the Dirac equation8 (iV-m>+(1)= v(l)#(l), (26) deducd from (25) by bpmting on both sides by iVa- m, athg the h, and using (12). This illustrates fhe rehtion b e t w m tht? p o h b of view. For m n y problem the total potenhl A+ V may be split conveniently into a h& one, V, and another, A, comiderd as a perturktioa, If K+") is de&& as in
+.
s i n e g* = #B. If a potenthi acts sommhere in the internal betwan O and T; K+ i6 repked by K+cA).Thus the first order e g s t on tbe transition ampEtude is, from (131,
-iS@(xi)~x+(z. 314(3)K+(3*I)@j(xr)hlhr.
(21)
Eqrwions such as this ean be simplifid and the which are inconvenient for rela;dsurfoce htwbls,
(25)
POSITRONS
755
(t6) with V for A, eqressions such as (23) are valid and u%ful with K+ replaced by and the functions f(t), g(2) reQl"ced by solutions for all space and time of the Dlrac Eq. (26) in the potential V (rather than free particle wave functions)
We wish next to consider the case that there are two (or more) distinct charges (in addition to pairs they may prduce IIIvirtual states), In a succeding paper w e d k w s the interaction between such charges. Here we amurnr: that they do not interact. In this case each particle behrlves independendy of the other. lnJe can expect &at if we have tvvo particles a and b, the ampfitude that particle a goes from x l at Is, to xa at while b gaes from X$ at t* to xp at t q 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~clesare 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. is clmr. There k l The generaEzation to ~ v e r a particles 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 s u p p w t4
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 reisstates 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. FEVNMAN
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 camplete and there are no divergence problems. Reai procemes are completely indepndent of what gaes on in the vacuum. When we c a m , 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 %curing in the vacuum are analyzeil in the next section, in which an independent method of abtsining P, is discussed,
h ddiition to these si_ngle loops wa have the posrsibilily 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 b p 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 contribuAn alternative way of obtaining absolute amplitudes tion to c, of single imps h -L is a coweqnence of the is to multiply aU amplituctes by C,, the vacuum to Pauli principle. For exawler consider a. situation in vacuum ampLitude, that. is, the absolute amplitude that which two pairs of prticfm are creatd. Then these there be no particles both initially and finally, We can wirs later destroy themselves m that we have two wsume C,= l if no potential is present during the h o p , The elwtrons ~ u M at , a given t h e , be interinternal, and otherwise we compute it as follows. It ckanged fanning a kind of figure eight which is a single dsem from unity because, for example, a pair could be Emp. The fact that the interchange must change the created which eventually annihilates itself q a h . Such sign of the contribution requires that the terms in C, a path would appear as a c l o d loop on s space-time apwar with alternate signs, (The exclusion principle k d i q a m , The sum of the amplitudes resulting from all also rapnsibfe in in similar way for the fact that the such dngle clo*d loops we tall, L, To a first approxima- amplitude for a pair creation is --K+ rather than +K+.) tion L is Syxnmebical statistks would l e d to The quantity L has an S n i t e imaginav part (from ',higher orders are finite). We will &&cmthis in n the succw&g For a pair could be created sily a t 1, the electson and connection with vacuum p h r h ~ o in p i b o a could both go an to 2 and there annihilate. paper. This h a no e@eton the noorrnalizstion c o ~ a t Tbe spur, Sp, is taken since one has to s m over aU for the probability that a vacuum remin vacuum is paible spins for the pair, The factor 8 arises from the given by P,= IG,j2=eq(-Z*reat part of L), fact that the =me loop could be comidered as s h d n g at either potential, and the minus sign r m l t s since the from (30). This value agrea with the one calculatrxf interactors are each iA. The next order term would beB directly by ~normaiizingprobbitities, The real part of tappears 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 consquentIy a value of P, greakr than unity which & p p = meaninglm ilthe quantities are interpreted as we lzave etc. The sum of aH such terms givm &.la 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? dkinis 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+~urwmh c u s d here for tbe l3irac eftf~trons.How this is done is its negative h m with &R odd number ~f otentisl intemtors d k u d in more detail in the succding p p r . The give sera ~by&catlyihis is &mu% faf each Lop the eloctmn a n real part of L corn= out nqative for this equation so 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 that in this case Bose shtktics a p w r to be requird it k k v e lib a posiitive chr& thus chmging the 54- of erreh for consistency.' 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. in any of the n poten(ials. 3% result after m m i n g over n by (131, (14) and using (15) is 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 pertwhtian 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 The term K&, I) 8etuay inkmte*jto are. L;""
-
757
POSITRONS
P<#, Bl@fis the fiXa&ek function and 8(@) is tfia 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 transform r is very simple, nameIy (ifi*(P-m)-Z that
= B ' ~ ~ - P ~ x ~ = P P ~ * S - P ? " ~ ~f i C f l ," P l r ~ C ,
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 inte~al
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 e n e r g (p,) and momentum (pLxB) spzlce L@-m and the t m e f o m of 6(2, 1) is a constant, The reciprod matrix and we select the component of momentum q = f t ~ - P ~ . for &-.1)-%n be interpreted as &+nt)w--&)-l The second order term (23) is the matrix element p-m*= (p-m)CpT"E) is a pure n m h r not involving between and un of 7 ntstrices. Eence d one wishes one can write
since the efectron of momentum pi may pick up q from the wtentil, a(q), propgate with momeatm jh+q is not a mac oprator but a functbn giitisfyiw (factor (ar+g-m)-" until it is m t & r d again by the potential, ~ @ s - P r - a ) , p i c b g up the remking momentum, #*-Pt-q, to bring the to?d to p%.Since all values of q are possible, one integrates over q. (a/axg,)(a/dx~~). where The irmtqrals (31;) and (32) are not yet completely The% =me matrices apply directly to positron prob d e h d for there acre poles in the in&@&& when lems, for If the time component of, say, is negadve p--ta2==0, w e a n define how these poles are to h the state represents a prsitron of four-momentum -#z, is an elecevalated by the rule that m s's tm&ed lo hawe an and we are desribing pair prcxluction if z ' ~ j s i & s i dn-egrstipte imgirsa*y p&. That is m, is re- tron, i.e., has p i t i v e time component, etc. T'he p b ~ b i l i t yof an event w h m mtrix element is 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 (z-i@lst) is proprtionaf to the absolute square. This iategrating on $4 first. If we call: E=+(mz$-$ae m y aIso be writ&n ( @ E @ u ~ ) ( ~ % Mwhere u I ) , .@ is M +pf+paa)c then the integals involve P4 essenthlly as with the operators written in op ite order a d exglicir J'exp(-z'~~(tg-Ir)]dpr(p*z- &?)-l which has poles at appearance of i changed to B times the complex p,= +E and pc= -E. The rwhcement of nr by m-z"B conjugate transpose of @M).For m a y problem we are m a n s &at E has a s m l l negative imaginary part; the not wnceraed about the spin of the hnal sate. Then we fust pole is blow, the sewnrf above the real axis, Mow mn slum the probabifity over the two 4 . l ~corresponding if h-tr>O the cxrrntour can k completed around the to the two spin directions, This i s not a complete set bewmicircie Lebelow the real axis thus gi.ving a residue from a u s e $2 has another eigenvalue, -m. To p r ~ sumt (?E)-%q(-iE(1g-t1f), If m i q over J 1 s b t m we can i m r t the projection operator the $4- +E pole, or 1%-tl spin with momentum pi will make transition t o h . The I+(r,t ) = (4a)-V(sP)+ (rnI8~s)Rt@~(ms),(34) mpresious are all valid for positrons when $'S with a U the --ib is kept with m bere tao the furzc~enI+ a p p r d e s where a=+(@-X*)# for P>xa and S= -i(#-P)b for rare for inhi& p l i v e and nepdve timm. W etay be w f u I "Iq(x I) is (2iLi)-t(Da(~ 8)-iD(x i)) wllrrrs 01 and D me thn: in gmeral aaaiyses in avaidiw wmpLica&ns fmm inffniteiy functtad dL.firrrxt by W. $at&, R@"[ Mod. Php. 13,203 (1441). remate fiudaces.
--a+=
i(E
-
-
negative energim are inserted, and the situatian interpreted in accordance with the timing rehtions d i m m d abve. (We have u d functions nar&lized ta (8%) .- l instad of the conventional f'LZ@u) = (%*U-) = l. On our scale fzZB%)=energy/m so the probgbilities must be corrected by the 8ppropriate factars,) The author has mmy pmpk to thank for fruitful convemtians a b u t this sugect, paxticuhrly H. A. Bethe and F'. J. Dyson,
Xexpl-i&tHdfl. AB is well known @(X,6) saws tbe Dirac equetion, (digermtiate * ( X ,6) with rapcct to l and use cummutation relations of N and Jlf Gotlsequentfy 9 0 , i ) must also sstisfy,the Rirac equation (digerentiate (41) with repect to 1, use (42) and integrate by parts). That is, if @(X, T ) is tbat saluaion of the Dirac egustion a t time T whirh is +(X)a t 1-0, and if we d d n e @*= JQ*(x)+(r)dk and @'*-f**(x)g(x, T)d% then Qr"*=Slf"*PI, or The principle on which the proof will be based can now be illustrstad by a Simple example. S u p m we h v e just. one electron iniriaily and tinally and & for
cr. Deduction fr@mSecond QwtizaG~ioa In this section we shall show the equivalence of this t h a r y with the hole t h m v of the p&tron "aording to the theary of w a n d quantization of the electron field m a given patential," the state of this field a t any time is repreanred by a wave function X mtisfying
We mi&t try putting P thrwgh the operator S using (m, SF*mFW*S,whmef in E"*=$**(x)f(x)dfx is the wave function j(x) a t 0. Then at T arising f r ~ m
.iax/ar=Nx, where N-.$ef(x)fu-(-iV-A)+Ar+mB)*(x)d% and %(X) h an operator annihihtmg an electron at pmition X, while * * ( X ) is the correwonding creation operatoor. We conkemplate a situation in wfrich at 1-0 we have present -me electrons in states represented by ordinary spinor functions fg(x), fs(x), . mum& nrthogonal, and some pasitrons, T b e are descriw as holes in the negative energy m, the efectrons which would nomally fill the holm having wave functions pa(xj, @*(X), . . .. We ask, at time T what is the amplitude that we find electrons in sates gr(x), g*(x), .. . and holes at gt(x), (i*(x),. .. If the initial and 6nal state vectors representing this situation are X, and xt respectively, we wish to calculate the matrix element
whwe the mend e x p r h n has, k e n obtirineci by use of the definition (38) of C, and the generai mmmu@&n relragon
-
-
We awume that the potential A differs from zero only for times lwtween O and T so that a vtzcuum can be defined at these times. IT rqresents the virruum slate (that is, all negative energy sates filled, all positive @ner@aempty), the amplitude for hadry: a vacuum at time T , if we h d one at l=O, is writing S for exp(- iATli"dl), Our problem is to evaluate R and show that i t is a simple factor times C*, and that the factor involves the K,") functions in the way dissussd in the previous swtians. To do this we krst express X , in terns of X* The opemtor
~ creates an electron with wave funcrion +(X).L i t w i @i~=J@*(x) X*(x)B)x annihilates ane with wave function #(X).Zence s a t e X. is X , = & * ~ Z *...PIP*. while the ha1 state is CI*G"S**. where F,, C,, P,, Q* are oprators deftnd Iike 9 in XQIQI. (391, but withj,, g,, p,, (i, repiwing (9; for fhe initkl state would reault fmm thc vacuum if we crfrated the elmeons in j a r jzj . arid annibitated t h m in p ~ pp, , . -.Hence we must 6nd R;..(X$ ..QsfQt*. . .G&$SfiLFf * PIP^. (40) To simpiify this we shall have to m commutation refationk betwwa a @* operator and S, Tn this end consider e~[-i&iEdI')cg+ Xexp(-t-i&H&') and mpand this quantity in terms of %*(X), giving f*"(x)#(x,l)&%, (which defines +(X,l)). Now muftiply n id this wuation by mp(+i&Wd~') . .~(-i&Bril" and h
.
-
.-
-
~ * * ( x ~ ~ f x l dm. . " r
r)*(r, r)d%
where we have defined @(X,1) by ~ ( x ,
(412
which is rr conwuence of the groperties of efr) (Xhe others are FGm -(;F and F*@= -&W*). Now x@*P*in the East term in (45) is tine cornpia exnjugate of PXB, Thus if f antained oniy p i l i v e enerw a m w n e n a , F"@ would wrnish and we would have vdu& r to a factor times C, But F" as work& cut hem, does enntda negative energy mmpnents creatd in the poltentht A and the method must be skghtly modified. Beliore ptting P1 through the o p a t o r we shall add to it another Wrator F" arising from a function Y ( r fcontaining DlJr negcJive energy compnents and so c h m that the resulting f has d y puziliv~ones. That is we want S(FW*+ Pm.,"'*) FWr*S, (44) where the "'"p~'' and "neg" serve W remindm of the sign of the energy mmpnents fontknect in the oiperators. TKis we can now use in the form SFpOlt~Fm~*S-SFer*l(l(47) In our one electron problem this substitution replxes r by two terms r = tx$f;FV,2Sx3 (x$GSr"~~,"'*xo). The first of these reducm to
-
as abve, for P , k x o is ROW bern, while the a o n d is nro since Ehe c a t i o n operator F.,,'"* givt?s zero when acting on the vacuum s a w m all negative enerdes are full, This is the centrat idca of the demonstration, The problem presc?nted by [M) is this: Givea a functien fm.(x) a t time 0,to find the amount, f.,"', of neggtive energy component w k h must be added in order that the soiution ol Ttimrs equation at time T will have only H t i v e e n s w mmponenq jIrg(II. This is a bou&ry value pmhlern for Pvhich ttie kennel K+td, is d&ped. We know tbe padtive emere commnents initially, fpoh and the negative ones h l l y (zero). The M t i v e ones &ally are therefore (wing (19)) fpoll"(xl) - J K + ~ A ) ( ~t)dfpae~~tld"rr, , where C-T, f1==0. Sidarly, the negative ones i n i k l y are
(48f
jJBf~&)~(X)
for emmpte G Wentzel Ei~tjechllgis d& Quarrlm~ d m f & k ( ~ r a n z~ i u t k k e ,Leipzig, 19431, C h a p
f r m a k e , a d tt =0. The fPoe(ns) is
THEORY OF POSITRONS s u b t w t d to keep in f,,"(xg) onfy those wava which return from the potential and not t h m arriving directly at IS from the K+(2, l) p r t of k',(r'(2, I), as 6 4 . We muld aim have written
759
The value of C,(lo-dlo) a r k s from th-e Narnstonian N60-ar~ which diEers from Hga just by baving an extra ptentiol during the short interval AC Hence, to first order in hie, we have
Therefare h e me-electron problem, r =Jg*(x)f,,'(afd*x.Cu, gives by (48)
as expect4 in accordance with the r m n i n g of the previous sections (i.e., (20) with replacing K,), The prmf is rmdify atended to the m m gmeml e x p r d o n R, (# which .lcan ), be a m l y d by induction. First one replwa Fa* by s relation such as (41) obGning two terns R=(xo**.-Qp*Qzf*..G&lFgpgSr*SI;(P*.. .PIPS* -(X$.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"rtima an exprmsinn with one tess eletron in iniriaX and ftnal state. Nett it is =changed with C*prociuetng an addition -Jg~*(x)ftpo7"(~)$% times a similar term, etc. WnaHy on rmching the Q* with w&ch it anticommutes it can be &mpfy m v e d over to juxawktion with X@* where i t gives zero. The s e n d term is imihrly b d l e d by moving Ptm,,*" thrmgh anti commuting PI", etc, until i t r w h m Pt. Then ir is exchanged with PI to produce an additional simpler term with a factor 7"Jpl*jx)Jtn,,"'(x)dt or FJPi*(xs)K+tA){2, I)J~~~(xI)$'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 when acting on X* '&us gively through the P's untif it. gives R is rrducd, with the q e c t e d facltlrs (and with attwoating $gm requird by the a l u s i o u prindpfef, to dmpler terms mnlaining two bss operatm which may in turn he furher rerluced by u i n g 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 ~ ae 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~geis 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 CVIs obtain& by normalization. However for photon fie& it L deairable 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 m w d i n g to m o n d quaotilation.
-
.
we therdare obtain for the derivative of CSthe exprmien
ex@)
b. Aodyds sf the Vacam Rablem We &aft almlate 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 w r m p n d i n g 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 mmpie& 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 ;
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 ~ in the current density, gives zero so we are left with t w terns *,m*@ABaeI and S,$SA?ir,,, The latter V.neg*BA*aeg is just .the q e e t a t i o n value of BA taken over aH negative energy states ( d n u s '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 sin the sea which we should hsve subtract4 from our original Hamifronian 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 h t 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~nce101, 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 f d u c ~ o nof the quations fw quantum electrod 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 q m t i o n can a m e n t I y be a n a l y d in emntiallv the =me way as that U& here for Dirac electrow.
PHYSICAL REVIEW
VOLUME 76, N U M B E R 6
13epat.lntd
SEPTIBABER
$ 5 , 1941)
R. P, F 4 Pkyska, C d Usimsicy, Iikft New Fwk (Receivd May 9,1949)
Xa this paper two things are done. (1) It is s b w n that a considwable 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 swiltc 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 nof 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 ~ n G r i v ete 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,
EEE paper should be considered as a direct canT tinuation of a profcerling one1 (If in which the motion of elttctrons, neglecting interaction, was am-
and p r m w h i y consistent, metbad is therefare availabie for the involving electrons and p h t o m , The siroplifiaaon in writing the e r c p h o n s 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 d a c u f 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 sof 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 ~ oEh 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.
positive eneqy ektrons are iavo1vd. Furaer, the effwts of longitudinal and tramverscl wava can k combined together. The separations previously m d e lyzed, by dealing directty with the so&aliionof the were on an unrehtiv&tic bash (reA~tedin the circumHamiltonLn differentbl equations. Here the =me kch- stance that apprently momentum but not enerw Es nique L applid to include interaetions and in that way commed in intermdkte s b t a ] , When the t e r m are to express in simpb t e r n the miofutionof problem in combhd and shpW&, the rektivistfc invaxhnce of quantum electrdynamics. the result is =g-eddent. For most practical ca1culations in quantum electreWe kgin by dimming the solution in space and time dynamics tht3 galutian is ordinarily in term of the Schrijdmger equation for particles interacting of a. matrk element. The matrk is work& out as an insbntaneously. The results are inrmdiateky generalthe successive t e r m cor- imble to d e h y d intctractions of rehtivistic ektrons expawion h pwers of &/h, responding to the inclusion of an increwkg number of and we represent in &at way the kws of quantum virtual quanta, I t a p p r s that a considerable simplifi- electrdmamim. We a n then fee how the matrix elecation can be achieved in writing down these matrix ment for any proems a n be written down directly. In elements for complex procmss, Furthermore, each term partkular, the rjeE-enew expression is written down. in the expansion can be written down and understood So far, nothing has k n done other than a resbtedkmtly from a physical point of view, simifar to the ment of c o n v c m ~ o electrodymmia ~l in other k m . spce-the view in X. I t is the p u p s of this p p e r to The~fare,the WE-energydiverges. A md%cation2 in de&h how this may I.re done, We shall aka diseusr; interaction k t w e n chaqes is next made, and it is metfis of handling the divr?rgent intgrtrlrj which shown that the seff-energy is made convergent and a p p a r in these matrix eIemen&, corrapnds to a canection to the ektron =ss. After The shpYIfication in the formulaeresults minly fram the msf correction is mde, other real pramsses are the fact that prevbus methods unnecmrijty ~ p a r a k d finite and i w n s i ~ v eto the ""width" of the cut-off In into individual t e r m pro that were claefy related the interaction,' physirsally. For example, exchange of a quantum Unfortunately, the mdAcation p r o p o d is not comBetween two elctrctrons there were t w t e r m 4 e p n d i q pletely satisf~torytheoretialfiy (it ke& to some &&on whkh electron emitted and which absorbd the culties of conwmation of energy). It does, hwever, quantum. Vet, in the virtual stam considered, timing seem consist-ent and satisfactoq to d e h e the rnatrix relsrtions are not significant. 0 b y the order of operators "or cr d i m d o n of this rnodifiation in elmial phy* gee in the matrk must be maintained, We have Been (I), P, Feyman# P b . Rev. 74 939 (IW), herurfter referred in wbicb virtual p& are R. to Ild k of the m e a d # and results wit1 be found in 8 b brief summ produced can be combined with others in which only R P. Feynmn, g Y s . Rev. 74, 1430 (%W, bnesftcr refwrd to as B. 1 R. P. Feynman, Phys. Rev, 76, 749 ft9&), Lrmfer &l& 1. 169
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~scan 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 equivdlent to tbme obtained by conventionalelectrdynamics 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 canventional electrdynamics was expressrtd in the Lagrangian form of quantum mechanics described in the Reviews of Modem lPhysics? The motion of the fietd osciIktors could be integrated out (as described in Section 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 w o r M 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 cornpiete 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 ~ dthat such a description cannot carry the conviction of truth wl~lchwould accompany 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 d The formuvarious meson theories is d k u ~ e briefly. las 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 i n t q r a 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 complementaryways. 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 exchange of quanta. The fields are so closely determined by the motions of the particles that it is just as well not to *prate the question into two problems but to consider the process as a direct interaction. Roughiy, the fieid paint of view is mmc pnrctical for probjems invofv-
(I~~(i,s'22$;~3~,"$~$F~437~~~3; ~~~~~~+~~
Rev. 75, 486 (1949). R. P. f i y n w n , Rev. M&. P~Y.;.2% 357 (1941. The apl"lbc d i n e Voi. U1,3 (226) 1949.
QUANTUM ELECTRODYNAMICS 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 interactioil, 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 Hamiltonian farces one to c h o w 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 temsequence 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 interveningperid, 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 selfeviclent. The &miltonian form of the equations develops the future from the instanhneous presnt. But "his is. t k view~ointof the thfiary af the S matrix of Heisentterg.
17 1
far difftzrent observers in relative motion the instanhneous p r w n t 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 mrdihed to include the effects of delay it1 the interaction and the relativistic properties of the electronswe 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-relativisticca* descrikd by the Schr6dinger equation (I, Eq. X). The wave function a t 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 .iwhife particle b at X; at 1' gets to rl,at i . Xf the particbs are free and do not interact this is where Kffeis the KOfunction 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 ~ tone , 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 to-lr 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~fcontains frequencim of both signs. Xt should ~) instead be replaced by S + f f p ~ - r Lwhere
~rs) This is to be averaged with ~ ~ a - ~ 6 + ( - l ~ e -which arises when tS
(2r]-t(4+(t-r)+ 8 + ( - f - r ) ) =&+(@-rg)*
this means rIe-f4(tgp) is replac& by 6+(s&$) where a general theoretiml se-rise. by this approximation. Xf it ~ , ~ ~ = r ~ r is; "the - r square ~~ of the relativistically inis not made it is not easy to study interacting particles variant interval between points 5 and 5, Since in rehtivistically, for there is nothing signgcrmt in chms- classical electrodynamics there is also an interaction ing 11=14 if X ~ + X ~as , absolute sinulitaneity of events through the vector potential, the eomg~leteinteraction at a disbnce cannot be defined invariantly. I t is or in the (see A, Eq. (1)) should be (1- (V,.V~)&+(S~$), thlliy to avoid this approximation that the wmpIicated relativistic case, 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 Hence we have for electronsobeying the Dirac equation, 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 &loat time C, where ra, and ybr are the Dirac mtrices applying to the R& order correction to K(3,4; 1,2) can be worked the spinor corrmpnding to pilrticies a and b, respecout emctly as was Eq. (9)of X by an obvious general- tively (the factor BD@b king abrjorbed in the definition, iation to two particles: 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>otwtingquantities involving the exchange of two where ts=.tl=.lo. If ROW the ptentiat were on at all or more quanta betwen two electronsor the interaction times (so that strictly K is not defined unless t4= ta and of an elestron with itself. It is a coilsequence of confE=l,), the frrst-order effect is obtained by intepating ventional eliectrodynamicf. Retativistic invariarlce is on to, wGcb we can write as an integral over both tp clear, Since one sums over p it contains the effects of and 1s if we include a ctelta-function 6(15-ta) to insure both longitudinal and transvem waves in a retaticontribution only when ts==fe. Hence, the first-order visticalily symmetrical way. effect of interaction is fcallmg to- t a x tar;) : \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~.1 ( S) ,~ ~ , ( 62)didra, , (2) is altered to hGt order because they c& exchange a quantum. Thus, "as'can go to S (ampkitude K+(5, l)) where d r .;.@zdl. ? It, and a like term for the eEect of a an b, I&s to a themy We h o w , bowever, in ckaical ejectrodynadm, that the aujOmb dctes instantaneous^, which, in the c h i c 4 tintit, exhibits interaction fhroueh halftaking the speed of light advanced and hall-rekrded potentiais. CluGmliy, this IS rquibut is delayed by a time aiitbin a closed box from which valent to purely reardd (e.g ste A, or J. A. Wheeler and K. P. Feynmsn as uaity, This su$gesls simply replacing rs88(ts,l in no light R w . M & . hys. ii, 157 (194511. Analagous t h o m n . ezist ; i (2) by something like r,-~606e-m) to the 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.
x
my
QUANTUM ELECTRODYNAMICS
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 hit 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_ycontained 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 t a t e 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. It 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
773
once (either in emission or in abwrption), terms like ff, Eq, (14)) ocwr on& when there is more than one quantum invoived. The Bo?;etstatistics 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). 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 prciblem. 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
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 + ( s r & ~ ~ X&(&, 2)dvr arnsmg from awe^$ equations C3tAA,=4xJ, *These considerations make it a p w E unlikely h%tb confrom B " c u f ~ e ~ t ' ~ ~ ( 6 )(4= 617 & ~ g (6 2) roduc~db pattention of f. A Wbsier and R. P. Fqynman Rev M&. Phya. title b in p i n g fmm 2 to d ~ k ils 2rt;e ofthe f.rt tK.1 I* 17, 157 (tM),b a t electmm do not aet on & e w i v a , wi8 be a satid= o/atlscj")4na(2,1). (5) 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 a superposition of free particle mlutions (actually it only contains f). The amplitude that g(2) is contained is m1cuXated as in (L, Eq, (2f)), The tlmgonaI element (g=f) is therefore J J ~ ( ~ ) B K ~ o ( z1)fij(1)8rt@s. ,
(71
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~iscan 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 and ~1 is a constant &-index symjbot, (8) becomes
w=n2),
the integrals exten&ng over the vofurne V and time intervai 2= Since K+($, 3) dqends only on the &Eerence the integmi on 4 of the =ordinates of' 4 and 5, $v= a result (except nmr the sudaces of the redon) i~dependentof 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
volume. If normaliad to volume V, the result would simpfy be proportionaI to T , This is expected, for i f the e E s t were quivalent to a change in enerw AE, the amplitude for arrival in f at f a is alter& by a factor exp(-idE(t%-&)), or to first order by the BiBmence - i f a Z : Hence, we have AE=Z [(1271~+(4,3 1 ~e1p(ip.~a)6+(IaI)dr(1 ~ ~ ) (9)
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 or since (&/m)(Bu) the left side as (M>(ar4%), 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 exponenthlty 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 c m 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;(nwhich 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 gin the mmentum and energy varkbles, rather than s p a end time. For this we shall need the Eourier Transfom of $(ss?) which is
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 mmif &e arc de~syd
pleE*Gam*
which a n be obt;ii~edfrom (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).
QUANTUM
775
ELEC'rHODYNAMICS
l0 1
fbf
FIG,5. Comptan scattering, Erl, (151. FIG.3, Ittirttarrvt correction to scattering, momentum space
trated in Fig. &fa),find the ntktrix:
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
For in this case, firstf2a quantum of momentum k is emitted (r,), the electron then having momentum PS--k, and hence propagating with factor @I- k-?)-', Next ~tis scattered by the potential (matrix a) receiving additional mamntum q, propagating on then (factor @- k-m)-q with the new mommtum until the quanturn 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 where we have used the expr~sion(I, Eq. (31)) for the equal to the expressions (16) and (17) given in B for burier transform of K+. This form far the self-enerm the same prceess, the various terms coming from resiis easier to work with than is (9)dues of the pies of the interand (12). The equation can be understood by imagining (Fig. 3) Or again if the quantum is both emitted and rethat the electron of momentum p emits (7,)a quantum absorbed &fore the scattering takes place one finds of momentum R, and makes its way now with mo- (Fig. .Q
w-
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@,,
for prol~agationof quanta of mrnenlum k is
(15)
the second &m arising b a u s e the ernhion of sz may also precede the abmqtion of el (Pig. Sfb)). One takm rather than W .That is, writing F+{#) -.- - ~ - ~ k C ( k ~ ) , 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 E v q int-al over an intemediate quantum which quanta are abmrbed or emittd depends on whether the prwiousty invofved a fczctor b k / R Zh now suppiied with time compnent of q is positive or negative. a convergence factor C(#) where 5,
TEE G Q ~ X C i E N G EOF PROCESSES WTE VIRTUAL QUANTA
Thew expressions are, as has been indicated, no more than a re-expresion of conventional qusntum electrodynamia. As a consquence, many of them are meaningkm. For example, the &f-energy exprasim (9) or (ll) 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 modificationof quantum eleetrodmmics analogous to the modification of classical electrodynamics described in a previous article, A, There the &(slag)appearing in the action of interaction where J(s)is a function of small was rephced by width and great height, mdif"ication in the quanThe obviom comespondin@; tum 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 frequenciies for &> tl and over only negative ones for I.r
The poles are defined by rephcing k2 by @+is in the limit 6-4. That is X2 may be assumed to have an infxniteshal 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 g d i e n 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 pupwe sfxall suppose herafter that Cfk2) 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 X cos(]R ~)dkrEKg(K. R), propagation of an electron while that quantum is in where g(KeK)is h---" times the density of oscillators and the held, vve can expect aU such integrals with their convergence hcrors to converge and that the result of d pwitive RI as (A, Q. (26)) may be a p r e ? r ~fox all such pracessrts will now be finite and detinite {excepting the p discas& below, Itb c l o ~ loop, d inlegr%Isarsover the moments in which the of the electrons rather than the quanta), The integral of (IQ) with C(P) X2(kZ-- h2)-koting where &"G(X)dX= l and G involves values of X large and dropphg terms of order d h , compared to m. This sixndy means that the ampfitude is (see Appendix A)
-
*This reIalion is given incarrecdy in A, equiitian just preceding lb.
177
QUANTUM ELECTRODYNAMICS
When applied to a state of an electron of Illamenturn P Vile mwt now study the remaining terns (13) and satisfying pzl=mu, it gives for the change in mass (as (14). The integrar on k in (13) can be performed (aftw mdtiplimtion by C(R3) since it involves nothing but h B, Eq. (9)) the i n t e p l (19) for the self-energy and the reult is. &low& to operate on the initial s b t e SE,(so that Ptlkr=ml). Hence the factor folloGring af#I-mf-%iii be just Am, But, if one ROW tries 10 expand l/(P1-m) We can now cornpteb the discussion of the radiative =(P~+m)/(nP-tlf~) one obtains an infinite result, corrections to sattering. In the integrails we include the since pla=mz. This is, however, just what is expected convergence factor C(kZ),so that they converge for physically. For the quantum a n be emitted and ablarge k. Xntqral (12 1 is also not eonverpnt because of sorbed a t any time previous to the scattering. Such a the well-known infra-red cabstrophy. For this reason proceps has the e8ect of a change in mass of the electron we catculate (as disrzusd in B) the vatue of the integml in the state 1. It therefore changes the energy by dE meuming the photons ta have a smaff mass Xm;,<<m<X, and the amplitude to first order in AE by -iAB.! where The integral (12) becomes 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 which when integrated (see Appendix B) gives (e2/2r) time, but at =me time far past suffered a scattering by a potential b. 11 we limit our discusion to the egects times 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
where fq*)t = 2%sin@and we have assumed the m;ctrix to operate between statetr of momentum #I and $z=#~+q and have neglected term of order &,,,/m, d X , and q*/kz. Here the onIy dqendence on the convergence factor is in the term fa, where r=;1n(XJrn)-+9/4-2tn(~ltm,,>.
(U)
As we shall see in a moment, the other terms (131, (14) give contributions which just cancel the ra term. The remaining terms give for smaI1 q, (24.
which shows the change in magnetic moment and the Lamb shift ils intepreted in more detail in B.'* "That the result given in B in Eq. (19) was in error
in which one is to integrate possibly over P7depending on deails of the situation). (If the time is long between b and a, the energy is very nearly dekrmined m that 8'2 is very nearly m2.) We shall compare the effect on the matrix (25) of the virtual quanb and af the chanp of mass Art&,The egect of a virtual quantum is
W=
whik that of a chan~eof mass can be written and we are interested in the digerence (25)-(27). A simple and direct rnethod of mrsking this comprison is just to evatuate the integral on k in (26) and subtract from the resuit the exprmsion (27) where Am is givm in (21). The remainder can be expressed as a muttiple -r(f2) of the unperturbed amplitude (25);
-~(191~)a(#'-
re-
r t d i y pointed out u, the author, h private communiration,
(281
y V. F. ~ e i w k o p faad 5. B. Rench, as their ai~utatton,a m - This has the same result (to this order) as replacing pleted simulaneously with the author's eady in LOIKJ, gave a (l and diflerent result. French has finally shown that a l h u g h the ex- h e potential$ a and 6 in (25) pression far the radiationless ~at&d:ringB, (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 3 8 8 The author feels unhepptly raponsble. used by the Phys. Rev. 7 ~ ~ ~ (1941)). result. He shows that the reiatiain Ln?k,,-l 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 dby this error, Tllis foolnote is spprwriately result now agrees with that o f 7 R. Fren& ind V. F, Weidopf, numbered.
m.
-
-%rwt)a
(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 infrared 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+~}2we have P'==((+E)# and we b k e t as very smalt, king of order P i 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 €4or 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-ctingon 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 efecwrwian is not exact becawe the substitution of btn opmteg on a state such 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 its eAmt (trapsout in the limit. We have used, to first order, the general expm&on (valid for any operators A, B) "fie
b tbe. integral in (19) L vaLd only if &t
) can
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 unnecasary, may facilitate analysis some~vhatby removing 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 &lis 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 sadsfactciry, (It is a k strong that a ~ t i s f a c t o qform for f+ can be found.) In the awlpb of the raaative corrections to sattering one type of tern was not considmed. The potentiali
779
QUANTUM ELECTRODYNAMICS
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-fi,, which qwntum scatters the original electron from slate 1 to state 2, The mtrix eiment for this procem (and the othm vvhich can be oblirined by reanan&ng the or&r in time of the vadous events) is
~ potential praducm the pair with This is b m u the 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~toxTV($)> over to the other eiedron, by which it absorbed (matrix element of 7, between states 1 an&2 of the ori&naI electron (%qy,%,)). AI1 momenta P, and spin slates of the virtual. electron are admitted, which m a n s the spur and the inte@ on d4p, are ealculatitd. One can imagine that the ctosd loop path of the pitron-eiectron produces a current which is the source of the quanta which a t on the ~ c o a ed l ~ t r o nThe . qmndty
is then chmacteristic for this prablem of ~ h r i m t i o n of the vacuum. One sees at once that J,, divergm badly. The modififation of S to f afters the amplitude with which the cumnt j,, will a E e t the scattered e l ~ t r o n but , it can do nothing to prevent the divergence of the integral (32) and of its edects. One way to avoid such difffmjties is apparent. From one p i n t of view we are consideriry~dl routes by which a given elelron can get from one lugion of spa=-tim to another, i.e., from the source of electrons to the apparatus which measuretr t h m . From this point of view the clmeif loop path I d i n g to (32) is unnatural. It might be: assumed that the only wths of meaning are those which start Irorn the source arid work their way in a cantinuous path [pssibly containing m n y time reveals) to the d e t ~ t o r .Closed Ioops would be exeluded, We have alrady found that this may be done for electrom moving in a fixed potentid. Such a suggestion must m e t -era1 qurrstions, howW ever. The d o 4 lwps are a conNquence of the w a l hole theory in electrodynmicts. Among o t k r things, they are require$ ta keep probability conmwed. The pmbbility that no p k is prduced by a potential is
FXG.5. V m u m piariaation ef-
fect on mLtenngz
(30)-
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 consewstion, 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 prmumption is that the diaculties of vacuum polarlzsltion axe not sia easily An alternative prmedure discusrjed in B is to msume t that the function K+(2,1) used above i s i n c a r r ~ 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 integrsI 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@,-$a+~)d~p&'pb and both p, and f i b get wnvergence factom. The integrat now converges but the result is unsatisfa&~ry.~ One e x p c a the current (31;) to be conserved, that is q,j,=O or qJ,,=O. Also one expects no current if a, is a gradient, or @,=g, times a constant. This leads to the condition J,g,.=O which is equivalent to qJ,,==O dnce I,, Is symmetrical. But when the eqnpresion (32) is integatd with such canvergmce f ctors it does not satisfy this condition. I& altering the kernel from K to another, K', which does not satisfy the Dirac equation we have iost the gauge invariance, ib consequent current canservation and the general mnsistency of the thmry. One can st?e this best by calculating J,d, directly from (32). The eqrasion within the spur Fcomes (P+q-rn)-lg(irCP-m)-Iy, which can be written as the difference of two terms: (P-m)-%# &+ @- m)-LyL' Each of these terns would give the same result if the integration d4$ 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 rcattering 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 i a w (33) there is a tern 4{k"--2@'+.fqrjbr for C(kJ) -Xt(&X'))', which is not gaugg invariant, (In irddician tfte chsrge rmonnslizationh -7/6 irddd to the iomithm.) 5
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 negativeiCy), 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
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> (2mj2it 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~eswith 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 containinga 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 proce~.~
-
+
-
In the usual form of quantum electrodynamics the tongitudinal, and transverse waves are given wparate treatment. Afternately the condition (MJdx,)'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 t3jJax, =0. That means at least ,/a%#:,)=O and in fact our so1utioq dso satisfies aAddz,--O, To show that this is the case we mnsider the aampiitude 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 mtrix 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. It dso vanishm in m r e camplex situsttions (menti&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 momentit q, (e.g., m m may be emisions or absoxptionsof the same or &Berent quanta) and cornider a matrix for the Lxansitbn from a state of momentum t;o $p such
Rev. 48,4f) (1935f.
B There are i q s mmpletely withntut externa.1 interactions. Far exrtnrpk, a pair is am@ v~rtualtyalong with a, photon. Next they annihslate, absorbin this pbton, Su& loops are disrqardsd on the gmads that tghey do not intern! anything and are &meby completely unobmablc. Any rndirect eEi?cu &bey may b e via tha ezciusjm prin+te have a b d y k n hdudtd.
the function G(X) satisfying m ( X ) d X = I ernd Then in the expression for JNFP the range of p inlegation can be extended to infinity as the integral now converges, The result of the integration using this method is the interal on dX over G@) of (see Appendix C)
&*C(h)XzdX=O,
with q2= b2 sina@. The gauge invariance is clear, since q,(q,q,- g*&,,) =O. Oprrtting (as it always wiU) on a potential of zero divergence the (g,q,- &,,@)a, is simply -$a,, the B"1embertian of the ptentml, that is, the current producing the potmtkl. The term $On(X2/nz2))(qsq. -g&,,)therefore gives a current proprtional to the current prduckg the potenthl. This would have the =me effect as a chnge in charge, so that we would have a Werence A(%) betwwn a2 and the e x p ~ m e n tally observe$ 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 ~9134effectsz1 of polruization of the vacuum, A. Uehlin , Phya, Rev, 48, S5 (19351, R. &rbr, Phys.
-
a2(i),Q
781
QUANTUM ELECTRODYNAMICS
aw UN ffsptM-' (@1-%)-'4%w h ~ e p ~ = f i (and ~ t +in~the ~ product, terms with larger i are written to the left). The most general matrix element is simply a 1inar combination of &m. Next consirier the matrix between states p. and #M+Q in a situation in which not only are the a, acting but also another potentiai a expf-@-%)where a = q, This m;;iyact prwioustoalla,, in which case it givm U H ~ @ , q+ nc)-8u,(Po+ g-m)-" which is equivaient to +a~n&,-E-q-m]-'ar since (Be+ g- m)-Iq is. equivatent to Cpo+ q- m)+ a equivglent to m rrcting on the X (P@-#g-m) 8s initial state, Likewise if it acts after all the potentiab k, is equivalent it give q ( p ~ - n t ) - % ~ ~ @ ~ - m ) - which to -aNn(p,-m)-%, since f t ~ - f - g - mgives zero on the final state. Or again it m y act between the pf&ntiaI ak and a&* for each K. This give
+
N-l
N-1
k-t
I-WI
Z: QN R[
9
-
(pl+q-nt>-fa,Cpk+~-m>-l k-I
xq@r-mI-"ak
XncirIentally, the virtual quanta interact through terms like 7,. ..r,k2dbk. Real proceseces correspnd to poles in the fomulae for virtual procmstzs, The pole occurs when @ S O , but it look at first m though in the sum on all four vaiues of p, of r,. -7, we would have four Ends of polarization instad of two, Now it is c l a r that on& two p g ~ n d i d to r K are effective, The usual elimination of bngitudinal and m l a r virtual photons (fading to an instantilne;~~ Coulomb potential] can of c o w be performed here too (although it is nat pzrticu:ularIyuseful), A typkat &rm in a virtual transition is 7,. y r P d * R where the represent some interve~irmgmatrices. Let us choose for tbe valuepr of p, the time 1, the? ciiretion ol vector part K, of k, and t m pqendicular directions 1, 2. We skI1 not change the exprwion for these two 1, 2 for these are reprewnted by transverse quanta. But we must find (76- * y t ) - ( y r r - * yx). Now k=kdrg-krE, where Xi"== (H;.K)l, and we have shorn a h m that k replacing the 7 , givm ~ e r oIIence . ~ k t r K is equivalent to K 4 r t and [yIrl.. (Yg* ' ((E(Z-ka"/ff2)(yt. - . r 3 ,
If @$-m)-"a 1-1
-
-
so that on mulriptying by k2d*k=d4h(k,2-Ka)-1 the net
However,
effect is -(7,...yr)d4R/flCIP., The m a n s just scalar 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, &r+ Q- m)-"q(gr-
m>-L
)-,- (pk+p-n)-E,
(34)
9. KLEfR \TOOWN EQUATEON
N-l
+a#
a @Cltl-m)-ba,-~N -1
N- l
If &e+~-m)-'@a,.. vat
The methods may be readily extended to partic1a of 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 ($5) r@+ is zero. I-lence any w ~ v eemitted will the eatin: on the c*rme care is rmuiret2 when 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 momentum -k, is disturbed b an added powriat k of matrix ele, taming have no dKt, for momentum + k (the r w e w d sign af rke momenta in the interments for em;i~ianand ~ h r p t i o nare si&lar. (we mediate factors in the seeond term X . . .k has no eRecr since we e ail R), Wmce as shown a b v e the result is have =id kittie more than that a potenthl A,= dv/azR wiU tater i n t ~ m t over but i n c e (k.. -rf+(x. .k)=kig(rr-* .rd-lr""(rzs ha no dect on a ~i~~~eiectron since a tIransformation sera, we can still cancIude (rrr; . .?K)"R8K4(yr*. -yd. $'=@q(-i@):ctr)tt removes it. It is easy to see ia The equations d i s c u d in this m t i ~ were n deduce$ from the formuia(ion of the Kfein CIordon quauon given in reference J, cmrdinate repremtation integrations by pab.) Section 1 4 Tbe fvnrrian in this r f i o n has only one component Thin has a uwful conwquence in that in 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@O 'n diretions rather than just the two special polarization ~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 d i r ~ t i o ne, is e,&f,, E the of the equations in momentum~prrcewill wrnato &rmal17 identrcd we how from the argument fC" t k for the spin 1/2- with the exceprian of t h m tn which a wave vmtor Xight denominator &-nr)-l h& been rat;ionalizr?d to (n3-n)W-m')--" above that g N ~ ~ For 0 * m ~ h r i ~ e&ht d BrOgresX since p no longer equal to a numhr, *-P. But p does equal ing in the S directi~nwe would ordharity calculate (p-p)@ so that (P-n)-1 may now be interpret& F (mp+d ~t)a-P'#~(P.P-~I)-lmmt. m@ impgm that %wuom .lu c* M,Z+M,S. But weaawejI sum M,2+M,2+lli( f ordinate space will be valid of the function K $2 1) is sven ari for p&, implies MP=@,~ i n f eqt= q, for flW2 quanta. (Z,t)== l;(iVz+m)-m-I(V$+ B2*)31+(2, 1) ;it6 This unpolar* tight is a r e ~ t i v i s t i a ~ ~&S y S; aU in vlrtue of the fact that the many fompnent wave in I) mtisfies 1(" (5 components for spin O 10 for invaknt EOncept, afld permiB some shPr&atiom in functio~ (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}, v
~
L
B
~
~
/
~
s
~
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 CfE ) = - XZ/(k2- F), Since the interaction occurs through the gradients of the ptential, we must use a stronger convergence factor, for exam@€ C
giving, 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), m that Together b, and a, carry momentum qb,+q,, 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 aPwrAs a further example the self-enerp of a particle of momentum p, is
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&),
0.
where the g,,=
4 comes from the A,A, tern and repre-
b.
C.
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&bilityarige$, (b), af simult&aeausinteraclian with two quanta a+. Thr: prctpagatian factor is now
(Pp-
r ~ @ ) - ~far a particle af momentum p,.
QUANTUM
ELEG
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~lsare 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 m w n s 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 t e 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 , yFar ~ ~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 reptacement of y, by other expressions (for example by k ( ~ ~ ~ e - r p 7 ~ with ) a subwquent sum over all p and v for virtual m a n s ) . 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 carreswn& 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 proce e s . 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 7 5 anticommutes with p? and p3 ovrating on the state 2 equivalient to K as is fix 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 psudovector coupling rvhich holds for low order processes therefore does not hold when the pseudoseatar theory is giving its most importa~tteffects. These theories will therefore give quite different rmults in the majority of prcrctical problems, Xn alcuhting the mrrectia~lsto 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. It 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 electrodynamim. 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~xndsquadratimlly 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.) It 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.
1949).
E"11)rs. Soc.
24, 3, 2 2 (Wahiqtan,
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, It is important here to use the method of calculation of Bethe and Pauli, that is, a virtual meson is assumed to have the same 'Lmass'Vuring 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~neousemission (or a b m ~ t i o n )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 vector meson field pokntiats ortisfy The author is particularty indebted to Professor H, where s the source for such m w u h is the matrix element of A. Betk for his expknation of a method of obtaining betw&n states of neutron and protan, By taking the divergence finite and gauge invari~ntresults for the problem of a/d+ of both sides, conclude that tttp,/ax, s.4ari-?as,fdz, so that vmuum polarization, He ia also gratttful far Professor the giginat equation can be rewritten as Bethe's criticisms of the m a n u z ~ p t and , for innumerr j s ~ f i - c ; " ~ p -.P~fr,+ci-"a/a.r(ds~/ax~)~. able discussions during the development of this work. The right hand side gives in momentum representation 7, -~**~,p,r, the left yieids the (@--&-g and finally the interaction He wishes to thank Profesmr J. Ashkin for his careful s , ~an the hgran&n gives the 7, on absoiorption. reading of the manuscript. 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 dterm (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 consented, 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 poteential 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 m m n s with a non-z@rodivergence of potential is to be avoided. (The correstion term p-%q,g gives zero in the neutral cm.) The asymmetfy in emission and abw ~ t i o nb 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 and accept the admixture of spin zero simply 7,.. 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
.
e r ,
droceedlng in t h way ~ find generaliy that arueIes of spin one r ( jwhieE, , for a free particle can be represented by a l o u r - v m ~ I
of momentum p satisfia q.rr-0). The promation of vixtuili prticles of momentum q from state v to C is reprmated by
In this appn&x a method will be iltustrated by which the mllltiplicatian by the 4-4 mtrix (or tensclr) P,,=(b,,-#*q.&,) X( m e 6rst+rder intmmtion (front the Proea equaann) simpkr integrals appearing in problems in electrodynamics can w i t f i n rlectmmagnetie potendai o rrp(-ih.s) corresponds to he directly evaluated. The integmk arl%ng in more cornpiear lead to rather eamplicated functions, but the study of nuttiplieation by the matrix I f , , = ( q z - a + q ~ - a ) 8 ~ ~ - q ~ p ~ - p 1prm~~ w b e p; and qg=qt+k are the mommta before and after the the relatias of one intqrat to another and their exprminn in interaction. Finally, two tentiak a, B m y act simulmmaaty, terms uZ simpler inlegrats m y be f d t i t a t d by the m e a d s with m&k (a-bR,+bIraI. given here.
-
78.5
QUANTUM ELECTRODYNAMICS As a tpimi problem eonsider the integral (12) appearing in
tb first order radiatianlem mttering problem:
j ~ ~ (k-m)-Ia&~ ~ 8 - -k-m}-IrPk-"d4RC(kl),
We Ben have to do inkmats of the farm
Jfl; ka; bL7ld%(V-L)*(A"-2pl.k-
At)-"
(la)
X ( V - Z p z - R - A ~ ) - ~ , (98)
where we shall take C(RI)to be lypicaliy -A2(8"-Xyi and d" (2r)*dRldk&b&k4. we first rationJEze factors @- k-.r)-i- U- k+Itl){(P-R}*-rn*]-l obtaining,
where by (1; R,; kOkJ we m a n that in the place of this symbol either ' 8 Or may s(;ind in diaTerenr In compkicatcrd problems &ere may be more factors f V- 2p6.K-d,i-* or other powers of these factors (the (&--L)-""may Fte considered ss a special case of such a factor with #,=a, &,=L) and further in the numerator. The potes in all the factors factors R e k,k,R,. am d e degnite by the assumptian that L, and the 4%have infinitesimal negative i r n a g i ~ r yparts. We shall do the integrals of sucmgve complexity by inducGen, W., start with hesimplest convergent one, and show
-k - t d ~ , k " d * k ~ ( @ l
S"vP@ k-f-mlailtt 1-
X((p8-A]*-m"-"@2-k)~-m33~1
(24
m e matrix e ~ r d o may n be simpiifid, f t a P W r s to h to do so a j t e the integraeons are pertormed. Since. AB==ZA.B--BA where A .B-A,B, is a, numhr mmmuting with all matricerr, Bnd, if R is any e x p r e e n , snd A a vector, since r,A-. --Ar,,+2A,, 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==k s r J ,
. f ; l % ~ ( ~ - ~ ) - s(8iL)-1, =
- (3i/32s)jeer 4r@dK(RI+ I;)-Mestahkishing (10a). We akw have fkod"R(V-L)-a=O k space. We write t h e results as
(318i)(1/3L)
from the symmetry in the
1
(8i ] (l; k,)dik(&-L)-ei. (l; O)&l;i, is far I; the (S; k,; k,k,) is rqtaced by 1, for JSby 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 , Fand 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 t-o s t h o s U& on th h s t order integralri, The factors @-k)*-m' may be writ&
-1:
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
by (&-l;)* and ot the end intel"hufi we can q l a c e grate the result with respect to L from m0 to X% We can for many p r ~ t i wpl u w s consider As very I s s e retative t~ m* or p. When the ~ d d n a lintegral 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'.
(loaf
For this integral is J(2x)*dR&E(k4#- K - K-L)-a where the vector R, of magnitude K==(W;.K)b is RE, 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
(Ifa)
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
wi)j '(1,, k
U,
RQR. )d'b(R"-2p.k-A)-' .- -U; p,;
p,p,-j~,,w+a)>w+&)+.
ff~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 by d some work of Sehwinger's involving Gaussian integrals). This r q r e n t s the prduet of two recipro~llsas a parametric 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*),
~4(i-z)dxln(x(l-z)*)=
--(1/4) find
(kf-2@,. ~-A,]"(Y - 2 p z z ~ - d z ) - g = x2lcdx(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
sv that substitution into (19) (after the U-k-m)-' replac& hy (p- k+m)(Re- 2p.k)'") gives (1%- ~&/8rlrpt@+nrlt2In(hg/=a1+2)
-p(in(ht/='3 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 abtained is a rather complicated camhination of rootsand lagarithms. Other integrals can he obtained again by pramet& differentia(18a) with respect to tion, For exampie difierentlation of (L&), ba or p*, gives
in (19) is
-#)l%
-.(@/8r)C8rn(tn(h2imZ)+1)-#(2 In(~~/nr")+5)7,
m.
using (4a:ita)tr, remove the 7,'s. ThisagrecPs with (201 of t k text, and gives the self-eneru (21) whea ft is r ~ l g by d 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-kj.-"l-2ps* k]-'i~f6 & ( @ - 2 h * h ) T 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
from (14a) where
-
vP1-1-(1 -rPz. We therefore need the integrah
(21a)
(8i)l(1; k,; k,k,)d4k(V-L)*(l-2k"k-A)*
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=@: 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 r o c e w of tncrea&ngly higher orders the wmplexity and difficulty increases rapidly, and these methads m n become impractical in their present form.
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 q u a 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 self-energy inlegrat f 19) is
that it requires that we find fusing the principle uf (8n)) the integral on L from 0 to XP of
W
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 The integrals on y give,
or prforming the integmt on I;, ss in (81,
M m t n g now that h+>m* we n w f e 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
QUANTUM ELECTRODYNAMICS These i n t e ~ a i son y were performed as fotlovrs, Since &=#tf q where q is the momentum carried by the potential, it follows from p$--);2=mz that Q=-# so that sinct? @,=P~+g(l-y), -y). The substitution 2y- l=ana/tan@ where B pu~=~m2-g~yjf is dehnetl hy 4mx sin2@=q" is useful forlr nawc*a/secV and @,-Sdy = j n ~ gsin2B)-Va where n goes from e to 4-8, Thew rrtsultsare subtituted into the original swtexing formula (2a), giving (22). X t has been simpiified by frequent use of the fact that p, oprating on the initial s a t e is m, and likewise #I when it appears at the left is replacable by m. (Thus, to Smplify: yltps~jb8~~-~PIQ#O by = -2@2-qfa@3+ql = -2fm-g)a(m+g). A tern, like qaq= -q1u+2(a.q)q is equivalent to just -@a ince -m-M has zero matrix element.) The renormalization term requ~resthe correqonding integrab for the special case 4-0,
-
C, Vacuum Palmizat-ion The expressions (32) and (32') for J,, in the vanrum polariza ikon lxrobicm require the caIculation of the integral, J.,(iifl=
-
5SSPEY.@-
t4+n)rp@+1~+r~>~4~
X (Q- &@lz- m?*)-"@+ f 1 ~ 1 7 - " , (32) Itbere ue have repiaad P by B- tq to simplify the cakculation mmewhat. We shall indicate the method of caieulstion by studying - -
the integral,
I ( 4= ~ P , P P S ~ C P -iq)2-ma)-h((P+
-
-
J-y : I P . P ~ ' P ~~p- 4- m".+i@l"d@iZ.
~(~a')
C3Oa)
'
Btrt the integral on ) will not be found in our list for it is badly divergent, However, as d i s c u d in Section 7, Eq. (32') we do not wish I($+) 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 derivative l'(m9+L) of I with respect to m%t &+L and later L from eero to dE~erenlialinlJ(%l, respect ta N10 find,
=Jiff
where we assume X*>>& and have put some terms into the arbitrary eonsbnt C' which is indepndent of Xvhut in principfecould 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, n e c e m y ta dlEerentiak his m* also when calculating I' and I f i 3 h e r e 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
-
ts@f@%+mr)@s-~z~@a-t-~a)@~-mt)f
t~)"f&~l-~.
The factors in the denminator, p-@,g-mP+jqs and fP+p.g ~ t 1 ~ 4q2 - f itre combined as usual by f8a"l for symmetry we sul>strtute .v-- g(1-f-rl), (1-2)- 4 U 1)) and integrate s from --l ta-f-l:
I ~ c ~ papde@w~ ~ R ?.pg-+-~-t-
787
(Pt.#s-mm~a)(pt-P4-"mrc)
- (~z.~a-mrmd(P~.Pe-~@O +(Pt.#~-mlmd(@~.lba-ms~z~~ (S@ where m,, m, are arbitrary four-vecbrs and constmts, I t is interesting that the terms of order X9nh* go out, so that the charge renormalization depnds onky fogaritbmically an ha. This is not true for Borne of the m w n rhm~.es Electrdynamics . h saviriowly unique in the miidnas of its divergeaa.
L), More C@mpi@8: PrabXenrs Matrix
compla prohiems =hnus that ad for lhe three iilweatjons; higher order carrections to the MQIHhr Katter-
~:p~-~d~,
This still diverges, but we can cfigerentiste again to get
I " < ~ ~+3 Bcp . p ~ " a ~ - @ p . gd-- ~+ip2)-"drr
--
::fl-lgi
(3 1 a)
( t ~ l q ~ q ~ ~ it (cip, ,- ~ ~ f d n
(where D- $(g-f)@+~g+L),which now convergesand 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 lwrithm ~ M n ong integnrting I, in (31aJ p r e n c s no probEem. We may take ~'(rn(+~f
(%)-'E
Ct~'g&~D-2+f 8.r bD3w-t-C&,
a aubsequmt integral on L and i i d y on problems, Thm rwults
pr-ts
+&,,C(A2+ma)h(h~m-2-f-f~-CfX2J,
no new
f32a)
FIG.8, The inkrxtlon betweem two electrons tn order One adds the contrribution af every figure ~nvoivingtwo vrrtual 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 -inP B~ e manner of Q, (4) and Fig, l. The matrix element for this prmess is proportional to (translating (4) to Rlomentum space)
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~tionsto mttering, which we have worked out. The term may be combined with the renormalimtion terms resultWe shall diwuss corrections to @?a) to the next order in @/h. ing from the Blgerence of the eFtects of mass change and the terms, (There is also the possibility that it is the electron at 2 which Rg.8f and &g, Figures ge, &h,and 8i are similarly awlyzed, 6naUy arrrva at 3, the electron a t 1 gdng to 4 through the exFiwUy the term Fig. 8c b ctearly retailed to our vacuum change of quantum of momentum PI-Pn The ampfitude for t h ~ s poiarizalion problem, and when integated gives a term proporprocess, (ricr,~tf(irar,ua)Ipa-P*}-~,must be subtracted from tional to (@,yBsp)(~l~,~l)lr&+. Lf the charge is renormaliad the (3Sa) in accordance with the exclusion principle. A imiiar situa- term lu(X/nt) in J,, in (33) is ontitttrd so t h e i s no remaining tion exrsts to each order so &at we need consider in detail only dependence on the cut-08, to the last, the subtraction of the corrections to (37a), ~er~erving ?he only new i n t e m b we requtre are the convergent i n t e ~ a l s the same terms with 3,4 exchangd.) (38a) and (39a). They can be simplified by rationdizing the One rmwn that (37%)is modified is that two quanta may be nominaha and combining them by (f4aI. For example (S&) mexchanged, in the manner of Fig. 8%.The total matrix element volves the factors (E- Z$t.k)-'(kf+ 2P~,k)-~k*(q'+ P- Zq.k)-'. for all exchanges of this type is 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 that+ the ~ intqrais q ~ ~nn~ ~ ~ ~ factor ( k - ~ ~ ~ k ) ~ ( kSO ~ d?5-now involve two factors and can be performed by the meshads as is clear from the dgure and the genera1 rule that electrons of given earlier in the appndix. The s u b q u e n t integrals on tbe momm;nmnt contribute in antpEtude (,P--m)-"etw~en inter- parameters s and y are complicated and have not been worked out acGons 7 , and that quanta, of momentum k mntribute @. In in detail. intepating on d% and summing aver p and P. we add all alkrnrnaWorking with charged memns there is often a considerable reliver, of the type of Fig. 8a. If the time of a b q l i o a , y,, of the duction of the numhr of terns, For example, tor the interaction quantum k by elweon 2 is tater thaa the absorrpttion, rp, of g- k, betwmn protons retrulting from the exchange of two merians only this comapnds to the v~rtualstate @S+& k i n g a psitron (m the term conesponding to Fig, 8b remains. Term 8a, for example, that ( 3 W contejns over thirty terms Ctf the conventional methd is imposrrible, for if the lrrst protsn emits a p s i t ~ v em e n the of andyL). second eannot abwrb it dircxtty for only neutrons can absorb In integrating over all the= alkmatives vve have considered all pvsitive mesons. w i b I e distortions of Fig, 8a which prewrve the order of events As a m n d milmple, mm'der the radialive correction to the along the trajectories. We have not included the p d b i l i ~ e s Campton scattering. As seen from Q, (1.5) and Fig. S this s-cattercarrapnding tn Fig, 8b, bowwer. Their mntribudon is ing is r e p r m t e d by two terms, so that we can cornider the carrectians 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
(sr~~~af(@rv,.dq-'.
i37af
+-
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 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 renormdization 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 t s 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 FIG.9. h d i a t ~ v emrrection to the Compton e t k r i n g term (a) of Fig. 5. Appendix D.
F, Bloch and A. Norrfsieck, Phys. Rev, 52, 54 (1937).
QUANTUM ELECTRODVNAMfCS momentum fit before B e mllision to tbat characte~scicof 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 em m n , 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 parng 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~agin 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 intera c t ~with the field rnd then rwbsorh the m w o (Fig.10a). The matrix far tfiig p m - is ( P ~ ~ f # t + q ) ,
780
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 =upling. r 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 m a n e r the lading faetor y6k in (4la) is written as - 2 g ~ s - y p @ s - &-M],the secand term keading to n simpler term con@ning no tpll-K-Mf-g factor and combining One 9implifies the rnRs and with a slmilar one from (-1. in (42a) in an analogow way. There finalty r a u l b terms like (4laf, (42aI but with m d o e a h r coupling 2dfy6 instad of where we have put I=k11-Q.m e change in sign arises km= ysk, no terms like (43a) or (&a] and a remainder, r p m ~ n t i n g r pseudwalar coupling, the virtuel m w n i s nestive. Finally there are two terms wiging the differenu? in eBwts af w u d o v ~ t o and m e p s e u b l a r terms do not d ~ e n dseensitiveiy an the cut-all, fmm the rdi part of the pudovector mapling (Figs. l&, 1Odf 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. c can Inteation of a protsn with an e ! ~ t m r n ~ e t jpknendal 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 U&ng convergence factors in Ibe m n e t d m m d in the stsction m m n s are netltmi. It is a n d q o u ~to I-he d i a t i v e wrrstions to on meaon theeries a& integral can be evaluated and the resuib the mtterlng of eIeetrons due to virtual photons. m e sum of the mmhind. E z p a n a in powers of q the timt k r m gives the W- magnetic mamenb of nmtron and protan for charged m a n s is netk momcmt of the neutron and is insensitive to the cut-o8t, the the m e as the proton m m n t dcuiatedl for the comaponding In fact it is mdily w n by camparing d i w r n s , next gives the Iseattering mplitude of slow electrons on neutrons, neutrd -R#. that for arbitrary q, the sfattering matrix ta f i s l W& in Ilrc? and depends logaritbmially on the cut-off, &r@mgdk powliol far a proton m d i n g to neutral m a n The qrEssions may be simplified and a m b i n d =mewhat hiore integrstion, %is makes the intepals a little m i e r and also &wry is qual, if the m m n s were charged, to the sum of the shows the nfation to the case of p s e u d m b r a q l i n g , For 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 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 coupling fnqlsting the masg diarence of ncutron and pmwn).
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)
