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
Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book and Perseus Publishing was aware of a trademark claim, the designatians have been printed in initial capital letters.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, elecmnic, mechanical, photncopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States af America. Westview Press is a member ofthe Perseus Books Group
Cover design by Suzanne Heiser
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)
This page intentionally left blank
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 = 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) 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 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,'" 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,
-
This page intentionally left blank
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
