Quantum Electrodynamics

QIJANTTIN{ ELTCTRODYNAMICS

AUANTUM ELECTRODYNAMICS ?3 486

.

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Transla.ted trom the German by

C. K. IDDINGS & M. MIZUSHIMA

LIBRARY MONTANA COLLEO€ OI MINERAL SCIENCE ANDTECHruNL(}&V BUTTE

Springer-VerlagNew

E I9rk. 1972

Heidelberg. Berlin

Gunnar I(?illdn Late Protessorin the Uniuersity ol Lund'

Prof. Carl I(. Iddings and Prof. MasatakaMizushima U nitsersitY of C olorado Deparfinent of Physics and,Astrophysics

All rights reserved No part of this book may be translat-edor reproduced in arry fo'tm without written permission from Springer'Verlag' @I972 by Springer-Verlag New York Inc. Library of Cottgre.. Citalog Caid Number:76'172529

Printed in the United Statesof America. . Berlin ISBN 0-387-05574-6springer.verlag New York . Heidelberg ' New York ISBN 3-540-05574'6Springer'Verlag Berlin ' Heidelberg

TRANSLATOR'SPREFACE

Kei115n's Quantenelektrodynamik provides a concise treatment of the subject. Its strong points are the careful attention to explanatory detail, the methodical coverage of alI the major results and the straightforward, lucid style . Certainly it wilt be a valuable reference for one learning the subject or for one who requires the details of the practical results. Of course modern quantum field theory has now grown far beyond its dramatic beginnings in electrodynamics and we have therefore included some references to introduce the reader to the more recent and more specialized literature . We have corrected some minor errors: wewould appreciate it if readers would inform us of any others which they find. We thank Professors paul Urban and C. trztdler for permissron to use the biographical material on Kii115n.We also wish to thank Springer-Verlag for undertaking pubtication of this edition by an unorthodox method, but one which will reduce the cost to the reader. In particular, we are gratefui to Dr. H. Mayer-Kaupp and Mr. Herb Stillman for their kind cooperation. FinaIIy, we t h a n k M r . M i c h a e l r e a g u e f o r r e a d i n g a n d c o m m e n t i n go n t h e f i r s t d o z e n s e c t i o n s a n d w e t h a n k M r s . J o a n n eD o w n s f o r e d i t i n q a n d typing the final manuscript. May i972

C. K. lddings M. Mizushima Department of Physics and Astrophysics University of Colorado Boulder, Colorado 80302

IN MEMoRIAM pRoFESSoR GUNNAR KliLrfN

On October 13, 1968,Professor Gunnar Kiill6n of the University of Lund, Sweden, died in an alrplane accident near Hanover. Undoubtedly Europe lost one of its most prominent theoretical physicists. Let me flrst briefly state the significant dates in his remarkable scientific career: B o r n o n F e b r u a r y 1 3 , 1 9 2 6 a t K r i s t i a n s t a d , K e 1 1 5 ns t u d i e d physics in Sweden and completed his doctorate at the University of Lund in 1950. Starting as an assistant professor at Lund from 1 9 5 0t o i 9 5 2 , h e g o t t o t h e T h e o r e t i c a l S t u d y D i v i s i o n o f C E R Na t C o p e n h a g e n0 9 5 2 - 1 9 5 8 ) , w o r k e d a t N O R D I T A( 1 9 5 7 - I 9 S B u) n t i l h e was offered a professorship for theoretical physics at the University of Lund. Moreover, Kiill6n made several journeys for the purpose of research and took part in numerous scientific conferences in about15countries, includingthe U.S.A. and the U.S.S.R. Let me now mention some details about his scientific work: Initiaily he studied electrical engineering but soon he changed over to physics, especially to the problems of quantum elecrrodynamics; in this field he achieved most important results in the years 1949 to 1955. His principal aim was the treatment of the theory of renormalization using, unlike other authors, the Heisenberg picture instead of the interaction picture and the relations now known as Yang - Feldman equations. Considering spectral representations for two- and three-point functions, he succeeded in separating the renormalization constants of quantum electrodynamics and in expressing them as integrals over certain weight functions; thus he could precisely formulate and try to solve the problem of the value of renormalization constants. Indeed, other authors are in doubt about his famous proof that at least one of the renormalization constants has to be infinite, but so far no definite answer to this question has been found. K?ill6nts authority at that time in the field of quantum electrodynamics is well illustrated by the fact that it was he who was requested to write the article on this topic in the Handbuch der Physik. In connection with his work on quantum electrodynamics, he began to study closely the analyticity properties of three- and four-point functions and obtained a number of important results, partially cooperating with Wightman and Toll .

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We must not forget a treatment of the Lee model, which Kli116n did together with Pauli and where they discovered and discussed the possibility of "ghost-states" . K6116nwas not only interested in the development of the general theory; he also treated many difficult concrete problems, such as in his works on vacuum polarization of higherorder. During the last years KdII6n performed fundamental work in the field of radiative corrections to weak decays. He took up the idea proposed by Berman and Sirlin in 1962, namely to take into account strong interactions by the introduction of suitable form factors. But there is one crucial difference from Berman and Sirlin: K2il15nuses on-mass-shelI form factors; that means quantities which are experimentally measurable in principle and can therefore be used as phenomenological parameters of the theory. By suggesting for the unknown form factor abehaviorsimilar to the usual ones, one obtains higher powers of the photon momentum in the denominator; infinite integrals do not occur any more. There are two important features in thls method: first we get finite radiative corrections (although no exact numerical results can be expected because of the approximative character of the formalism), and secondly, an estimate of the cutoff is possible. This estimate shows that the assumption A**pin the bare particle calculation was a very good approximation. Kiill6nts result -- finite radiative corrections by means of strong interactions --is in striktng disagreement with works of other authors, who included the modern concept of current algebra in the calculation of radiative corrections in 6-decays. Though K61t5n did not solve the problem of the influence of strong interactions on the convergence of radiative corrections in weak decays (by means of his form factor method), it turned out to 'an important controversial question in this way, still lacking be a final satisfactory solution. Kei116nwas one of the first who used reduction formalism, dispersion calculations and spectral representations in alI his works , methods which became standard tools in modern physics. Surely Kdll6nts works have contributed much to the fact that field theory is applied in elementary particle physics more than ever. Furthermore, Kiill5n has earned considerable merit in the field of elementary particte physics as the author of an excellent book in which many problems of strong and weak interactions are treated. Here, just as in his conference lectures, K61I5nproved his outstanding pedagogical talent. In his book he has shown excellently how much about mathematical methods and detailed calculations should be presented, enough to clear upthe connection between theory and experiment, but not so extensively that the presentation could be spoiled. r hnno ihat trl a certain extent I have been successful in doing r

rrvyv

lx

justice to the personality of Gunnar Kiill6n and his position in science. The reader will certainly agree if I emphasize again that his early death undeniably has left a gap among the most out_ standing theoretical physicists of Europe. Paul Urban

These remarks are a condensation of those appearlng 1n ',particle Physics", Acta Physica Austriaca, Supplementum 6, Vienna, New York; Springer-Verlag (1969) .

GUNNAR KALLEN IN MEMORIAM

G u n n a r K i i l l 5 n w a s b o r n F e b r u a r y1 3 , 1 9 2 6 , s o h e w a s o n l y 4 2 years old when he died in the fatal airplane crash on October 13, 1968. In spite of the short span of time in which he was active tn physics he left behind him a large number (about 60) of original papers, conferenoe reports, lecture notes, and monographson many different subjects of modern physics, in particular in the domains of quantum electrodynamics, quantum field theory in qeneral, and elementary particle physics. It wilt not be possible, a n d i n t h i s c i r c l e a l s o n o t n e c e s s a r y ,t o m e n t i o n a l l t h e s e p a p e r s here, but I shall try to give an outline of his main contributions to our science in the different stages of his nineteen years of activity in physics. As so many other physicjsts he started his career as an engi_ neer. Twenty-two years old, he came to Lund to pursue hi.s studies of theoretical physics at the University. With amazing speed he caught up wlth the problems and soon he was working at the front line of our knowledge at that time. The main subject of interesr among the theoretical physiclsts in Lund and elsewhere at that time was the new method in quantum electrodynamics which was initiated by Kramersin 1947,and which seemedto make it possible to evade the disturbing divergence difficulties, inherent in the 'of quantum formalism electrodynamics, by a renormalization procedure. In the following years this program was successfully carried through by Tomonaga, Schwinger and Feynman by makinq use of the so-called interaction picture. r6tt5n was fascinated by this difficult subject and by the challenge it represented. His first p a p e r a p p e a r e di n t h e H e l v . P h y s . A c t a i n 1 9 4 9 . I t c o n t a i n e d a treatment of the higher approximations in the vacuum polarizatron. This problem was suggested to him, during a short visit in Zirich, b y W o l f g a n g P a u l i w h o w a s m u c h i m p r e s s e db y t h e y o u n g s t u d e n t , s quick and independent mind. Kiill6n, on the other side, admired Pauli immensely and took him as a model for his future work. The relations between the older and the young physicist developed into a Iife-long warm friendship, which also led to a fruitful collaboration between them in the later years. A-fterhis return to Lund, Kiill5n set himself the task to carry through the renormalization program without the use of the inreraction representation which he regarded as an unnecessary math_ ematical complication. In a series of papers leading up to his

x1r Inaugural Dissertation in 1950,he was able to show that the ideas of renormalization can easily be formulated in the original Heisenberg picture, and that many of the calculations are simpler and their physical interpretation more transparent in this picture. In these papers the notions of free "in"- and "out"-fields were defined clearly for the first time, and a method was developed which ln the literature often has been called the Yang-Feldman method. The reason for this.is probably that Yang's and Feldman'spaper appeared in the Phys. Rev., while K6iI15n'sfirst paper on the sub* ject was published in fuk. f . Fysik. Since these papers appeared nearly simultaneously and were produced independently, there is no room for any priority claims (and Gunnar would have been the Iast to make such claims), but one thing is certain: K5115nmade much more extensive use of his method for practical calculations, and soon he was also recognized by his colleagues everywhere as a master in his field. His briliiant appearance at international conferences, starting with the Paris Conference in the spring of 1950, contributed much to this. His elegant way of presentinghis points of view and his sharp and witty dialogue in the discussions made him an excellent advocate for his ideas, which evoked the admiration of his older and younger colleagues. One of the latter wasA.S. Wightman who later wrote about the early work of Kdll5n: 'At that time I was trying to puzzle out the qrammar of the language of quantum field theory, and here was Kiill6n already writing poetry i n t h e l a n g u a g e ." Gunnar f5:Il6n's connectlon with CERN dates back to the very first years of this organization. Already in 1952, when the site in Meyrin still consisted of a collection of deep holes in the ground and a few shacks , Kiill6n became a Fellow of CERN's Theoretical Study Group, which at that time was placed at the Niels Bohrlnstitute in Copenhagen. I remember vividly his appearance there, which brought exciting new life to our group. He gave a series of admirable Iectures on quantum electrodynamics, which clearly showed his superior mastery of the field and his exceptional gifts as a lecturer. Simultaneously, he pursued with characteristic energy a plan which he had conceived after the completion of his Doctor'sthesis. The current renormalization theory was based on a series expansion in powers of the fine structure constant and, although each term in this expansion was finite and showed a surprisingly good agreement with the experimental results, the convergence of the series had not been proved. Thus, it was still an open question whether renormalized quantum electrodynamics could be regarded as a consistent physical theory or whether it only represented a handy cookery-book prescription for getting useful results. The answer to this question was of great principle importance, but also so difficult to obtain that it required all the courage and tenacity of a Kdll5n to attack and finally solve the problem.

xiii By means of the formulation of the renormalization theory he had given in his thesis, he was able to define the renormalization constants without making use of perturbation theory. Tn a series of papers in Helv. Phys. Acta and in physica he showed how this can be done and, in a final paper in the proceedings of the Danish Academy (which later was reprinted in special collections both in Japan and U. S, A.) he proved that.at Ieast one of the renormalization constants had to be infinite. Thus, he had come to the conclusion that renormalized quantum electrodynamics could not be regarded as a completely satisfactory physical theory, in spite of the success of the perturbation theory version of the theory in accounting for the experimental results. On the other hand, the latter circumstance gives good reason for believing that thepresent formalism may be regarded as a limiting case of a future more complete theory. Krill6n was a Fellow at CERN's Theoretical Study Group from October I, 1952 to June 15, 1953. During this period, his professional ability and his personality had impressed us so much that we naturally tried to get him on the permanent staff of the Study Group. AJter he had finished a second longer stay in Zi.irich, he joined our staff in October I954, where he remained until CERN's Theoretical Study Group finally moved to Geneva in September 1957. Thereafter, he accepted a chajr as professor at the simultaneously established NORDITA in Copenhagren, where he stayed until a personal professorship was created for him at the University of Lund at the end of 1958. Thus we had the privilege of having Gunnar with us as collaborator in Copenhagen during more than five of his perhaps most productive years. It is impossible in a few words to describe how much we owe him as a constant source of inspiration, as a teacher, and last but not least as an always alert critic. The ruthless honesty and objectivity of his criticism, which soon became legendary, recalled that of the young pauli. It has even been said that Gunnar modelled his style on pauli, but this was only partly true. I rather think that the similarity in their reactions was due to an inherent kinship of these two original personalitres. In Ziirich KtiiISn and Pauli had started a fruitful collaboration which was continued after Gunnarrs arrival in Copenhagen. It resulted in a paper "On the mathematical structure of T. D. Lee's model of a renormalizable field theory', which was published in the Proceedings of the Danish Academy in 1955. AJthough the Lee model is non-relativistic, it is of great interest as an illustration of what might be hidden in the more complicated formalism of quantum electrodynamics. The advantage of the model is that it contains a renormalization of both the coupling constant and the mass, and still is so simple that it allows exact solutions. The main result in the just mentioned paper was the surprisinq discovery that the renormalized Lee model contains an unphysical state--the "ghost" state--which has a negative probability. It is

xrv quite possible that the formalism of renormalized quantum electrodynamics also contains such unphysical states, which further supports the view that quantum electrodynamics, when taken 1iterally, does not represent a consistent physical theory. A fortiori, this holds for the current meson theories which, for obvious reasons , do not lend themselves to a perturbation treatment. Therefore, in the following years and in particular after his retwn to Lund, Kdll6n joined in the trend of research, which has been called the axiomatic way, and which was being pursued at several places in Europe and America. Instead of investigating the properties of a definite formalism, the idea was to see how far one can go by startingl from a few general physically necessary requirements, such as relativistic invariance, causality and positive energty. With his usual energy, t<5tt6nthrew himself on this seemingly infinite problem, which consisted in investigating the mathematical properties of the vacuum expectation values of the product of an arbitrary number z of field operators. In collaboration with Wightman, Toli and several of his students ln Lund, he obtained many important results, in particular for lx = 3 and n= 4' However, the general "to-point function"turned out to be so complicated that the problem could only be partially solved. This is probably the only problem, attacked by Gunnar, which he had to Ieave without having obtained a complete solution. In parallel with these more mathematical investigations, he also worked on problems which had more immediate physical applications. I am thinking of his calculations of the radiative corrections to weak interactions, in particular to the nucleon-decay and the electromagnetic and nucleon form factors, a work which inspired his pupils in Lund to interesting investigations. After KilI6nts appolntment as a professor at the University of Lund, much of his time was occupied by teaching and guidj'ng students, and his natural gifts as an educationalist were brought into full play. His review articles and monographs, in particular his article onQuantumElectrodynamics in the Handbuch der Physik (1958) and his book on Elementary Particle Physics (1964), are lasting witnesses of his pedagogical faculties. He gathered around him a large number of Swedish and foreign pupils, who benefited immensely by his profound knowledge, his lucid lectures and his objective criticism which fortunately was linked with a deep sense of humor. The latter most essential characteristic allowed him in the course of the years to develop a totally harmonious and wellbalanced personality. A contributing factor in this respect was undoubtedly his happy family life with his charming children and his lovable wife Gunnel. She was an intelligent womanwitha strong personality. The bravery and courage which she showed when her husband suddenly was taken away aroused the admiration and compassion of all her friends. On the day of the funeral she said to my wife: "AJter this I am not afraid of anything in the world, my only ardent wish is that I may keep my health". This

,xv wish was not fulfilled; only half a year later she followed her husband into the grave. In the light of history, Gunnar Kiill6nts appearance in the world of physics was like a shootlng star. It was short, but so ardent, so shining that his name will be remembered, not only by those who knew him personally or even had the good fortune to become his friends, but also by the coming generations of physicists. He will be sorely mlssed.

C. Mdler

Thls biography ts taken from the proceedings of the Lund Conference on Elementary Particles, 1969; G. von Dardel, editor.

TABLE OF CONTENTS

C H A P T E RI . 1. 2. 3. 4.

G E N E R A IP R I N C I P L E S

FieId Operators, State Vectors, Periodic Boundary I Conditions .3 Different Pictures in Hilbert Space Lagrangian, Equationsof Motion, and Car,onical . .b Quantization . .I0 T r a n s f o r m a t i o nP r o p e r t i e s o f t h e T h e o r y

FIELD C H A P T E RI I . T H E F R E EE L E C T R O M A G N E T I C q T.Adranoe 16 F r r n c t i o na n d C a n o n i c a l F o r m a l i s m . 2 0 o f P h o t o n s F r e e H i t b e r t S p a c e T h e 6. 7 . C o m m u t a t o r sf o r A r b i t r a r y T i m e s , t h e S i n g u l a r 26 Functions . . . 32 M e t h o d F i r s t 8. The Subsidiary Condition: . 38 9 . T h e Subsidiary Condition: SecondMethod 1 0 . T h e Problem of the Measurement of the Electric and , 43 Magnetic Field Strengths 1 1 . T h e E l e c t r o m a g n e t i cF i e l d i n I n t e r a c t i o n w i t h 46 ClassicalCurrents, . lsYf

v..Yv

CHAPTERiII.

r

s

THE FREEDIRAC F]ELD

1 2 , E q u a t i o n s o f M o t i o n , L a g r a n g eF u n c t i o n , a n d a n 55 Attempt at a Canonical Quantization . 13. Quantization of the Dirac Field by Anticommutators . 5B 62 1 4 . T h e C h a r g e S y m m e t r yo f t h e T h e o r y S p a c e . t i n 1 5 . A n t i c o m m u t a t o r sa n d C o m m u t a t o r s .64 The S-Functions. 1 6 . T h e D i r a c E q u a t i o nw i t h a T i m e - T n d e p e n d e n t , .69 External Electromagnetic Field . CHAPTER]V.

THE DIRAC FIELD AND THE ELECTROMAGNETIC THEORY FIELD IN INTERACTION. PERTURBATION

.75 L a g r a n g i a na n d E q u a t i o n s o f M o t i o n . 79 P e r t u r b a t i o nC a l c u l a t i o n s i n t h e H e i s e n b e r g P i c t u r e . 8 5 The S Matrix Treatment of Quantum Electrodynamics by N4eansof 89 a Time-Dependent Canonical Transformation .93 2 1. C a l c u l a t i o n o f t h e P - S y m b o l . T h e N o r m a l P r o d u c t . 98 2 2 , A G r a p h i c a l R e p r e s e n t a t i o no f t h e S - M a t r i x 1 7, 18. 19. 20.

):VIl I

23. The Physical Interpretation of the F-Functions CHAPTERV.

.

103

ELEMENTARY APPL]CATIONS

24. Scattering of Electrons and Pajr Production by an Fvfornal

Fiald

,q,

Qaal-*orinn

26.

Bremsstrahlung and Pair Production by Photons in an

27. 28.

^{

T idht

l.rrr an

v ! l f Y r r L ! f g r J U J v v 4 v r r .

Fvtarnrl

I'lanl-rnn

Fiald

Scattering of Two Electrons from Each Other Natural Line Width .

CHAPTERVI.

r07 .II2 Il6 122 125

RAD]ATIVECORRECTIONSIN THE LOWESTORDER

29 . Vacuum Polarization in an External Field. Charge I32 Renormalization . 3 0 . R e g u l a r i z a t i o na n d t h e S e l f - E n e r g y o f t h e P h o t o n . 139 3I. The Lowest Order Radiative Corrections to the I44 CurrentOperator. . 32. Mass Renormalization . 150 33 . The Magnetic Moment of the Eiectron . 154 3 4 . T h e C h a r g e R e n o r m a l i z a t i o no f t h e E l e c t r o n S t a t e . . 1 5 7 35. Radiative Corrections to the Scattering in an External 162 Field. The Infrared Catastrophe . . '' e+-,,^+r,-^ ^f L^ -.,^-^^^3 6 . T h e H y p e r f i n\ ^e S f t+ h tructureo e H y - , v v ' r r nn oL^ v' . r r r . . . 1 6 7 37. Level Shifts in the HydrogenAtom: The Lamb Shift . I74 38. Positronium. . .L82 39. A Survey of Radiative Corrections in Other Processes . 190 CHAPTERV]].

GENERAI THEORYOT RENORMALIZATION

40. General Definition of Particle Numbers . . . 195 4 I . M a s s R e n o r m a l i z a t i o no f t h e E l e c t r o n . . 198 4 2 . R e n o r m a l i z a t i o no f t h e D i r a c F i e l d . E q u a t i o n f o r the Constant try'. .203 4 3 . The Renormalization of the Charge . , 207 4 4 . General Properties of the Functions II(pz) and rd(p'z),.212 AE The Physical Significance of the Functions Jf (22) and fr(p') . Connection with Previous Results 216 AA 2I9 Charge Renormalization for One-EIectron States 4 7 . Proof that the Theory Contains At Least One Infinite Quartity . 223 48. Concluding Remarks . . 229 INDD(

. 231

aUANTUN{ ELECTRODYNAMICS

CHAPTER G E N E RA L

PRINC]PLES

field Operators, State Vectors, Periodic Boundary Conditions The general mathematical principles of quantum electrodynamics or any other quantum field theory are copied from those of the usual nonrelativistic quantum mechanics of (point) particles. Hence quantum electrodynamics also deals with a state vector in a "Hilbert space"and with a set of linear operators in this space, The latter are the dynamical variables of the "field operators". the theory and correspond to the classical fields, in the sense of the correspondence principle. As an example, certain operators correspond to the classical electromagnetic potentials. The fourdimensional space-time coordinates xt: x, rz-- !, xs:2,x4-- i xo: i,6tr, which we shall often designate simply by x, are not operators in the sense of the nonrelativistic qudntum mechanics. Rather, they must be understood as "indices" or "labels" for the field operators. Thus with each point , there are associated a finite number of field operators (eight in quantum electrodynamics), which we shall designate as q"(x) in the first chapter. The index a distinguishes tha rrarintrc fialdc. fnr ovemnla il- r'lietinnrlichoc l - h a f n r r r n- n-r . n p o nents of the electromagnetic potential and the four Dirac field operators. The quantities p*(x) and go(x') must be regarded as two independent operators when they are referred to two different points x and x', Since each finite volume Z contains an infinite many operators or number of points x t our system has infinitely an infinite number of degrees of freedom. This is an important difference between a field theory and the usual mechanics of a point particle, where the number of degrees of freedom is always which we shall later enfinite. The mathematical difficulties counter are always closely connected with summations over an infinite number of degrees of freedom. For practical calculations, it is usually convenient not to work directly with the field p"(x) in x-space, but instead to employ the Fourier components of the field as the independent variables. For this purpose, we imagine the fields as being enclosed in a iarge we requirethe fieldstobe cube of edge Z and volume V:1s,and periodic in r-space with period Z . Under these conditions, the field operators can be written as sums: I.

(i.l)

v * Q ) : , , t . -I e i \ ' v , ( h ) V VT HArA ,A sfands

for a fouf -vector

with

comnnnaniq

2 5 .- , f '

of c

, 6ni

^

-

-

.

.

a

a

/

^

G. Kallen, Quantum Electrodynamics

Son

I

the product px is to be understood as the scalar product of the ...* !axa. In a similar way, we shall designate vectors:Px:ft\* the cnrrarc of a forrr-rrec.for blnv b2. This latter arranfifrz is ihere-1

foro

nnqiiirze

for

snacc-like

r

.

7.erfi fnr

linhf

-likc

YggrrgrL]

4U

. and

ncaatirze

u:r

r r v Y v ! l v \

for

time-like vectors. When we use the spatial (three-vector) part of a four-vector, we shall designate it by boldface: p. The summation in (1.1) is to be taken overall possible values ofthe spatial components of the vectorf, Because of the periodic boundary conditions, only those values of p are allowed which have components satisfying the condition

f r : n r 2 r L ( i : 1 , 2 ,r ) , where

the

n.

are

nnsiti\/e

or ncnafirrc

intodarq

(r.2) Tn o anoral Y ! r r e f v ! ,

aS

r r r

a

consequence of the equations of motion, the fourth component 24 is a many-valued function of p. Equation (l .l) must or .po:-ipn, therefore be summed over all these possible values of fo. The number of different values of po is determined not only by the physical theory but also depends on the representation employed in Hilbert space. In Sec.2 we shall give a more detailed discussion of the various possibilities for this representation. I n E q s . ( 1. 1 ) a n d ( l . 2 ) w e h a v e p u t f r : l . This is only a matter of the units employed and does not have any physical significance. Likewise, we shall often take the velocity of light c equaltol and thus obtain a common unit for length, time, and reciprocal mass. F v rr t o

cxannlc.

vi>errrts+v

if fhc

rrnii

of

lcnnfh

fhc

ccntimpicr

iq iaken

as

the

basic unit, then the unit oftime is the time required for light to travel a distance of I cm. and the unit of mass is that mass with a Compton wavelength of 2n cm. These units are often referred to in the literature as "natural units". E^r

!,dr,, lrrra

-,n1116pq

1z

a 1l tha

rosrrlts

of

the

are of physical interest must tend to finite limits. pens that a sum of the form

iLhr r ee ovrr v J

w vvh i C h

It often hap-

s: + Ztte)

( 1 .3 )

occurs in a result where either l(d i" independentof V or tends to a f i n i t e l i m i t a s V - > a . W e s e e i m m e d i a t e l y f r o m E q . ( 1. 2 ) t h a t the number of states with spatial vectors p lying between p and p+dp is given by (

"d . ' b' . \ .^2t, 1\ '*ar 'b^"rdj ^hr ,zd b " : l, 2^ nv l.t^ t Thus it follows that as V-->a,

(r.4)

the sum S tends to the limit

t*o;,n I o'ottnl.

0 .s )

T h e s y m b o l 6 t . p i n ( l . 4 ) a n d ( 1. 5 ) m e a n s t h e t h r e e - d i m e n s i o n a l

Different

Sec.2

Pictures in Hilbert

3

Space

Later we shall also use four-dimenvolume element in 1-space. sional volume elements and designate these simply as dP (nq! as d,ap). Different Pictures in Hilbert Space It is well known that in ordinary quantum mechanics the time evolution of the system can be treated in several ways. For ex2.

amnio ertrP.Lv

nno

aan

o v rm r r Pn . Llvn f rz

a

nicfrrro u

s

ytvLsr

uzhoro

tho

n n o r aet nL rv < r vvvr

aro

fjry1g-

independent quantities, and aIl motion of the system is described by means of the state vector. Alternatively, one can regard the state vectors as constants and describe the evolution of the system by a time variation of the operators. In the first case, we have the "Schroedinger picture" and in the second, the "Heisenberg picture'r. These two possibilities are also present in field theory. In a way, they are extreme examples of the more general possibility in which the operators as well as the state vectors are treated as time-dependent quantities, and the precise t'distribution" of the time dependence between the two is a matter of convenience. In the Schroedinger picture, the time variation of the state vector is given by the Schroedinger equation

(2.1)

o!]#):HWU))

Here lg(l)) is the state vector, and the Hamiltonian operator H is a Hermitian operator depending on the dynamical variables. Ina f i e l d t h e o r v . F f d e n e n d s r r n o n f h c f i c l d o n c r a f o r s . T T n d c r. e r t a i n conditions, 17 can be time-dependent, but then the time must occur explicitly and not implicitly in the field operators. In this case, the system is not closed but is interacting with external sources. v y v l s u v r u r

T.afor vze qha lI

freorrcnilrz

cnnsider

su grvrr cl h

q vstcms: J f

ir rnr

C v r rhg a y |n

- T ! , ,

fOf

simplicity, we shall limit ourselves to closed systems, i.e. , to time- independent Hamiltonian operators. of The generalization the results of this chapter to systems which are not closed can usually be made without difficulty. We may therefore assume

!,ls!\")) : 0 .

(2.2)

at

In a field theory, .H usually contains the field operators at all points of three-dimensional space and can be written asan integral of a Hamiltonian density #:

H:ld3x,tr(E"@)).

Q.3)

in this picture, Although the operators are time-independent "time derivative" of an operator F(n) can be defined by

F(u) : i lH, F(n)): i(H F(r) - F(n)u) . This operator is

obviously

not the actual

derivative,

a

Q.4) but does

G . K i i t l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec. 2

have the property that its expectation value is the time derivative of the expectation value of the original operator:

(,pQ)| h @)tv Q)>: * <,p (t)| n @)ly (t)).

/,

c)

The proof of (2.5) follows immediatelv from the definition (2.4) and the Schroedinger equation (2.1). As is clearly evident from this discussion, the spatial coordiand the time f,o are treated in quite different nates ," (n:t,2,3) Because of this, the relativistic covariance of the theory ways. has been lost. However, it is often quite useful to employ this covariance in order to prove general results from considerations of symmetry, and sometimes it is even necessary to use such in order to qive meaning to mathematinvariance considerations It is appropriate to introduce anicaily undefined expressions. The is clearly exhibited. invariance this picture which in other Heisenberg picture is completely symmetric in a11 coordinates. and there is no Here the state vectors l?) are time-independent, the equations ( 2 . I ) . a r e t h e r e I n s t e a d , t i k e schroedinger equation o P e r a t o r s , f i e l d t h e o f of motion

!'!)

:ilH,F(x\l

(2.6)

dt

T h e E q s . ( 2 . 6 ) a p p e a rf o r m a l l y t h e s a m ea s E q . Q . a ) ; h o w e v e r , here we are dealing with time-dependent quantities, so that Eqs ' (2.6) are the actual differential equations which can, in principle, be used to describe the time evolution of the system. since the Hamiltonian operator commuteswith itself , it follows directly from (2.6) that its time derivative vanishes. Accordingly, in the Heisenbergpicture also, I{ is time-independent. we can establish the connection between the schroedinger and Heisenberg pictures with the formal solution of (2'1):

l v ( r ) ): t - n o ' l r p ( o ) )

(2.7)

If we designate the operators in the Schroedinger picture by F(O) , we see that the canonical transformation F(t) :

eiHtF(o) e-i,t

( 2. 8 )

gives the operators in the Heisenberg picture. The operators F(l) in (2 . B) obviously satisfy the differential equations (2.6) with the Inthe boundary conditions that both pictures coincide for l:0' (2'7)' o f q u a n t i t V i s t h e p i c t u r e , v e c t o r t h e s t a t e l r p ( O ) ) Heisenberg i n the u s e d Besides these two pictures, a third one is often in time long a for used been has Strictly speaking, it literature. been years has it recent in however, quantum electrodynamics; particularly emphasized and has received the name "interaction

Sec. 2

Different

Pictures in Hilbert

Space

5

representation" in the works of Tomonagal and Schwinger2,3 . To construct this picture, we begin with the Schroedinger picture and transform the operators by a canonical transformation which is not generated by the operator for the total energy, but only by u purt n f v !

i+

r L ,

U

\A/a

r r 0

rrrri+a

Fr@ :

( 2. e )

eiH"tF(o) e-n'o' ,

where

( 2. r 0 )

H:HolHr.

In this picture the operators as well as the state vectors are tlmedanonr]onf

Tho

lattar

nrrantif

iac

fhan

caJ'i cfrr

:

rrQnlrrnad'

-- *rnger

equation" of the form i

2

u,lvwUD

: s i H oHt r e - i H oltv , r ( t ) ) : H , . * ( t ) l r l * @ ) , ( z , t r )

and the operators F* obey the differential equation

3P:ilHo,F*(t)).

(, 1r\

The advantage of this picture is that with a suitable choice for I/o , Eq. (2.I2) can be solved explicitly, and so part of the problem can be solved formally. The differential equation (2.1I), which obviously contains the remainder of the problem, is usually so complicated that the solution of (2,I2) has not really accomplished very much. Yet, in the historical development of modern quantum electrodynamics, this decomposition of the problem into two parts has played an important role. This is chiefly because the decomposition (2.10) often can be done so that ,[1, , the so-called interaction energy, is a three-dimensional integral over an invariant density function. The corresponding statement is valid nejther for H nor I/o . in a sense , the differential equation (2.11)is invariant, and considerations of invariance and symmetry can be applied to it and its solutions. Tomonaga and Schwinger have particularly stressed this invariance; they do not write Eq. (2.1I) as a time derivative, but introduce a system of spatial surfaces and consider the changes in the state vector with infinitesimal variatrons of these surfaces. The resulting formalism is extremely elegant and has played an important role historically. We shall not consider it further because we are not concerned with the historical development, and because there is no real advantage to the use ofthe spatial surfaces. Moreover, it is clear that one has real

1 . S . T o m o n a g a , P r o g r . T h e o r . P h y s . I , 2 7 ( 1 9 4 6 )a n d l a t e r articles. 2 . J . S c h w i n g e r , P h y s . R e v . 7 4 , 7 4 3 9 ( 1 9 4 8.) 3. (Translator's note) The original term "interaction representation" is now generally replaced by the term "interaction picture".

G. Kd116n,Quanturn Electrodynamics

Sec. 3

covariance only if all four coordinates are treated in the same way, and this is the case only in the Heisenberg picture, which is therefore best suited for considerations of invariance. The transformations (2.9) and (2.8) have been chosen so that This prescripall the pictures related by them coincide for t:0. and if one starts with tion obviously involves some arbitrariness, the differential equations (2.II) and (2.1), this arbitrariness appears in the boundary conditions. It is often convenient to choose the boundary conditions so that the interaction picture and the Of l:0. oo rather than for Heisenberg picture coincide for l:course this is actually done by introducing a suitable Iimiting process. Although this requires special mathematical tricks, the physical interpretation of the theory is simplified so much, at least for scattering problems, that we will tolerate these matheLater we shall return to this point in detail. matical difficulties. -3. I n Lagrangian, Equations of Motion, and Canonical Quantization this section we shall consider only those theories which We therefore assume can be derived from a Lagrangian function. that there is a density function

ql-

*

\rd

( r . \' a q " ( x \1 \--i

a.r'u

l

and that it is an invariant function of the field variables g"(r) and their first derivatives with respect to the four coordinates ,r . integral From this density we can form the four-dimensional

L:

av"u)\ r d x 9 ( o - 1 9 . A * , I t.

J

(3.r)

\'^'

which is the Lagrangian function.l In a classical theory the equations of motion are obtained from the requirement that the integral (3.1) be stationary with respect to variations in the fields g"(r) : 4^

/-

\]a 4

a9 Ot,,

a9

--:-;-T opa\x) u" t u

(3.2)

apq\xl

In a classical theory it isalso possible to constructa canonically conjugate momentum ze,(*) [abbreviated n,(x)7for each variable, n-\xl:-T-:,

ag

^ oVa\x) u__

I

( 3. 3 )

ofro

@) defined as Z: the "action".

In customary usage tlre "Lagrangian" is known w hile the integral A--ldxt' d ' " x 8 , J

1S

Sec.3

Lagrangian, Equations of Motion, Canonical Quantization

and hence to calculate

the classical

Hamiltonian

function

u^Y."\.( 3 -yl : H (r* H : I a "* 1 7n *(r)0 (e) :.'*' .4) * , {' -", d x k ' o \ J l? I 9t*' in (3.4) it is necessary to express as a functi on of no(x) ( 3 . 3 ) by the use of tq. and to remember-that it is this expression which is used in. 11 . Thus the Hamiltonian function is understood to be a function of the momenta, the field variables, and their spatial derivatives . We must assume that the time derivatives of the field variables can actually be eliminated in this way. In the quantized theory we shall now require that the field operators and momenta, which are defined by (3.3),obey the canonical commutation relations

l n " ( * ) ,p e @ ' )- ) - i 6 * B6. ( r * n ' ) , : o ln"(*), nB@')7:lv"@),VB@')l rn

r

niafrrra

urhara

rhp

ontrratorq

^rc

f i m c - d eunst|r/ ns d r reu n e tr .r L r : : : l : " h o t h

(3. sa) (3. sb) fimeS

x,r and rL must be t1re same. This is the canonical quantization prescription, which we regard as a postulate.r Just as in ordinary quantum mechanics, it follows from (2.8) lor (2.9) in the case of the interaction picture] ttrat, if the commutation relations are nbeved for one time then fhcv a]sn hnld for ^0 - ^0 -n arbitrary times . Equation (3.4) will now be regarded as an operator equation, and the operator .EI defined there will be used as the Hamiltonian operator for the quantized theory. In principle, this prescription is not aiways unique, since the result can depend upon the order in which the operators are written. In actual use, however, this does not give rise to any essential difficulties. We shail now show that in general the quantum mechanical equations of motion in the Heisenberg picture agree formally with Eqs. (3 .2) and (3 .3) if the latter are taken as operator equations . To do this , we calculate the commutator of 11 and nr(x) . Since two momenta commute with each other (for equal times) it follows that t\' - H , n " ( x ) ) :" - 1- ' 1 1a E P t ' ' t *r o /\ x \ l - { ( x ' t , r r " ( " ) l.f ( 3 . 6 ) t ;,, , -) ldsx'1 1 ) t7 " B r ^ t u^o 'i'!"

Obviously the commutator of a momentum and a function of the coordinates is equal to the derivative of the function with respect I. J. Schwinger has attempted to deduce the canonical quantization prescription from a variational principle. We shall not discuss this point further here, but refer the reader to the original rrrnrL.

Dhrzc

P a r rv

'

A) ::-l

qll

/lqC,l\

G. (att6n, Quantum Electrodynamics

B

to the coordinates multiplied by the commutator of the momentum Using this resuit we can compute the last and the coordinates. term oI (J . oi: 3

i6(r -r')+ , lg(*'),n"(x)l: ,17;,, ,Yc\^) A.gspg?

i o x hA@-n')>,---;--------,. "., = adv!t:! 0*'u

_' ) /-t -I

\3./)

AE l.aEB(x'tn (,\l 0 E B 1 x ' 1| a t ' o ' " " ot ' " t l ' ^ a-raxo

From Eq. (2 .6) we have

?norx\* ag - , 1- A 0^'o 0go(r) L a^u

Ag A E o l * 1f " oxh l a- 'o: 'o, ( x ' ) ,l X I . 7 t*-"( 1x ) l . I oro

o aW). r, ,',!,"0" 4?'@') l,r.

In a simiiar way we obtain the commutator of l/ and g"Qc)z

l H , E " @ ) 7 - - _ , {{ -- ;o!'v* "' \ : l L 6 @ * n ' , ) * l n B @ ) x 1,..n, x'o:to x ,"," (r)] ls(r,),e,(r)ll,l lav_pl:'t

: >;#rrlff, ts(r'), v"?)t ,,

"

E.(")],

_

( 3 .l o )

Ar,o_

g4! : - u?".{,r,,1|!34'_, r /s- Jf 'a",,' s",(")1. (3.11) " In,,,', ' +9+ b,'o c.yo o)'o l'"0'-' ,08p0') ll P x!:/o

L

"

7xi

I

a'p:(*\ appears on the right-hand^side, Axn 28"\x) it is to be regarded as the same"function of no(x) ,9o(x)and # as one has in the classical theory. Equation (3.1I) must then determine the corresponding function in the quantized theory. It obviously has the solution that the classical and quantum mechanical functions are equal. That is, (3.11) is equivalent to Eq. (3.3) conIt then follows that (3.8) is the sidered as an operator equation. This concludes the operator equation corresponding to Eq . (3.2). proof. The difficulties which arise from the order of the operators Since it in the Hamiltonian have not been taken into account. will lead to no serious difficutties in the applications, we shaII not pursue this question further. Trlct as in the cla qsica I iheorv - ther:e are also conservation We laws for energy and momentum in the quantized field theory. have already shown that the conservation of energy in quantum In Eqs. (3.6) to (3 .lI) , when

J

qp

L

u

r

Sec. 3 Lagrangian,

Equations of Motion,

Canonical euantization

9

theory is a trivial consequence of the fact that H always commures with itself, and therefore that the time derivative of the total energy must vanish. As in the classical theory, this conservation law can also be proved by direct calculation, starting from the operator equations (3.2) and (3.3). Also as in the classical theory, the three spatial components of the momentum are defined bv

pa:- I d'r4r"@)W

(1,1r\

and, in the usual way, we have

lPa,v"@)): i I d.'x, 6(a _ n,)3!!y1 : ; aq_,(,) oxh ?xn ro:z'o

( 3. 1 3 )

n,(x)l: -,r{,!', n.(*') lPe, #oi 6(t - u,): , 9W, .

(3.14)

From these Iast two equations, it follows that any operator F(r) which is constructed from gn@)and zo(r) satisfies (3.1s) lp".F(x\j:;!r?) 0'n I n t h e H e i s e n b e r g p i c t u r e E q . ( 3 . 1 5 ) c a n b e u n d e r s t o o da s a c o u n t e r p a r t o f E q . ( 2 . 6 ) . I n t r o d u c i n g t h e f o u r t h c o m p o n e n to f t h e m o mentum by the definition

Pt:iPo:iH,

( 3. 1 6 )

we then have

l P , , F ( r \ 1 : ; a ox" !(x)

( 3. 1 7 )

for lt:I.to 4. The Eqs. (3.17) are relativistically covariant , and will be very important in what follows. In Sec. 4 we shall derive them in another way. lfwe insert the energy density ff in (3.15) and use the periodic boundary condition, we obtain

l P oH, '1J d: x ni I a "x - 2 * : 0 .

( 3. i 8 )

As in the classical theory, the components of momentum are conserved quantities under the quantum mechanical motion of the system. In a similar way, any two spatial components of the momentum commute with each other. Thus it follows in qeneral that

lP,,1): o.

( 3. 1 e )

In principle it is therefore possible to choose a representation ln which all four quantities We shall designate { are diagonal.

10

G . K a 1 I 6 n , Q u a n t u mE l e c t r o d y n a m i c s

the state vectors of this representation by the corresponding eigenvalues of \ show that {or an arbitrary F(*) in (3.17),

t

a+n

h(a) rp

.a

_ py)l(alF(x)lb) : ?plf, (ol lp,,F(x)llb> In this representation the fore given by

lt a \ --/

pI

ox"

lt /" r/ \

(an

t

i)na

a e uf .v . , aan vgrr

L

a s .nr rs ' l

rr au:sdvi *l rr jr

( a l F ( x ) I b ) . ( 3 .2 0 )

x - d e p e n d e n c e o f e a c h o p e r a t o r is there-

"' ( a l F ( x ) l b ): ( a P 1 u S, o | @ ' i ' - P ' ?,) where the quantities (alFlb) we shall refer to this remark.

(3. 21)

are independent of r.

In Chap.VII

Transformation Properties of the Theory the original In the canonical formalism developed so far, Certainly the relativistic invariance of the theory has been lost. equations of motion (3.2) and (3.3) have the desired covariant 4.

form

aq

lona

as

fhc

T.aaranaian

iq

an

inrrariant

Hnwerrcr, r r v v ! v

v v r

/

the

canonical commutation relations (3.5), which allow the proper transition to the quantized theory, are not covariant because the two times ,0 and ri are assumed equal. In this section we shall further investigate the properties of the quantized theory under In particular, it will appear that Eqs . Lorentz transformations. ( 3 ^ 1 7 ) n o t o n l v F o r ma c n \ / A r i ^ n t s r z s f c m h r r t a l s n t h a t t h c v h a v e for the transformation properties of a fundamental significance the whole theory. We assume that we have two Lorentz frames, n and r'and that ifa point P has coordinates * in the first system, its coordinates in the other system are given by \w

.

,1,

/

rrvL

vrrrl

av1

rrr

rYrr!

xp:

v

1

f i p | . e , , , , x 1, 6 , , ,

( 4. 1 )

Here , and in the following, we sum over repeated indices . For simplicity we further assume that the quantities F,, and bu are so that their higher powers may be neglected infinitesimal, Since the Lorentz transformations form a group, it is sufficient to study the tansformation properties of the theory under infinitesimal In order that (4.1) describe an actual Lorentz transformations. transformation without stretching the coordinate axes r rp, rflust be chosen antisymmetric. The transformation properties of the classical fields are known and have the form

VL(x'): L sp,S,,,oB Ve@@'))I q"(x (x')).

(4.2)

The left side of this equation gives the field funptions in the new On the coordinate system as functions of the new coordinates. right side are the field functions in the old coordinate system,

Sec. 4

Transformation Properties of the Theory

evaluated at the same point Rrr

cnlrlinc

/1

l)

f^.

P which is present

1l

on the

lAff

cid6

x",

xu:xL-ep,x:-6p,

(a 7,\ \

4

.

v /

one can likewise regard the right side of Eq. (4.2) as a function of the new coordinates. We shall now consider two points p and P' which are chosen so that the point P in the first system has the same coordinates as the point P' in the second system. The field functions g,(x) at the two points therefore differ by amounts dV"@) which are given by

: l, uu,Sr,.oBvB@) 6v,@) * ry: @,- ,L): 1

:

^

7o-(l\

tu,Su,,oBqP@)

( e u , x ,l 6 r ) . )

(4.4)

\I

\

In a similar way we find the change in the quantities a.g

(4.s)

a4"VL 0r, in going from one system to another: c

/ \

I

-

o x t d p \ x ):

;

^

t

l-Tho -^*o.+^ L rl rv

An-,.(x)

E b s t , , p a n B u \ x )-

( e t , x , * 6 ^ )* e * n , , ( x ) .

/ ? 3 ) a r e s p e c i a l c a s e s o f t h e s e z r - . .. l

. . . v . r r v .

v/

vr L

r yvvr

vr

Lr r Er c

J04l

.J

(+.a)

Thtr last Iltg

t e r m i n ( 4 .6 ) a p p e a r s b e c a u s e t h e v e c t o r i n d e x p r i n ( a . 5 ) m u s t a l s o be transformed. In particular, for the canonical momenta,we obtain r-

/--\

o x r d \ x ):

-

I

z

.

E ) . v5 t , , B u n p \ x ) -

)n.(xl

;;

( e ^ ,x , l

6 ^ )- i e n , n o , ( x ) . ( 4 . T )

Now if we first carry out the canonical quantization in the original coordinate system and then subsequently quantize in the new coordinate system, it is necessary that g*(z) and no@) as well as q"(x)*6q"(x) and n"(x)16n"(x) satisfy the commutation relations (3.5). The two procedures of quantizing are only equivalent if there exists a Hermitian matrix 7 for which q"(x) *.68"(x) : n"(x) i

6n"(x) :

eir gn(x) e-dr , e;r vo1a1t-n,

(4 9,\

(4.e)

with the same matrix Z in both Eqs . (4 . 8) and (4 .9) . Since we have assumed that the Lorentz transformation (4.1)is infinitesimal, we can also take T infinitesimal and obtain

6q"(*) : i lT, q"(x)),

( 4. 1 0 )

Sec. 4

G. Kdll5n, Quantum Electrodynamics

(4.11)

6n"(x):ilT,n"(x)l'

-he existence of this matrix can be shown most simply by giving :r explicit expression for it. We wish to prove that Eqs' (4'I0) and (4.11) are satisfied if we choose the following form for T:

r:

gp(x) * x,€,n""(4ry! + I Ia'xlt up5,,,"an"(x)

-1-

dr)(""@) * i (ennxn* W

6en,(x) ry!)

Iv,,t

runs only from I to 3' The summation over the index hin(4.I2) We shall introduce the general convention that Latin characters like ft , I , etc. can take only the values I to 3, while Greekchar acters like pl , 'v , elc. can also take the value 4. The proof f.or(4.I0) and (a.li) is simple in principle, but somewhat laborious . We first give the calculation f.or 9,@). With the help of (3.5), one has

ilT,q"(x)l:

t )x , a , n L t l : * i @ n n x u t 6 n ) x ) ro,su,,,avp@

From (3.3) , we conclude that the last two terms in the sguare b r a c k e t s c a n c e l . T h e o t h e r t e r m s c a n b e g r o u p e dt o g i v e t L t , P q \ x ) J:

1^Ao-(x\"?o-(x\

T

u u u J p , , " p V p \tx )r , a , p

r;,

- o,,-

V.I4)

bli,

T h i s p r o v e s ( 4 . 1 0 ) .T h e c a l c u l a t i o n f o r z o ( r ) p r o c e e d s i n t h e s a m e way: i lT,n*(x)l=|

,u" Su,,fonp@)+ e,enn(x))* i (e+axu{ 6n)x *?, ^ ova@) , app@)\ - n B \x) 0 x ^ + A E -+;WA- r9*T " A x o l _

a s" G l

a q ,1 ,1

)-

- Ju,o1'n' xt lo r 6t)#;1

uu-t

- au4!

:

(4.15)

: - + ep,sp,,BonB(x) * x,quW - 6 rw 'i : ,n ,n n )(x)

* i bnnxe* 6a) ry!

: -

- orff

t *, r,u#? + ep,Sp,,pone@)

-

- i en,n,,(x) .

Sec. 4

Transformation

Properties of the Theory

The last form for the right side of (4.15) is identicat side of (4.7) , and this proves (4. ll) . The matrjx Z can be written in the following form:

r : fasx { 1e,,, x,. | 6,,\ P' | J l'"" I

",

I3 fn

fho

rinh+

(x1a!,"(*)- t' -g a^. - " * J| + Axu * * r , S r , , o p n n ( xv)' t 4 j . ]""'

Because of the antisymmetry of -

V

ep, (5,a, op n"r(x) np(x) |

is equal to zero. fni

l nrarina

-

e,n , the expressionl Spr,,Bn",(x) gp@))

(4 .r7)

We can rewrj.te the last term in (4.16) in the

ura rr.

; n"n@)9B@)* S,+,"p nou@)qe@)+ Z to, (Sp,,op pu@))= - i er,1r,: * Spa,oBn',(x) o , . , : t Z q L f o t , @ , n x o* 6 ) l - i ( e , nx n * 6 , ) a ftln^,: , A ., :i;;llnn,@,nxnt 6,)f- i (e"o xn| 6)

n

I I I \/ 4 a "t"ni l [

I

fttr^,.J

Ohesymbol fr", is antisymmetricin the first two indices.) From ( 4 . 1 6 )a n d ( 4 . 1 8 )w e o b t a i n T - - i I d . 3 x T n , 6,x ,

(4.19)

9lra? + 6t,gTr": - nnr@) f, l^u,,

(4.20)

6xu: errxr* Otr,

@.ZI)

with and

Fromthe equations ofmotion (3.2) it follows that the tensor {,, onrraf i nn satisfies

the continuitrr

!*,':-#qhp,:o

(4.22)

The three -dimens iona i inte gra ls

(4.23)

Pr:-ifdsxTn,

]-

F. l. Belinfante, Physica, Haag 6,887

( 1 9 3 9 .)

14

C. fdtt6n,

Quantum Electrodynamics

Sec' 4

(4.20) are therefore constants of the motion. since the last term in d i v e r g e n c e ' the a t h r e e d i m e n s i o n a l a s b e r e g a r d e d p : 4 c a n for with the momenta defined in (3 '12) and ( 4 . 2 3 ) i d e n t i c a l a r e o f 1, (3 . 16) . If we now put tr,: 0 in (4 .2I), then with the help of (a ' a) a n d ( 4 . 7 ) w e c a n s i m p l i f v E q s . ( 4 . I 0 ) a n d ( 4 ' 1 1 )t o

o!';,? , lP,,E*@)7: -- 'i,444 . lP, n,(x)l

(A 'A\

lA

1\\

them as Here we have recovered Eqs. (3.i7) and can now regard transi n f i n i t e s i m a l u n d e r t h e o r y o f t h e i n v a r i a n c e t h e expressing For this reason we shali also refer to the operators Po lations. as displacement operators . The conservation laws (4.22) and (4.23) can also be deduced under in other ways. since the operator 11 is obviously invariant ( 4 ' 1 1 )t h a t every translation , it follows from (4 .10) and

: 0. 6H : i V, Hl : i lPp6r,,l7l

(4.26)

from (4'26) The quantities D, are arbitrary, and we can conclude derivatives time vanishing have so that all P, commute with I/ and differenp o s s i b l e t h e i f i s o n l y T h i s s y s t e m . in every Joordinate This consideration is partictial conservation law (4.22) holds. but ularly important because it is useful not only for translations i n f o u r d i m e n s i o n al f o r " r o t a t i o n s " f o r m , also, in slightty modified h a v e w e t r a n s f o r m a t i o n s s u c h space. For

aH : +

le, 1,,,H) : - i e4,P,,

(4.27)

where

Ir,:'i

f d ' ' x( T n rx , T n ,x , ) ,

(4.28)

or

lH,I,,l-

-

6arP,* 6a,Pr.

(4.2e)

operators Since the operators !u, are formed not only from the we have gd and zo but also c'Jntain the coordinates r explicitly'

- Ta,6r) : o' 0", (Tnu6,n lld : ilH, I,f I ff

(4.30)

must If Eq. (4.30) is to hold in every coordinate system, there also be a differential conservation law here:

-f, {r^, xo- T^,xr): o.

(4.31)

Sec. 4

Transformation Properties of the Theory

15

T h r o u g hc o m p a r i s o n o t ( 4 . 2 2 ) a n d ( 4 . 3 1 ) w e o b t a i n Tur-T,u:0, i

a

+ha +6hc^r

T r ur

ic

crrmmafrin

symmetric energlF-momentum density the tensor

r+ ig

(4.32) Often

feferred

tO aS

in order to distinguish

@,,:- nn,@) s, ry! + ap,

the

itfrom

(4.33)

w h i c h i s k n o w n a s t h e " c a n o n i c a l " e n e r g y - m o m e n t u mt e n s o r . T h i s Iatter quantity usually gives the same dlsplacement operators as T, and therefore can also be interpreted as an energy-momentum density. For the construction of the angular momentum fu, ,however, it is essential to employ the symmetricaltensor Tun.

CHAPTER THE

FREE

ELECTROMAGNETIC

Iaqranqe Function and Canonical Formalism The Lagrange function of classical electrodynamics from

FlELD

5.

is obtained

g--lFpF,,.

( s. 1 )

Here .{,, is the electromagnetic field tensor: Fta: - 44: 4.2: - 4r:

i E h'

H a , a n d c y c l i c p e r m u t a t i o n s'

( s. 2 a ) ( s. 2 b )

In this Lagrangian we introduce the potentials Ar(x) as the dynamical variables, rather than the field strengths. The field strengths are glven bY -pt ' r -_ a A , Q \ 0*u

_AAu@) 0n,

tc.J,

Regarding the potentials as the variables in the lagrange function 7 we obtain the equations of motion according to (3 .2):

.

- t"!,1"!,\ :0. - !A-,(,\) : I- e,, -a ( aA-,(t) P.ul axpottt 0x, I Axu

7xu \

(s.4)

This formulation of the classical theory is obviously Lorentz inThat is, it is invariant under varlant as well as gauge invariant. the transformations

Au@)-->A,(4 + 4+ Neither the field strengths nor the equations of motion are changed In the classical theory this invariance by these transformations. under general gauge transformations is often reduced by requiring that the potentials satisfY

a 4 : @:)o . ofrp

With an appropriate choice of the gauge function A(x) in is always possible to achieve this; however, the gauge

( s. 6 ) ( s. 5 ) , i t function

Sec. 5

lagrange

Function and Canonical

Formalism

is still not uniquely determined by this condition. possible to carry out further gauge transformations, functions which satisfy the wave equation

Iz

It is always but only with

Z A(x): o.

/c

7l

In this gauge, the equations of motion (5 .4) simplify to

Z A , ( x ): s . If we attempt to go to the Hamiltonian form of the theory, starting from this Lagrangian, we find that the canonical momenta (3.3) are Tvok):iFnu@). The momentum conjugate to Aa(x) vanishes /\ v( . J9/ ) n ea s rnrnr rnvi, h - ,a

identically

and Eqs.

oA"(zl

-c-n. vr e d

for

all

Ar"

d e v e l o p e di n C h a p . I i s u s e l e s s . with the following Lagrangian: g:-LP 4 The equations momenta are

(5.9)

of motion

.

Thus the

whole

method

Instead of (5.I) we can start

a A ' ( t t ')-3- 4 A rar@ - ) 'ttt P ^ttt - r Z A_, (5.8) are obtained,

(5 .10)

and the canonical

n e @ ):

i Fno@),

( S. I l )

na\x) :,

. --;; 0A..(x\ .

(5 .12)

In the Hamiltonian formulation of this theory, we do not obtain Eq. (5,6) as an equation of motion, but only the weaker condltion n

04,(x) ofro

\c ..r\t.l

If we do not consider the most general solution of these equatrons but only those which satisfy the initial conditions

- 0a4; ,:( x )

02A,(x)

Z;a-"

:0'

for all c'

( s. 1 4 )

and for some fixed time (e.g., xo:O), then from (5.13) it follows 0- 4 " ( x \ that the quantity a';,, must vanish for all times . As desired , we then recover the usual theory of electromagnetism. In the quantization of the electromagnetic field, we shall first ignore completelythe inltiai conditions (5.14) and use only the Iagrangian (5.10). For 5o: x'o the canonical commutation relations become

18

G. Kli116n,Quantum Electrodynamics

Sec.5

A,(x')l: g, 14,,(x),

(s.is)

I 93# - t aL;:?): .l, lA,(*),', 4(x')l: lo,t,l, [o,f 2P] : I ( s . 1 6 ) r =t6pn6(u-n') )

4#] :fo utu), :lo u@),, r#l no(x1l u@), LA :fon(i, r#l na@,)f lAn@),

:l#, n1(x')l lno@), :[#!, nn(x,)f lnu@),

:,

(s. r7)

: i 6(a_ &,),

( s. 1 8 )

o, \fl: aff]: o.

(s. Ie) (s.20)

T h e E q s . ( 5 . 1 5 )t h r o u g h ( 5 . 2 0 ) c a n b e s u m m a r i z e d :

yAr(x), A,(x')f: g, | 0Au@)

ltif

(c . z.r./

, -.,,1 - i 6p,6(n- n'), , d,(''))"

| 0Au@)

1A,(x')l

(5.23)

a t o1 : " '

t--a;'

(5.22)

It must be emphasized that the commutation relations (5.15) to (5.23) are valid ina Heisenberg picture only for equal times ro and ri. For what follows, it will be of interest to carry out the quantlzation in momentum space. We therefore introduce the expansion ( t . I ) f o r A u @ )z

AP, (.x ) : + t yv

+

eih"Ar(k).

$.24)

From (5 .8) it follows that in (5 . 24) only those ft can be present for which

hz:kz_ kE:0.

tc. rc,

Equation(5.25) has two solutions: ho: la

,:

,

*lF

i

/tr

, A\

and we can write (5.24)as

Au@):*

yrT

> \eh"tr(rt)| e-ih'Al (Ql,

(s.27)

where ho:a

(s.27a)

Sec. 5

Lagrange Function and Canonical Formalism

lg

From the requirement that ,4u(r) is a Hermitian operator and z{^(,') is an anti-Hermitian operator, it follows that Ao(k) , iAn(k) and Ai(k) , iAt(It), respectively, are Hermitian conjugate operators. Moreover, since Ar(x) is a vector, for everylc therearefour independent "possible polarizations", which are conveniently described by using a set.of orthonormal, but otherwise arbitrary, " p o l a r i z a t i o nv e c t o r s " , l ! r , ) , : I , . . . , 4 . I f ( 5 . 2 7 ) i s t o b e t h e m o s t general possible solution of the equations of motion (5.g), we must sum over the four possible directions of polarization:

Au@): #ZZ vv

4

t)\

(k)-y e-ta,e*a)(k)), (s.28) .,+^ leihraQ)

h ^ _ L l l2 a

: 6u(. 4) e('x')

(s.2e)

In (5.28) we have introducedan additional factor llZa in the cie_ nominator. This is only a matter of notation with ho special sigTha ahni sives the commutationrelations for aa)(k) a very ,r*or."ioJl'.28) It is not necessaryfor the vectors af;) to be independentof.h,

nifinanna

and it is even advantageous choice:

to make the following

lc-dependent

ef,t: sf,): ef):o ,

(5.JUal

et')k,: ejDk,: g ,

(J.JUDl

e/ ar\ ,: ;

hr

elto):

,

( s. 3 0 c ) ( 5. 3o d )

O '

pl4) v4 -,. a

(s.3oe)

In this case we shall refer to transversely ar

I , t)! L

*n

nryi+r,Ai-rl L U .l nr O n g l L U O t n A . L yh ^vl .^.ru. ir?r^af u e r vi rnrn

rf nv-r

^7-. 3 ,

polarized light for i:t and

to

scalar

polarrza_

tion for )":4. We now see that if we were to have Eq. (S.6) as an operator equation, only transversely polarized liqht could be present. Since Eq. (5.6) has been ignored so far, we have totake into account all four polarizations. with this choice ofunit vecrors e@ , in addition to (5.29) , we have

, P ( i l 'Pvl ^ ) _ L-u I

A uuy' '

f\ vq

'

?ll

v+/

Because of Eq. (5.30e) where the right side is I (rather than z, for example), Eq. (5.31) has formal Lorentz invariance. The r e a l i t y c o n d i t i o n s f o r 4 t i r ( l c ) a n d o , r \ A \ ( k )a r e t h e s a m e a s t h o s e

t?,

^ !,

\J.zdl,

(lc) and Atr (1"). we geI

FromEq. (5. 2l) , with the hetp of Eq.

G . K 6 1 1 6 n ,Q u a n t u m E l e c t r o d y n a m i c s

20

.

^(,1)^{.1')

lA, (x),4, (x'))"": "r - # Z Z #' k,k' 1,1'

V@u

Sec.6

r e')-i (ur a')*6 y { ei\k +t''

i (hr + tt'u'J+ i (- t a')zo @ (It), ox lt') (k' * x lal^) (k), 6l]') (tc')I + ela* )l *n r" y tt') (h,)l or" s-i(t'r-Ht\ri(a-u') eiltsa-IE'n')*i\a*a') I + VA) Qt),

(s.32)

x la*tt\ (It), a0')(tc')l) : 0. Since all

a,l are positive and since ( 5 . 3 2 ) i s t o h o l d f o r a r b i t r a r y

l-.ima<

annclrrr'la

rnra

v a ) ( k ) , a Q '()k ) l : i a x t ^()k ) , a * t t ' )( k ' ) l : o , )") latD(Iq , o* tt')(k')) : c0 (k) . d*, r, . The quantity a way that

cl^x;\(tc)can depend on ,1,, i'and

(s.33) (s . 3 4)

Ic but only in such

(_ t ) . (It) : 6t1').) c(4,()

(S.35)

I t f o l l o w s f r o m E q s. ( 5. 3 3 ) t h r o u g h ( 5. 3 5 ) t h a t ( 5. 2 3 ) i s i d e n t i cally satisfied. To get the specific form of cwe use Eq. (5.22):

-i6,,6(n -"'):+>

V # f r>,

thus

' il t'\ : :t 2i ' \ ^.\ 2 @ \? i ^' )" c- /Q- ) " )\(-k- )l -2-e i k ( xi - x(. ',crt.rr o /

c ( ^ ^ (' )I t ) :

6ty.

/R

e "\

Equation (5.34) then takes the simple form bat 0t), a*\^')(k')):

6t t,. 6nn,.

6.

The Hilbert Space of Pree Photons We obtain the Hamiltonian operator from the Lagrange function

/(

lLnv /)

\ w .

^fl'or

cnmo

gr LU r uv r r r !

narfial

intanraiinnq.

ysr

un"'gl: aA!@) oA-P(x\ H: + [ ar*llltkL * Oxh 2 J oxh OXo Olo I I :-+

> I Jf - oa' ! " * S r , u , a t ^ t ( n ) - r - i * r o ' * t t ) ( k )( 6) .x1 ) n' ,?^' f* lf

x ledh'"ao') (k') -

:

e-ih'r oxlt') (k')l 617,(a a' + kk') :

o*ttt(k)}. a '1t t> {at^t1*1,

The last term on the right contains the anticommutator {o, o*}:

Q ' a *I

axa ,

(6.2)

aan

A

The Hilhcrt Snaec af Free Photons

We introduce the Hermitian operators tions

qf) and fl!) vv the defini-

l

q(^): ,,.'-(attl * a*Q)) , y 20)

'p.,': - iv+

[ , o , ) , : t , 2, 3 ,

,J

( a @- a * Q ) 1

q@): h : i pta)

2I

( o. 3 )

- ,a * G ),)\ @@ (6

A\

-t a*to1.J /4 ,.*,

The o p e r a t o r s p a n d q s a t i s f y t h e u s u a l c o m m u t a t i o nr e l a t i o n s f o r a l l ), and ),' :

lptA),qe'tl: _ i 6t t,.

to . J.l

In these variables, the Hamiltonian becomes 1a

) H : + 2 l Z ( p r , ,* a z q t t ) 2 , , _ ( p ( 4 l).2r q (-q r ) I . Ic tl:1,

(6.6)

l

Thus the Hamiltonian is a sum of independent harmonic oscillators, The eigenvalues of the energy of this system are

tr-

; {g

,na) - n Qt) @)}a.

(a.t)

In (6.7)the zero-point energy of the oscillators has been omitted. This is allowed here because the observed quantities are only energy differences, Formally this is apparent because we can aiways add an arbitrary c-number to the energy. The commutation relations (3.17) are not changed by this, and if the c-number rs time-independent, the shifted energy is stitl a constant of the motion. One can also change the order of operators in (6.6) or (6.1) so that the zero-point energy automatically vanishes. This ordering is not determined by the correspondence principle, and therefore we have no particular reason for either choice. We shall designate the state of z particles of given ft and given polarization direction simply by ja) , momentarily suppressing all particles of other polarizations and Ic, for simphcity. From (5.38) , (6.1), and (6.7) we obtain

/4.aa* | n) for 7 1 a , \ H a* | n) : lH, a*f I n) { a* H t *> - [(,1+ (6.8) l ( t n ) a a * l n ) 1 6 1) , : 4 . 1

Sec. 6

G . K 5 1 l 5 n , Q u a n t u mE l e c t r o d y n a m i c s

22

Hence the state vector a*ln) is also an eigenvector of the Hamil2, 3 and with eitonian, but with eigenvalue (n ! 1) a fot )':I, ( n : 4 . h e l p o f Eq. (6.8) we can t h e W i t h genvalue l)a for .1 express all eigenvectors in terms of the vector I0), , the state representing "no particles " :

(6.9a)

lne)S:cf\la*a\)"lo)for 1+4,

(6' 9b) tna)> cf) lata))"lo). The normalizationconstants cf) can be determinedfrom the condition I : (n{Alln{Al:r. For ,l* 4, we obtain

(n0)| ntt\s: I cr)P
: n lcf;) l0): n lax(t\1n-r l2(01latt)1n-r ffi

(6. ro)

,

with the solution

t,f)t:#. For i:4,

.

( 6. l l )

the calculation is similar and the result is the same:

n tnrl (n$)|n@\: l rlf)l, (- {)"(oI gax la@)lo>: " H,+, I cr*-tl' I trr"(n nl )t: _ f i .

The most general eigenvector of (6.I) direct product of vectors (6.9):

(6. 10a) (6.11a)

can now be written

as a

t'(')(t)l:':l(*) rtua)Qc)>:Ir r0). (6.12) nh n ft t'-l'lg,#(*) ^:r k A:r V t l 4 )( k ) I In (6.12) and all subsequent discussion, l0) is to be the eigenvector of the coinplete Hamiltonian which represents the state The arbitrary phases which could have apwith "no particles". peared in (6.12) have been taken as zero, since they have no significance. we have found that the quantized electromagTo recapitulate, netic field can be described by means of a system of oscillators

Sec. 6

The Hilbert

Space of Free photons

23

and that the energy of such an oscillator isquantized in the usuat way. This is the old (light) quantum hypothesis which is in a sense the source of the quantum theory and which we have re_ covered here as a consequence of our formalism. From (6.12), matrjx elements of the operators aV)(l&) can be simply determined. obviously these operators are diagonal in all quantum numbers other than Ml^J(li . we again suppress alt indrces with other quantum numbers and so obtain

( n l a t \ 1 +n t ) : ( n * t l a * t t ) 1 n S : l n + i for A+4, (6.r3) ( n a t l a t a t l e : - f u l a , F t 4 ) l n +r ) : (6.14) l;+1. The other matrix elements are zero.

The operators

lgrr)(rc)_ d6(^)(k) a0)(Ir) 1,rtal(k) :

for ,t g 4 ,l

_ a(4)(k) e* t4)(k)

|

are therefore diagonal in the representation (6.12) and have the eigenvaiues n@(lt). They are the operators whichgive the number of light quanta of given polarization and given momentum and are usually designated as the ',number operators". In a similar way, we call the operators 4lA)(k) for ,tr{ 4 and - a,t(4)(tc) tfre "annihilation operators" and the quantities oxttl(k) for 1{ 4 and ala)(k) the "creation operators ". By (6.15) we can write the enersy as

H: >,{i 1ur{rr (k) 1- (,r]r

lurtar 1k)}a.r,

to . .Lo/

w h e r e t h e z e r o - " p o i ne t nergy isagain neglected. This distinguishes ( 6 . 1 6 ) f r o m ( 6 . 1 ) a n d ( 6. O ) l u t , a s a l r e a d y n o t e d , t h e r e i s n o physical significance to this difference. As an illustration of the interpretation of these results in terms vo rf

f ru^ qn rf r^L q , l ri ng hr rt L tf 1

Li tL !i *q

infarocf

inn

t,--n a , -n, ,*t p g t e

the

tgtal

m6menlgm

and the angular momentum. For the first quantity, we obtain from ( 3 . 1 2 ) , ( s . 1 1 ) ,( s . 1 2 ) ,a n d ( 5 . 2 8 ) ,

a!'\ ?4L a!" u!'\ :* p^: ' R - - [ a'* * ' " {t(a!L +i Ax1 7xp , " 0*, 0*nJ J

\" \Ara

: - [ a" ul- z,: ox60xt J -

J

\ |

- f tu.rtl (rl ke{at]'t (k)t a*a)Qqat^I4q \ ' axtt) 1 z f> ,i" \

a(^)(Is)a0) (- k) ,-ziaro-

I

o,*lt)(k) a*Q) (- k) ezi.,,\.

)

Since the operators a{,t)(Ic) and aV')(-Id commutewith each other, the last two terms in brackets are symmetric in lc . The sum over both terms vanishes because of the antisymmetry of Ao , and Ee.

G . K l i t l 5 n , Q u a n t u mE l e c t r o d y n a m i c s

24 th

I /t

clmntrfrd<

Sec. 6

t6

Pn: t Z {a1t (lt) , o* tt)(Ic)\kn

( 6. l B )

,t, A

With the introduction write this expression

of the number operators (6.15) , w e c a n a l s o as

(k)} . Pr:4uu {oi,",', (Ic)- N(4)

( 6. 1 e )

in (6.19) the term with the zero-point momentum has been dropped because of the antisymmetry in Ao No special ordering of the operators is necessary to obtain this result. It follows from (6.I9) for )":4) in complete that each photon has a momentum lc (or -lc accord with the corpuscular interpretation of the radiation field. In order to calculate the angular momentum, we begin with the symmetric energy-momentum tensor (4.20) and decompose it into two parts:

(6.20)

T u o : T J o , \1 T ) t ) ,

,tr:H',H-n,#+ 6,,,s ,

(6.2I)

A^6r,) rn: - iilo"r,^* (Aud,tHl.

6.zz)

In a simllar manner, we write the angular momentum as

(6'23)

Iu:I,(f\*Lf),

Ili):t[d'sx(r]ltx,-rll)r,): l (.6 ' 2 4 ) - - t f A r * l a !(^r:, a n ^ * i L A n \ ,x i r-t ' a-- 7ar-r A ' *. . 9 n '^)l \ '-- ^xri .r tF 2*,J ( o x 1\ u i U)-t J

Ili): -;! a'*{r,file,r,*An6i* l}.to.zsI +)- .,*rln,Fne*Aa6iaff The two expressions (6 ,24) and (6.25) may be simplified considerably. From appropriate combinations of the terms in (6.24), we obtain after integration by parts

+ qoxi ^): ! " - * , 9dxt ! n!ox1 ! o ) : I a " * {' *oxh , ' ! !^0x1 f a ' rl.{"*dxy ,au laxiu! n- * , :arh J J ) |

J

tA fC) -' : - a,,Ia,*#u nn+[a'*en'#:"

For the expression (6.24) it then follows that

: - ;I a'x{.,#H -,,## riro\

+ e,4,1.rc.27)

Sec. 6

25

The Hilbert Space of Free Photons

In a similar way, we can transform (6.25) to

j 'A- +. .F,\ - -A, ' t Fn,l: ' - ' l,,l.,no, * ' "x l.s,#, - A, - - r 94 tt l | . rc \ . , , ) -i- l' as r.rf) t l : ;' JI -a", ax, , l--, ", J l'-t 0*a (

f

I

The iast two terms in we have tIt i; , : -

P

|

^,

(6.27) and (6.28) cancel upon addition and

aA, [ a- " "' * { r , Aa. ,r oA0,x 1 - * ',-91 7. xAo , 0' !x,; -,A- -,tu?l xoo - A- -,rp7)x.o J ' t",

J

zB)

.

(6.29)

W e i n t r o d u c e t h e e x p a n s i o n ( 5. 2 8 ) i n t o ( 6 . 2 9 ) . T h e f i r s t t w o t e r m s s(t'l: d,r,r,as a factor. This operator is contain the express !6y1 B(1) therefore independent of the possible polarizations of the photons. Consequently, it must be interpreted as the orbital angular momerF tum. Fromthe last two terms we get r

(

2z

| 6tx ]e, Ti J l'axo

'l')

I \_. 11\rr'\r. /r\ e ] - 'e' i ' L \ u ' ' , a * l ^ ' ) j- { a t t " t , 4 t ' ( ' r :) } ] ;,/, "*i^" (6.30) : i z el^)e\^')lat6l^')a0) _ at6(ila@'ll.

- A i' :dl tx o )| :

lc,1, A'

,

Equation (6.30)is certainly not diagonal in the representation introduced earlier; however, a simple transformation enables us to d i a g o n a l i z e s i m u l t a n e o u s l y o n e c o m p o n e n to f t h e a n g u l a r m o m e n tum (6.30) and the energy. To do this, Iet us write

o.:i;@Q -iaP)), ,- :

i;

@ft\-Yi atz)1-

to. J.r/

/A

?t\

The new operators a* and a- satisfy the usual canonical commutation relations

1,

(6.33)

l o , ,o _ f: l a + ,o l l :

. . .: O .

(6.34)

the energy becomes

In these variables, H :

l o . ,a f ) : l a _ ,a l l :

Za

(af a* * a! a- |

k

q,*(z)a@ + a(4)a*ln)) ,

and the component of the angular momentum in the direction

(o. Jc,,

of. h

J J

/(B)(Ic):

af, a* -.a!

a_.

to.Jo,

26

G . K 5 1 I 5 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec. 7

Therefore, if we choose a representation in which the operators N_:ala_, N*:ala* N ( a ) , a n d r V ( 4 )a r e d i a g o n a l , w e s e e f r o m , (6.36) that this component of the intrinsic angular momentum or or scalar "spin" can have the three values zero (longitudinal photons) and +l (circularly polarized photons). If the square of the spin is caJculated from the above expressions, it can be shown that the scalar photons have spin zero and the others spin one. Indeed, serious objections can be raised against the formalism developed here. The energy is not positive definite; the last term in (6.7) has a negative overall sign. Furthermore, the theory we have quantized here is evidently not the "right" Maxwell theory, since the classical Lorentz condition (5.6) does not hold in the quantized theory. We shall later see that these two points are connected and that if we take account of the problem of the Lorentz condition, the states with negative energy will not occur. Before further discussion ofthese points, we wish to study a different problem in Sec. 7, but one which will be of importance in introducing the Lorentz condition. Commutators for fubitrary Times , the Singular Functions The canonical quantization method prescribes the commutation relations between the field operators and their time derivatives only if the time coordinates of both the operators are equal. For unequal times there is no immediate recipe. In this section we will show that the commutators can be calculated for two arbitrary points from the solution (5 .28) and the commutation retations (5.3B) and that the result has a relativistically covariant form. Thus if we compute the commutator of Ar(x) and A,(x') at two completely arbltrary points r and x' , we obtain 7.

d,(*')l:-i I ++ vih"at^) + LA,,(x), Qt) x,F's..7'2llaa' (i')(k,)] : + e-ih" a*0)(k) , sih'r'aQ')(k,) + e-ih'/ a*

'(f)'!l

-,')- e-ihtx-x')f sr,ul" Lr

:

1I vkj

:

6u, d3h - f 1r;ne-,') - -s-ih(x-x't1 r '. l2n)sJ 2uL-

With the introduction

2u

of the

- ht): d(,t,): d(rcz E q . ( 7. i ) c a n b e w r i t t e n a s

( 7. 1 )

d-function

;q

[d(tI't - AJ+ d( relt ko)),

A,(*')f: - *k LAu@),

6(hz) e(k), I dkr,oro-o

l'7

'\

\t.J)

Sec. 7

CommutatorsforArbitraryTimes, Singular Functions

27

where

r(l):#

(7.4)

The relativistic covariance of the commutation relations is clearly evident from the form (7 .3) . Furthermore we see that the commutator of the two field operators is a c-number even if the times are not equal. This is not a consequence of the canonical quantization but a special property of a field theory without interactions. We shall later be concerned with the general problem where there are two fields interacting with each other and then the commutator is no longer a c -number but isa q-number if the two timesare nor equal. The right side of (7.3) will appear very frequently in what folIows, and we therefore introduce the special symbol

A"(*')l: - i d,D(x, - ,) , S_Ao@), where the real function

D(x) is definedl

D(*):

#

by

4 p r i h6r @ ,e) ( k .)

I

(7.s)

( 7. 6 \

The function D(x) has the following properties:

D(x):o a D @ -\ - d ( c ) l ?xo

- '--'

I

for

Because of relativistic invariance, xo:O, but, in general , f.or x2 >0. equation

fio:

O.

( 7. 7 b )

)

D(r) vanishes not only for It satisfies the differential

2 D(x): s

(7.8)

and can be defined as the solution of. (7,8) which has (7.7) as initial conditions for xo:O . Equations (7.7) and (7.8) can be shown to hold by means of the integral representation (7.6). For ,o:0 the integration over fro can be carried out immediately and, on the basis of symmetry, the result is zero. From this we have (7 .7a). if (7.6) is first differentiated with respect to ,0 and Alternatively, then ro is set equal to zero, the integration over Ao gives l;and the three-dimensional integral gives the spatial delta function on the right side of (7.7b). Fina1ly,Eq,. (7.8) follows from (7. 6), l. In the literature, other notations than that of (7.5) are also used. In recent years the notation of (7.5) has been used most frequently, and we shall commit ourselves to this. In the older Iiterature one can find the symbol /(xl used instead of D(x) ; however, we shall reserve this for a more general function. (See below.)

28

C. fltt5n,

Sec. 7

Q u a n t u mE l e c t r o d y n a m i c s

b e c a u s e h 2. 6 & 2 \ : 0 . Subsequently we shall be interested in the anticommutator of This quantity is not a c-number but a q two field operators. number. We calculate its expectation value for the state l0),obtaining

(o l{Au@), A,("')}10): l. t

e? etx') a(r')} {eih'+ih'z'
2lfaa; f,"-o.,^'

l,tZ' Ol r*ih't' q0 e-ih't (O | {a*tr),a(r)}I 0> + [ + eih I 1arit,a*(r'))l0) | e-ih l - l s - i h * - ; e ' z('O ; 1 a * t r ta, * ( / ) )l 0 ) ) . F r o mt h e r e p r e s e n t a t i o n s ( 6 . 1 3 )a n d ( 6 . 1 4 )w e o b t a i n u .10)

( o 1 1 4 t aa,a ' ) j l o > : ( o l { a * Q \ ,a x $ l } l 0 ) : 0 , ( o l { a * @ ( k ) , a v 0' )c ) } 1 0 ) : 6 t t , 6 * u ,

A a n d1 ' * 4 ,

for

(o | {o* G\(k), a\^\(tc')}| 0) : - 6**,6^n ,

(7.11) (T .rz)

and we can write (7.9) as

I,'.", 4,1

i j'rli) et^t - eLat (ol{Ao@), estfx A,(x')}lo): r " / - v Ii r t. l ! " , I I l(r) (yt : x leih("-"')| e-ih(*-t')1: (dr, - 26u&,a,) where the function p
p
(7 ta\

The principle difference between (7.14) and (7.5) is that the signfunction e (&) is lacking in ( 7 .14) . Because of this , the function p
-

lA/a shall

retttrn

to

fhis

noint

later

,

vhsr ur

f

ri n rr

fhis

setltion

we

wish to study only the properties of the functions D(x) and Do(x). B e c a u s eo f E q . ( 7 . 7 ) , t h e f u n c t i o n D ( x ) c a n b e u s e d t o f i n d t h e solution of the inhomogeneous wave equation

tt,p(r): | (x),

(7. rs)

Sec. 7

C o m m u t a t o r sf o r A r b i t r a r y T i m e s , S i n g u l a r F u n c t i o n s

LJ

with the prescribed initial conditions tp(x): u(r)l I

_?-v: (axl\i l t axo for an arbitrary solution is

time

T.

' I I )

for

xo: T

One can readily

( 7. 1 6 )

show that the desired

ao(1-o/) u 1u'1].lt .v) u| \(x\=[ , ' dx,'D (x - x')| (x') -[ au*' [o O - x') u (a') + T

"0_*.

The first term on the right side of (7. 17) contains a four-dimensional integral in which the llmits of integration are shown only for xo . The integration over the three spatial variables is over the whole three-dimensional space. In what follows we shall employ this notation quite frequently. This integral alone is a solution of (7.15) . Both this integral and its time derivative vanish for xo:f. By itself, the last integral in (7.17) is a solution of the homogeneous wave equarion fbecause of (7.8)] and has the correct initial values. Together these terms give the desired solution. In many applications , we shall pass to the limit in I-> - oo (7 .I7). With the assumption that the three-dimensional integral ,!,\o)(*), we obtain tends to a finite limiting value ,o

,t @) : rrot(x) I

I dx' D (x - x')l (x') rto>1x)- I D o@ - x,) l @,)d,x,. (7.r8)

I n ( 7 . 1 8 )w e h a v e i n t r o d u c e d t h e r e t a r d e d D - f u n c t i o n the definition D11@):{_;,,

for xolo ,I for

xolo

. )

Do@) by

,r.r$

If no limits of integration are written, for example, as in the last term of (7.18), the integration is to be taken over the whole fourdimensional space. By analogy to (7.19), we can also introduce an "advanced" D^(x) by the definition D-function

for D^@):{^? . lD(") for

xo) O

,]

( 7. 2 0 )

x 0< - 0 .

Half the sum of D*(x) and Dn@) i s o f t e n d e s i g n a t e d b y D ( x ) i n the literature:

D (*) : $lDo(*)-f Da@)1.

l'7

C1\

30

G. Kdlt6n, Quantum Electrodynamics

Sec.7

The last function is therefore related to D(r) by

D ( " ) : - + -# D @ ) :

-

t'ol

) e@)D(x).

(7.22)

It will often be useful to have the Fourierrepresentations for D(r), Do(r) , and Dn@) , analogousto (7.6) and (7.i4). Startingwith D(x\ , we write 6+@

e(x\: ,\:'

lxol

I

o'

n J

z

, i n f' t x ^ \ : "'

t rn

P [ !! r','o. J

r

(7.23)

0-@

From this we obtain

D(*):-rnt,p ,',,"# Io: I6prihx6(k)e(k):

l

-: : [ 6* p" "r 'i n , p I o z' 6 " \('t-r - ( A \ "^u*'z ") /2t) . l . o ! ' , : = ] -( pz n ) n J-[ * n , ", u' )u , . l ( 7 ' 2 4 ) en)aJ J lho+rl Tn (7 .23) and (7 ,24) the letter P in front of the lntegral sign indicates that the principal value is to be taken. With this, we have the integral representation for D@) . From the equatlons Dp(x) : D (*) - *D (*) ,

(7.2sa)

D 1 @ ) :D @ )+ * D ( * ),

(7.zsb)

we obtain representations for D^(z) and

Do@): '

Dn@) z

( 7, 2 6 a ) dkeih, # * on 6(h,)eft)|, {n D A ( x ) :- ^ 1 _I,ono r , u , { ohi ,' _ i n 6 ( k ) e ( k ) ) . (7 .26b)

rr',, I

l2n)"J

For completeness,

")

\

we give a few definitions:

D @ ( x ) :l , t o ( x )- i D < 1 ) 6 ) : #

[

a n , o u " a 1 n ) ] l r ( f rt)l+i 0 . 2 7 )

a n r " ' 6 ( h \+ t u ( A ) -1 ] t ( 7 . 2 8 ) D-(x): ) P@)-liD
D6@) and is

Sec. 7

CommutatorsforArbitraryTimes,SingularFunctions

3l

other by different contours of integration in the complex Ao-plane. For practical purposes, only the complex representation for Do@\ is essential. From (7.29) one sees directly that this function -Ls given by

ur\x):

2

I

f ,,

eiht

i edn 1an-pz I

(7.30)

c7t

where the path Cp is shown in Fig. l.

F i g . 1 . T h e p a t h C r i n t h e c o m p l e xA q - p l a n e . F r o m t h e r e p r e s e n t a t i o n s( 7 . 2 4 ) , ( 7 . 2 6 ) , a n d ( 7 . 2 9 ) , i t f o l l o w s that the four functions D@) ,D*(x), Dn@) , and D"(*) satisfy the inhomogeneous wave equations

J D ( x ) : Z D n @ ) : A D e @ :\ - 6 ( x ),

( 7. 3 1 )

J D p ( x ) : z i 6 ( x ),

(7 '1r\

Instead of Eq. (7.15) we shail consider the more general wave equation

(J - *')tp(x): l(x)

(7.33)

with a mass tn and introduce a system of singular functions for this equation. We shall denote them by /(x,tnz), etc. or by /(r) if we are discussing only a single mass. We have, for example, (J * m') /(r):

(Z - m') /tt)(x) : o ,

A(,):u#:tl 'i:.::-

6(n) i

'* xo:o'

( 7. 3 4 )

(7.3s)

(J - *') z(r) : (J - *') Ap(x): (J - mz1 \@) : - 6 ( r ) , ( 7 . 3 6 ) Z l x y : - i u ( * )A ( * ) ,

etc.

( 7. 3 7 )

5L

Sec. B

G . f 5 1 l 5 n , Q u a n t u mE l e c t r o d y n a m i c s

From the previous integral representations, we can easily find the representations for the various / -functions. To do this, we replace

by 6(k'+mz) and.O# rn PE+A 6(k'z)

everywhere.

In order to obtain the r-dependence ofthe various /-functions, the evaluation of the integral representations given here can be These expressions do carried out in terms of Bessel functions. not have any qreat significance for subsequent applications, and usually it is simplest to work directly with the integral represenWe shall not state explicit formulae for the singular tations. l The functions , but instead refer the reader to the original work. We therefore give the result is simple only for the case m:0. result for the functions D(r) and D@(x):

D(*): -

* ' @ ) 6 ( * ' ),

p J, . D$\(x\ : -12n'

(7.38) (7.3e)

The function D(x) not only vanishes outsidethe light cone but also and there jt is inside it. It is different from zero only f.or x2:0, The function pttl(r) is dif f erent as singular as a delta function. from zero everywhere and becomes singular on the light cone like In a theorywith x-2 , These results hold only for the case tn:0. a particle mass different from zero, the function A(x) is also difHowever, the strongest ferent from zero inside the light cone. singuiarities on the light cone are the same for the functions D(x) and tot ptt)(x) and /(t)(r). and /(r) The Subsidiary Condition: First Method As we have frequently noted, the theory which we have developed so far is incomplete because the Lorentz condition (5.6) has This condition is essential in the not been taken into account. classical theory if the equations of motion are to have the form (5.8). It is quite clear that we cannot simply interpret (5.6) as an operator equation because thls would contradict the canonical commutation relation (5 .18): 8.

|

24

tv',\1

nn@')f," ,'": i lnn@), lAo(*), Tl,"

=,;: i 6(r

- *'). (B. t)

In order to obtain the classical theory in the limit, it is not necessary that alt the classical Maxwell equations correspond to in the quantized theory. It is sufficient to operator identities require only that the expectation values of the electromagnetic 1. Comprehensive discussions have been given by l. Schwinger, P h y s . R e v . 7 5 , 6 5 1 ( 1 9 4 9 ), a p p e n d i x ; a n d b y W . H e i t i e r , Q u a n t u m Theory of Radiation, Third Ed., p. 7I-76, Oxford,l954. In Heitler are interchanged, and a different choice of the terms D and / sign is used in the definition.

33

The Subsidiary Condition: First Method

Sec. 8

field quantities satisfy these equations for every physically realizable state. It is sufficient to limit ourselves to the states lrp) which satisfy the equation aA..' tt $.zl aa lv): as a subsidiary condition.I Equation (8.2) is equivalent to [ 4 ( 3 ) ( r*r ) i a r t t( t c ) ]l V ) :

o,

la'rtsl(Is)I i'a*@ (lc)llV) - o .

(8.3) ( 8. 4 )

The subsidiary condition is thus only a requirement fcn longitudinal and scalar photons; it has no effect on the transverse photons. An altowable state vector lg) can be written as a product:

(B.s)

l , p ) : l , p ) L ll @ * ) , ,c

where l?r) contains only transverse photons and where | @*) can be expressedas l@n) :

)

o , , " n , . l, n @ ,n @ > S :)

n\s),ntr)

a,,"nn,!IHl1to>.

n\,.,tu\t)

(8.6)

Yn?)ln@l

S u b s t i t u t i o ni n ( 8 . 3 ) a n d ( 8 . 4 ) g i v e s

- 1,nQ))+i{86 + 1 lo,u,,,ull@ln'.t1 l n @ , n @+ i ) ] : 0 , Z o,*,,,rl/n@+T I n- i lW

(8.7)

pG)- I )] : o. (8. 8) In@t,

The most general solution of (8. 7) and (8.8) is d,nl)ntt\:

where the constant requirement

c 6nril, rtnt (-

i)""'

,

/R

q\

c is to be determined from the normalization

({D4l
( B. r 0 )

The subsidiary condition (8.2) therefore furnishes an exact prescription for the mixing of longitudinal and scalar photons which may be present in a physically realizable state. In a sense, these degrees of freedom of the field are eliminated and only the transverse photons distinguish the physically realizable states from each other. For many applications it is therefore sufficient to regard l91) alone as the state vector and to consider the dynamical variables to be atr)(ft) only for .1.: t and 2. With sufficiently careful calculation, one can actually obtain correct results in this manner. For example, the total energy (6.7) is just equal to that l.

E . F e r m i , R e v . M o d . P h y s. 4 , 8 7 ( 1 9 3 2.)

34

G. fdll5n,

Quantum Electrodynamics

Sec. 8

of the transverse photons since, because of (8.9), the energy of the longitudinal photons is exactly eancelled by the energy of the scalar photons. The total energy of a state (8.5) is therefore positive definite, and the state of lowest energy--the vacuum-has no transverse photonsand only the mjxture (8.6), (8.9) of the others. A similar result occurs for the spatial Pi and f ;, whose eigenvalues can be expressed in terms of only the transverse photons. In particular, it turns out that for the component of the angular momentumin the direction lc only the two values +l or -l can occur although, since the total spin is one, three orlentatrons would be expected. This is a special property of quantum electre dynamics with a vanishing photon mass and Ls a consequence of the Lorentz condition. No matter how attractive this method looks at first glance , there are certain mathematical difficultles hidden in it.l For example, the subsidiary condition (8.2) appearsto be incompatible with the commutation relations (7 . S) . From (7.5) we have

W!,

: -i uLD(x,A,(*)f x) ,

( 8 .u )

and therefore

@' - x)fo <' " ,t P t ld 3r u4 ! ,A" ,' J("*' ) 1 t D :- ; * n dx,

(8.12)

instead of zero, as one would expect from (9.2). This difficulty is resolved by the remark that the vector lOr,) in (9.6), (g.9) is not normalizable and therefore that (8.10) is not satisfied for any finite c In applying (8.2),we do not obtain zero insteadof (8.12), but rather an lndefinite expresslon of the form 0, oo . This musr be defined by taking some limit.Z A useful formalism for this purpose can be constructed if we ascribe a small rest mass p to the photon and therefore modify3 the equations of motion (S. g) to (tr-p')Ar(x):6.

(8.13)

The commutation relations for the operators A,(x) are only changed b y t h e a p p e a r a n c ei n ( 7 . 5 ) o f t h e s i n g u l a r f u n c t i o n s / ( x , - r ) w i t h I . S e e F . J . B e l i n f a n t e ,p h y s . R e v . 7 6 , 2 2 6 ( 1 9 a 9 ) F ; . Coester a n d J . M . J a u c h ,P h y s . R e v . 7 8 , I 4 9 ( 1 9 5 0 )S; . T . M a , p h y s . R e v . 80, 729 (1950). 2 . R . U t i y a m a , T . l m a m u r a ,S . S u n a k a w aa n d T . D o d o , p r o g r . T h e o r .P h y s . 6 , 5 8 7 ( 1 9 5 1 ) . 3 . I a m i n d e b t e d t o P r o f. W . p a u l i f o r t h e s u g g e s t i o n t o u s e a s m a l l p h o t o nm a s s f o r t h i s p u r p o s e . F . C o e s t e r , p h y s . R e v . 9 3 , 7 9 8 ( 1 9 5 1 a) n d R . ] . G l a u b e r , p r o q r . T h e o r . p h y s . 9 , 2 9 S ( 1 9 5 3 ) have given formulations of quantum electrodynamics as the limiting case of a theory with a non-zero mass.

Sec. 8

The Subsidiary Condition: First Method

mass p rather than the functions D(*'-r) a scalar fietd B(z) and a new vector field I

B (r)

p

. Now let us introduce U,(x)by the definitions

AA,,(r\

( 8. 1 4 )

o*, /

Ur(*) : A , , ( x \ * t

35

aP@\

(8. rs)

ozu

These satisfy the equation of motion (8.13) and, in addition, Uu@) identically satisfies the operator equation field

B@)-rLan@):o'

ry*

the

( 8. l 6 )

Because of this, only three components of Uu@) can be taken as dynamically independent. Decomposing the'new field into plane waves, we write

(Je : uo) (tg) " * uu'L'ut') @) @)]+ F 1la {"' IA*'n) * e-in*[Atf,u*o)(tt)+

= -i un@) : =fr Ito@) \frYV,rA: -R (\"t

I tlT

I

1,,,,

,f,zr-(s)(k)]],

(tt)+ e-ih, u@r u*@) (tc)1,(8.18)

Y - - l - r , t i t- ' ' b ( k ) + e - i h ' b * ( k ) f tl;ll

a

,c v'w

'

( 8 .r e )

with h2+p2:la2-r;uz* p2:O ,

(8.20)

alO.

In r-space the new fields have the commutationrelations A'(*')l:i/(x' I B (x\,B (x'\1: '',"o=u' *- .-,u;lAu@)'

- x)'

uu:!.!'J : - pl,lo"@), B(x')l /@'- x): o, '. ^? lu,,(*), 0 x , l+ p!. O ^p L F' J L r'\ ,'

(t,(*')f : \u,(x), yur@), A,(*)l: - o(u,,# d*)

(8'2I)

(s.22)

o @'- x).(B. 23)

From this, we find for the operators u{D(k)and b(Ie),

pttt (tt),u*tn (k')l: 6^^,6n*,,

(8.24)

lb (k), b* (tc')l: - d* *. .

(8.2s)

36

G. Kiitl6n, Quantum Electrodynamics

Sec.8

AIl other commutators vanish identically. We see that the roles of the creation and annihilation operators for the B-field are re.rareod

,

Iv 9 q

irrqi

J g g L

tharr

rarara

fnr

y'

t/v\

. ^ 4 \ - i

a hn-t * - - " e .

We shall now discuss what restrictions on the state vector are necessary to enable us to take the limit p,+0 and recover the usual Maxwell theory. For this it is necessary that the expectation a4'0) value of aIl products of ouantitie vanish in the limit. This 0,, " we ensure by requiring that a physically realizable state contain no " B-particles ". Hence it is exactly true that

: - rt(vtB(x)t,p): o.

<,pt !*tD in the limit

Moreover,

riq(,./l ryL .^p

trr+u

p-+0

( 8. 2 6 )

, we have

on;!i'' (vlB(x)B(x')lv): l?): Plim/2 +O' : ilj5r'4ft) @'- x) :0,

I)

etc. J

ta ' a L.t7 '\ \e

the physical We shalt now imposethe addltionalrequirementlthat state containno longitudinal" l/-particles", i.e., that u\3)(It)lut)>:o ,

( 8. 2 8 )

b*(lc)ltp): 0.

(8.2e)

F r o m ( 8 . 1 4 )a n d ( 8 . 1 5 ) a n d f r o m t h e e x p a n s i o n s ( 5 . 2 8 ) , ( 8 . 1 7 ) t o /a \v

IO)

. . r gl

trra

nlr'l-ain

,

b(k): -

liatorntlttli), | t*lr*,tk) aaJ(k):a7)(k), A:i,2, ? uB)(rt):

p

(8.30a) (8.30b)

(r")l. (8. 3oc) (k) * ; I' k I b(rc): pl," - l* ot')(k) + i I tt 1atnt a.3) 11

The vector l@L) in (8.5) therefore satisfies

la attta i lkl au)ll@r): o , L l k l o * @+ i @a * t a ) l 1 @ r , ; : 9 .

( 8 .3 r ) (8.32)

t. This requirement is certainly not necessitated by our previous postulates. If longitudinal U-particles are allowed, it can be shown that they have no interaction with the Dirac field in the Iimit p+0. S e e F . C o e s t e r , P h y s . R e v . 8 3 , 7 9 8 ( 1 9 5 I )a n d F . J . Belinfante, Phys. Rev. 75, I32I (1949). In later applications to coupled fields, these particles can play no part; therefore we shall ignore them from now on.

5ec.

o

The Subsidiary Condition:

First Method

37

In the limit p-+0, (8.31) and (8.32) obviously go over into (8.3) and (8.4^). With a non-zero mass the solution of the equations is normalizable and can be written in the followinq form:

I@rc): o,

o ^A,F

\l)^"'

d,t,1, a<,t I n$),n\t)',.

/a

??\

In this way we get an expression for the expectation value of the commutators (8 .I l) which has the " correct " value (8.12) in the limit. Such a calculation with unnormalizable state vectors is not especially attractive, but it is scarcely any worse than many other The fact that in this way things which we will have to do later. one obtainsan infinite resultfor the expectation value of the anticommutators (7.9) is a little more serious. If we restrict ourselves to the state lrpo) where no transverse photons are present, we have

: (!r, l rpo) Qpol {Ur(*),(1,(*')} 1,,o x p o x v l

- x) ,

kpol{n@),B(x')\l',p): 7l\t)(x' - x) ,

(8.34) (8.3s)

Qpol {Ur(*),B (*')}lqo): o,

(8.36)

and so

kt)ol )\l,p}: (u,,, {At Q),A,(x' fr 4Ol

no Q'- t').

(8.37)

F o r 1 t , ' - > 9 t, h e r i g h t s i d e o f ( 8 . 3 7 ) b e c o m e s i n f i n i t e . W e s e e , h o w ever, that this singularity occurs only in a "gradient". Consider, for example, a gauge invariant expression such as the integral [[ d,xdx' Fu,(x, x') Qpol{At,(x),A,(x')}lrtto)

(8.38)

with 0 \,,(x, -_ oKp

x''1 _

0 Fr,(*, x') _ n o1,

l|a ?q)

We can integrate by parts before taklng the limit p+0, and the last term in (8.37) then contributes nothing.r With these requirements we can compute as if the limit of the expectation value of the anticommutator were qiven bv

(\)ol{A,(x),A,(x')}l,po): 6,,,D\r)(r'- *).

(8.40)

We shall derive this result inSec.9 by another, more systematic, method. We have seen that we may work with only the transverse photons for gauge invariant expressions and that the longitudinal and scalar degrees of freedom of the electromagnetic field may be completely ignored. Obviously the gauge invariance of the theory

1. F. J. Dyson, Phys. Rev. 77, 420(1950).

3B

G. Kii115n,Quantum Electrodynamics

Qan

q

has been lost by the special choice of the subsidiary condition, and clearly an infinite gauge function was chosen in (8.37). At best, this is an inelegant point in the theory, and it would be preferable to have a formulation where there is always the possibility of different gauges and where we could avoid the transition to unnormalizable state vectors and infinite gauge functions. Second Method The Subsidiary Condition: The procedure given above for treating the Lorentz condition (5.1) in the quantized theory is not the only possible one_. Many other methods have been suggested by different authors.r In this section we shall concentrate on a method which has been developed by Gupta and Bleuler.2 with unnormalizable state vectorsarise because The difficulties are present in (8.3) and both creation and annihilationoperators (8.4). A condition like (8.2) can be satisfied without difficulty operator; however, if creation operators are for an annihilation also present, one is led to a system of coupled equations like (8.7) and (8.8) . The solution of such a system necessarily entails many particles, and these are not state vectors with infinitely seen above. It is clear always normalizable, as was explicitly that the vanishing of the expectation value 9.


(e.l)

is ensured by (8.2)i however, that equation is not necessary for the vanishing of (9.1).We wish to modify the subsidiary condition (8.2) in such a way that oniy annihilation operators are involved. We cannot do this simpty by throwing out (8.4) and retaining (8.3) unchanged. Certainly this would ensure the vanishing of (9.1), but since the roles of creation and annihilation operatorsare interc h a n g e d f . o r ) , : 4 , E e . ( 8 . 3 ) a l s o c o n t a i n s c r e a t i o n o p e r a t o r s. I f we do not want unduly complicated transformation properties for the theory, we must choose the following representation for at4l and

4*

(4) .

(n * | | a*6\ln>: (nl atdI n I t'2: lfn { r,

(e.2)

instead of (6 . l ) . By this choice , all a* (i)are made Hermitian conjugates of oll) , and hence all the operators A,(x) are made selfadjoint. Certainly this contradicts the reality requirements^ for the classical electromagnetic potentials, but the introductionr of H e i s e n b e r ga n d w . P a u l i , z . P h y s i k , @,w. 5 6 . I ( 1 9 2 - 9a)n d 5 9 , 1 6 8 ( 1 9 3 0 ) ; F . ] . B e l i n f a n t e , P h v s. R " Y ' 8 , ! , e a i O g ' s i i l . c - . v a i a t i n , D a n . M a t . F v s . M e d d . ' 2 6 , N o . 1 3( l 9 5 T J . 2 . S . G u p t a , P r o c . P h y s . S o c . L o n d .A 6 ! , 6 8 1( 1 9 5 0 )a n d 6 4 , 8 5 0 ( 1 9 5 i ) ;K . B l e u l e r , H e I v . P h y s .A c t a . 2 3 , 5 6 7 ( 1 9 5 0 ) .S e e a l s o W . H e i t ler, Quantum Theory of Radiation, Third Ed., Oxford, 1954, pp . 9 0-103. 3. Such an operator was introduced by P. A. M. Dirac in a q u i t e d i f f e r e n t c o n n e c t i o n . S e e P r o c .R o y .S o c . L o n d . A ! 8 0 , 1 ( I 9 4 2 ) .

Sec. 9

The Subsidiarv Condition:

39

Second Method

a "metric operator" 4 enables us to get around this objection. With the use of the metric, the norm of a state vector isdefined as

( e. 3 )

Normlrp) : (rpl rllrtt). In order that the norm always

be realr

must be Hermitian:

{

( e. 4 )

n* : tl' So that we can ignore some trivial qulre

numerical

q':rtrl*The expectation

As a consequence of this, qarilrr

h r ra v r z e

a

we also re-

1'

/q

('l

value of an operator F is now defined by

F:

rq r4 rl

factors,

rtral

a Hermitian

trync.fafion vJ:yvvLssr vr r

(e.6)

QplqFllt)). rralrrc

operator does not necesthiS is just what we

and

want. With these new definitions, the norm of a state vector is not aiways positive . We can divide the state vectors into three classes. The first class contains those vectors of positive norm, and these can be normalized to I . The usual probability interp r e t a t i o n o f t h e q u a n t u mt h e o r y c a n b e g i v e n f o r t h e s e v e c t o r s . The second ciass contains vectors of negative norm,which we can take as -1, and the third class contains the null vectors. For the iast two classes of state vectors there is no probability interpretation, and we must therefore require that every physically realizable state belong to the first class. Fortunately, it will be seen later that our form of the subsidiary condition is compatible with t h i s r e q uj r e m e n t . In order that the expectation values for Ap(r) be real and the corresponding quantities for An(x) be pure imaginary, we require

Q t l qA n @ ) l ' p: ) k t l A t @ ) r t * l , t: ) Q p l A u @ n l), p ),

( s. 7 )

l A u @ ) , r )o: ,

( e. 8 )

Q t l q A n @ ) l v ): - ( . p l A n @ ) r t l y,t )

( 9. 7 a )

and

(e. 8a)

{An@),ri:0. G o i n g o v e r t o m o m e n t u m s p a c e , f r o m ( 9 . B ) a n d ( 9 . B a ) w e have

l a ( 4( k ) , T ) : o , {a@(Ic),rt}:o.

(e.e)

1+4 , ,

( e. 1 0 )

40

C . f i i t t 6 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec.9

From (9.9) it follows immediately that 4 is diagonal in the quantum numbers ntz\(k) f.or .l$ 4 , and has matrix elements I. Tr rnr lL' hr ar v rr av Pnr r 6 v s e n t a t i o n ( 9. 2 ) , E q . ( 9. l 0 ) i s s a t i s f i e d b y (nltl 1r, n' @)>: d;$ o,o,t (- 1),,n,.

( e. 11 )

For ? we therefore have the solution (alrtlh:1-

( e. l 2 )

r 1 " L 6n o' o,

where the quantity nLo)is equal to the sum of all the s c a l a r p h o tons in the state l4) :

nf,r:Znf,i(k).

(e.13)

tc

In this metric, a few of the previous formulas are changed a little because we now have to put N(4):a*G) aQ) . For example, instead of (6.7) and (6.19), we obtain 4

H : Zar ) lftir (lc; ,

( e .r 4 )

Pu:ZAe X N(.1)(ft).

/q rcl

,t

k

l:L

t:t

We now return to the subsidiary eondition. In momentum space it has the form [4(3)(tc)! i a(at(k)f lrp) : o.

( e. r 6)

I n t - s p a c e ( 9 . 1 6 )b e c o m e s aAL+\ @) '

/q r7l

-;lv):o'

In general the symbol F(rr(x) denotesthat part of the operator F(r) which has only positive frequencies, i.e. , wlth r-dependence g i v e n b y e i h' . w i t h t h e u s e o f ( 9 .8 ) a n d ( 9 .8 a ) i n ( 9 . 1 7 ), w e h a v e

, , t a A r ) @_ ?aA l t @ \ n: \ :ee t l n l P 9 : 0 , ---a

\?t\-iA

1,1

ont

(e.tB)

and therefore hAt+t l y\ t^ , (x\ < v t n0 4f" (rx \f l D : Q t t r0 Avl ;h : x t r p*) ( v t r t x f i # l y ) : 0 . ( e ' r e )

I n o u r n e w f o r m a l i s m ( 9 . 1 7 )r e p l a c e s t h e c o n d i t i o n ( 8 . 2 ) . I n t h i s w a y (9.1)is satisfied, and the connection with classical theory ensured.I also be shown that several factors +* @ have the expectation value zero, It is simple to make the necessary revisions . For example, for two factors , aAlY) ( u , t, z A l , j , ( x ) a A t f , @ ' lV t)^ . ._ \ e p l n 4 90x, ): A*i. It i, p':) : dr, AxlaAti)(x) -a,nf) ; i ' l l t1x'11 p): , , - \,, t t, ,.-l t ( , p l , t l , t ) . D e , Q , _x ) : 0 , e t c . tL-;,

Sec. 9

The Subsidiary Condition: SecondMethod

4I

Equation (8.2) had only the single solution (8.9), but the new subsidiary condition (9.16) has many independent solutions. As before, we write an arbitrary state vector as

lrp):lrl)n@*) h

r

e.Zl)

and consequently Eq. (9.16) is only a condition [as (8.2) was previously] on the quantities l@p) . As a fundamental system of solutions, we can choose

ip1>, il,,l:li,l; ,

:

( 9. 2 t a )

( s.2r b)

-r..-

_ >.(i),ll (:) ln _ r,r) t@(")> ' . Y\//' --'

( 9. 2 l c )

:

Obviously all the vectors (9.2I) are mutually orthogonal, ( o t t 1 , l @ t r ' ) :; 6 .

ia

lo

)t\

rfo

t2\

More important is the norm of the vectors l@t't);

( o a t , ,| @ a t:sI

( - r ) '( : ) : u " r , f---O

\ r /

None of these norms is negative, and only the vector l@(0)) has a non-zero norm. Because we require that a phystcally realizable state satisfy (9.16) and have norm I, the allowed vectors are of the form

(s.24) with arbitrary coefflcientr ,(r)(k) . If we now caiculate the expectation value for the electromagnetic potentials ln the states (9.24), we obtain (assuming for simplicity that there are no transverse photons present)

I <,VlA r l p \ x ) l 'V ) : f r l,=!1r'u' 7 yr,

(o^lq ats) (k) l@*) * ItL")

1- ef (o*l q a$)(k) l@r)l f e-ihx lef)(Qxl rt o*tz)(k) l@*) f

(e.2s)

i *) (Ailqet6(4\(ts) l@e)l). Fromthe esuations

a@lc,@>:lipa-'t; , :;)fi1Ett"-rtS a@l@(n)> ,

(e.26) ( s. 2 7 )

42

G. Klill5n, Quantum Electrodynamics

Sec.9

w h i c h f o l l o w r e a d i l y f r o m ( 9 . 2 1 ), w e f i n d ( A k l U a @( I l l Q ) :

I ( @ ( 0I)1 1 l Q t " - r t S \ 6 c r " 1t

,,,

* I ,* tn')<(D(n'J l 4l@t"-ttS Vn cv\: co (k) , | , , h+0 tu'+o

( @ n l q a t a(\k ) l Q ) : i c , 1 ) ( r q .

(e.2e)

Using this result in (9.25) gives ,A t.) (\ uw| ' lt n : -':'' - - lA t \ ",t(I fr ) I u/ ) bxt,

( e. 3o )

t

with

;

,-ihx -t,. >, ,,1 - L lI l v 4R lt 'l *) , , , 3 1 -r * t , r\ -1-tl c" , 1

-A- (\x. \- :/

c o ) ( k \c i h , rf .

1 \ _9 . 3 1 )

Thus we see that with the appropriate choice of the coefficients c in Q .2a), we can obtain every gauge function A(x\ which satisfies the wave equation (5.7). This possibility did not exist in the previous method where a definite gauge function was chosen. On thc nfhcr hand. if follows that the different state vectors (9-24\ o

cnrroqnnnd

nnlrr

fn

v v ] l v v y v l r v v r v q q y u r o g ^ P L 9 L v v

diffcront

choicoq

nF

narrno

^hd

it

ic

\ J

'

' = l

ovnonfod

that physically significant results are independent of the "mixing in" of longitudinal and scalar photons in (9.24). Although only t h e c o e f f i c i e n t s c ( r ) ( t c e) n t e r t h e r e s u l t ( 9 . 3 1 ) , i f t h e e x p e c t a t i o n value of a product of potentials is calculated, it turns out that the other coefficients play a similar role. The states (9.24) are not eigenstates of the Hamiltonian if the quantities c do not aIl vanish. However, if we calculate the expectation value of the energy of such a state, we get

(k)<@t)t),1,4( QtlnH le):x {+l,n.ra @c@') +: c*@) *

T

, (nfti a ngtl: Z@@E) + "ft\. I

This expectation value is equal to the energy of the transverse photons alone and is independent of the mixing in of longitudinal and scalar photons. If we regard the expectation value as the observable quantity, then a1l the states (9.24) are equivalent. In many applications it will be useful to arranqe our states so that they are dctual eigenstates of the Hamiltonian. For example, we must then define the vacuum as that state where no transverse photons are present and where no other particles are mlxed in, i . e . , where I @r) is given by (9 .2Ia) . The vacuum is theref ore the state where no particles at all are present and which we have previously denoted bV l0) . By this, a special gauge has again

Measurement of Electricand MagneticField

Sec. l0 lroan

nhnqan

and

it

iq

that

dArr-a

Strengths

rrrhara

( o l 1 1 A , , ( x ) l o()o l A " ( x ) i 0 ) : 0 . With this definition

haarrrca

tha

d u , D ' , (' )x ' ,-

x). (9. 34)

(8.40), and this result differs from

This is the same equation as 1?\

(e.33)

of the vacuum, we immediately obtain

r 0 | r 1{ . 4 , (, x | , A , ( x ' ) ) l 0 ) : 1 o | \ A , ( r ) , A , . ( r ' ) \ o ) :

t/7

43

ranracan+^+inn

fnr

-(e)

and

,{<(4) ic

nhannarl

\ , . r u l v v v g g v l l v r l Y 9 g .

We obtain corresponding results for the spatial displacement operators 4 and for the (spatial) angular momentum components I Jitl

T.ikcruisc.

fhe

exnecfation

rralrres

arc

indcncndenf

of the

particular mixture taken in (9.2A); however, the state vectors ciocncfafcs

nf

v rv L rrJ u s L v e

thc

onor^tnrq

v

nnlrz if

vr r r l

all

ggqffigjents

c

afe

are

zefo,

For the angular momentum, we again find that the component in the direction lc can have only the two possible values +l and -1 fnr rvr

a u

yn rh rr jr uc li vn sr l. L

/i \r

a

f r : n c r r a r cv ov )r ! v /

yn Jh rn vf nL nv r r .

We must note that the states (9.21) do not form a complete set. This is obviously impossible because otherwise the condition (9,17) would be an operator equation, If we wish to do a sum over "intermediate states" as, for example, in a matrix product like

(e.3s)

( a l A B l b ) : Z Q I A I z ) ( z l B ) b >, l')

then we must include all states (6.12) in the sum over lz) . This remark will turn out to be essential for a theory with interactions. We should give a detailed discussion of the Lorentz invariance of the new method , Qt u formai level , this can be done in a quite satisfactory manner.l For us it is sufficient to state the two most the definition of the important consequences of the formalism: vacuum and the expectation value of the anticommutator are obrri nrrelrr

nnrzarianf

Rrz nnmnaricnn

w r v g l r f . U l v v r r l y g r g r l v L r v r l \ J . v r l

fha

rarroc

frrnctinn

(9

3ll

de-

pends upon thec(tc) and can therefore depend upon the coordinate q r r su Li p n s r:r

r 1

if

the

c

artr

nnf

rrerrr carcfttllrr

ch6ggn.

ThiS

iS ngta

Sefious

objection, since the final results are independent of the gauge function. We will be content for now with these simDle remarks. I0.

The Problem of the Measurement of the Electric and Magnetic Field Strengths In the usual quantum mechanics of (point) particles, the com-

trrrtaf^r

af

h

and

a

iq

n^n-\/Anichinc.

lP,sl: - '

(10. r)

This has the well known consequence that one cannot simultaThis is the neously measure I and q with arbitrary accuracy. L . 5 e e , I O r e x a m p . L e ,1 . J . b e l l n l a n t e ,

f nys. Rev. vb, /uu (.tv54J.

G . f i 1 t 6 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec. l0

content of the Heisenberg uncertainty principle, which saysl that the uncertainties /p and Aq must satisfy the inequality

/p.lq21,

( 1 0. 2 )

since the mathematical formalism in quantum electrodynamics rs just the same, we should expect that a simildr uncertainty relation would follow from (7.5), ( Z.SA;. Because the electromagnetic potentials themselves do not have a direct physical significance (only the field strengths are observable quantities), we must dif_ ferentiate these equations. This gives the uncertainty reiation between the field strengths:

- t[r,. lF,,(*),40(x')l: 4q-

6,^4q-ar,ffi7^+ I (to'') -^!n-l ru -",. i u^vu^e

Introducing the usual

E and

)

)

H in (10.3) according to (5 .2) gives

lE,(x),E"(x')l: lH,(*),H.(r,)l: i lAf ;'t _ Ay.il t lE,(r),Er(*')): LH,(x),Hr(*,)l: ilAy;') - Ayrtl, l E , ( x ) , H , ( x ' ) JO : , l E " ( x ) , H r ( x ' ) l :- L H " ( x ) , E r ( * , ) l : i l B y ; ' ) _ B y ; i l

00 .4) (10. s) (10.6) ,

00.7)

and cyclic permutations of these equations for the other field components. Here we are using the notation

- *(#"AY;'):

#q)l| uo;*,0-,)f,

(ro. B)

: -+ AY;') dwlru a;- "o- 4],

(10. e)

uwt*,0- 4],

/ 1n r n \

: - +#r6[i BY;') r:la'-nl

.

( 1 0r. l )

The right sides of these equations contain quantities which are as singular as the Dirac delta function and its derivatives . unlike (10.2) ' it is clear that these equations do not appry directly to experimentally attainable situations. Instead of field strenqths at a single point, we shall consider their averaqe values orrl, u finite region of space and time. This enabtei us to inGliJt" rott, sides of (10.4) and (10.7) and get meaningful expressions. we have developed a formalism, and it is essential to understand that only such mean values are involved in the interpretation of this forat

d

1 . A c t u a l l y , ( 1 0 . 1 )i m p l i e s t h e s h a r p e ri n e q u a l i t y Z p . l s > l / 2 , suitable definition of the u n c e r t a i n t i e s i s u s e d . T h i s f a c t o r

of. 2 i s n o t s i g n i f i c a n t

for us.

Sec.l0

M e a s u r e m e n to f E l e c t r i c a n d M a g n e t i c F i e l d S t r e n g t h s

45

m a l i s m . A s a n e x a m p l e , w e c a n s h o w f r o m E q s . ( 1 0 . 4 )a n d ( 1 0 . 8 ) that

- VF?), : ; SIYP e"P'1) le,g1, e,1x1:# Io', I dxoE,(x) ,

( l o. 1 2 ) ( 1 0. 1 3 )

nT1

: n;rr At;,1) a'* [ a"*'I o-,1 dx'o Tff,t ' v r ' z l t ' z JI, tr, l,

( 1 0. l 4 )

Tz

From (10.12)we obtain the uncertainty relation

/ E G ) . A q - V ) - t W F -A W t.

(101 . 5)

This rcsrrlt nrts no restriction at aII on the measurement of a The minus sign on the right single field strength at one point. From this, it follows that two composide of (10.15) is crucial. nents of the field strength can be measured to arbitrary accuracy This is the imif the two regions (Vr,Tr) and (Vr,Tr) coincide. portant difference from the usual quantum mechanics , where two quantities with the same time coordinates do not always commute, Similar results can be proved for the other components of the field strengths by using Eqs. (10.5) through (10.7). From these equations we also see that the mutual interference of two measurements can only occur when the two regions can be connected by light signaIs. It is therefore expected thdt the uncertainty (I0.15) comes An electromagnetic disturbance, produced by a about as follows: measurement, propagates with the velocity of light to the other There it influences this other measureregion of measurement. The first term on the right side ment in some uncontrollabie way. can be interpreted as the disturbance of region (2) by region (l) and vice versa for the last term. It isthe difference of these two terms which appears, rather than their sum, as one might at first expect. These considerations are quite formal, and we ought to invent gedanken experiments which explicitly exhibit this measurability. Although this is not a simple problem, it has been exhaustively Limitations of space prevent discussed by Bohr and Rosenfeld.' so we us from going into the details of the measuring apparatus, We shall remark that must refer the reader to the original article. the discussion of Bohr and Rosenfeld has completely confirmed the equations given here. In order to make the best measurement, it has been shown that the test particles, which are used to define Rather, should not be elementary particles. the field strengths,

, an. Mat. Fys. Medd. 12, No. 8 l . N . B o h ra n d L . R o s e n f e l d D ( 1 9 3 3 ) ,a n d P h y s . R e v . 7 8 , 7 9 4 ( 1 9 5 0.) A s h o r t s u m m a r yi s g i v e n b y W . H e i t l e r , o p . c i t . , p p . 7 6 - 8 6 . S e e a l s o L . R o s e n f e l d ,P h y s i c a , H a a g 1 9 , 8 5 9 ( 1 9 5 3 )a n d E . C o r i n a l d e s i , N u o v o C i m . S u p p l . I 0 , 8 3

(res3).

46

G. Kiill5n, Quantum Electroclynamics

Sec. ll

they must be macroscopic bodies with large, uniformly distributed chargesand masses,' One should think of them as a large number of elementaryparticles, held together. Moreover,the test bodies must be rigid; i.e., the whole body must be set in motion at the same time and must not bend. Because all forces have a finite velocity of propagation, this is no trivial probJem! Bohr and Rosenfeld have shown that, in principle, one can construct such objects. The solution of the problem is not particularly slmple and can scarcely lead to anything practical. The essential point is straightforward: The present formulation of the quantum theory cannot forbid the existence of such test bodies. On the other hand, this makes it clear that quantum electrodynamics cannot attack the problem of the elementary particles because macroscopic bodies are essential for the whole interpretation of the theory. For this reason, we do not expect that the value of the ** , can be determined from the * theory. When we later study the interaction of the electron and the electromagnetic field, we shall regard the charge as an arbitrary parameter and even allow a time variation of the charge. For the reasons stated above, this formal procedure does not contradict the basis of the theory and will turn out to be quite useful i n g o i n g t o t h e l i m i t d i s c u s s e d i n c o n n e c t i o n w i t h E q . ( 7 . 1 9 ). charge of the electron,

lI . The Electromaqnetic Field in lnteraction with Classicat Currents As preparation for our later discussion of the interaction between photons and electrons, we shall now consider the much simpler problemz of the interaction between photons and a grven classical current denslty i*@) . The Lagrangian denslty for this nrnhlam

.

i c

s:-Io,,o,,-i+rui*o,i,.

For the potentials,

we obtain the equations

nA,@): _ i,@) and the Hamiltonian

onpratnr

(ll.l)

of motion /il

rl

iq

I. In an earlier work of Landau and peierls the measuremenrs are discussed in terms of elementary particles (as test bodies), and a contradiction is found to the equations given here. See Z . P h y s i k 9 9 , s 6 ( 1 9 3 1. ) 2 . A s i m i l a r p r o b l e mw a s f i r s t c o n s i d e r e d b y F . B l o c h a n d A . N o r d s i e c k , P h y s . R e v . 5 2 , 5 4 ( 1 9 3 7 .) S i m i l a r q u e s t i o n s w e r e l a t e r consideredby many authors. See, for example, W. pauti and M. Fierz, Nuovo Cim. !!, 167 (1938); W. Thirring and B. Touschek, ; . J . G l a u b e r ,p h y s . R e v . & ! , 3 9 5 ( t g S I ) ; P h i I . M a s . 4 2 , 2 4 4 ( 1 9 5 I )R M . a n d F . R o h r l i c h , H e l v . p h y s . A c t a 2 7 , 6 1 3( 1 9 5 4 ) . J. Jauch

47

Interactlon with Classical Currents

Sec. ll

! t +' atxp ! u !0*6 ! o l - ''H : l , 2f a " r l ! ! t 9Lxo J

)

10ro

J

'/

\ au\x)' / ' \ d'"x|P\t(1

(1I' 3)

In general the components i*@) witl be tline-dependent. Because the Hamiltonian is not invariant under time translations, there will be no conservation law for the total energy. We cannot expect stationary states of the Hamiltonian. If we require that the currents vanish for ro->- oo at a rate fast enough so that the integral

I D(, - x')ir@')dx'

01.4)

-@.

converges, then a solution of (II.2) is found to be

' )' x " A r ( * ) : A P @ ) ' l! D * ( x- x ' ) i r ( x d

(11's)

The operators Alll (r) satisfy the homogeneouswave equation

JAf,i(x1: s ,

(11.5a)

and have the same commutation relatlons as the total potentlals A u @ ) , b e c a u s e t h e l a s t t e r m l n ( I 1. 5 ) i s a c - n u m b e r . T h e y a r e therefore ldentical wlth the operators ,4r(*) for the free fields which were lntroduced earlier. Using the simplifications of the method of the indefinite metric, we have

lAlP(r),Alot(x')l : - i 6p"D(x'- *) ,

( r r 6) .

(o 11Z1ot (r),A[o) (r')]lo) - 6p,Dt1t (r' - ,) ,

(u.7)

T h e E q s . ( 1 1 . 5 )t h r o u g h ( 1 1 . 7 )g i v e t h e ' s o l u t l o n o f t h e e q u a t l o n o f motion (11.2) and the canonical commutatlon relations. As can be seen frorir (ll .5) , the opera{ors /t0) (x) correspond to the initlal values of the complete operators A,(x) f.or fro+- oo. Iheyare frequently called the "in-fields". If we now compute the energy for this solutlon, we obtain, after integration by Parts,

H : Hp)(AIp)+ E(ro)+ l r r ra,l\9rxt airql I U,. ,, ) !a , o + J a ' x J a * ' 1 " ^ t ; )D" o ' @ - x ' ) - A r l t @"!.g_!)j I a , , o, ) . H \ o\ ")p( tAt o z) \:J[4 " " 1 37ox f'a J"'"1 o

a n t'trt a Af) v\aet) v) 1.( 11.e) +' 0 * n )xn I' 7xo

*')D*(x-r'{3i$-ai#J:,"t *1 E(xo):+IIo.'d*" oxo I axo f d"xDo@I 01.10) n'

aio@,\ airV!,\l_ A''r

A''i

I

- - ' t t t \ ' - tDo@_ *,)f,(*). - ' - I o*,i,@) [ ar, J

J

)

48

G. IGi116n,Quantum Electrodynamics

Sec. ll

The term ( 1 1 . 9 ) i s o b v i o u s l y a c o n s t a n t o f t h e m o t i o n , a l t h o u g h the time d e r i v a t i v e o f t h e o t h e r t e r m s o f ( 1 1 . 8 ) d o e s n o t v a n i s h i n It is easy to show that dH

-j;

: -

dx'1. (u.11) [a'* \fffea'@)+ I n*@ x')iu@')

We now require that the current vanish for xo++oo, as well as xo+ - oo From (11.11)it then follows that the expectation value of the energy tends to a finite limit for /o+:l oo fortime-independent state vectors. As is seen from (ll. B), the energy operator for .fo-> - oo is simply

lim H:H@(A@\ \

f , o - - @

(r1.12)

t '

P

If we choose a representation where the operator (ll.12) is diagonal, we can characterize each state vector by the particle number operators, which are constructed from the operators AfJ(x) by the methodof Sec. 9. The operators Ar(x) coincide with the AII)(*) for ro->- oo under the requirements we have imposed here. Inthis limitthe energy is given by (lt.t2). The particles constructed in this way are therefore to be interpreted as those which are present in the system as .t0-+- oo . They are called the "incoming particles" or the "in-states',. It is obviously quite possible to study other solutions of (11.2). For example, we can consider

A,(x): AIP' @)* I Dn@-,',)i,(x') dx,

0 1 .1 3 )

T h e o p e r a t o r s i y l ( r ) a l s o s a t i s f y t h e h o m o g e n e o u s w a v ee q u a t i o n

niftlxy:s,

(il.14)

and have the commutation relations

@, Ata@')l: - i 6,,D(x'- *). lAlP,

/lr rql

F o r t h e e n e r g y , t h e r e a r e e x p r e s s i o n s s i m i l a r t o E q s . ( 1 1 . 8t)h r o u g h (1i.10), except that all the retarded D-functionsare replacedby advanced ones, and the fields AIP@) are replaced by lr!(x) In this case, for xo+* oo, we have

lim A,,(x\: A\9ttrl P

to++@

'

t

t

lim t1 : gtot1rtot1

x o .- + @

( r 1r.6 ) (il. r7)

If we choose a representation in which the operator (ll. 17)is diagonal, then it is quite clear that the particles we have constructed are those which are present in the system as r0=>f oo Subsequently, we shall call them the "outqolnq particles" or the "ours t a t e s " a n d 1 1 o t( x ) t h e " o u t - f i e l d s ' , . - f n g e n e r a l , t h e s t a t e s c o n -

Sec. 11

49

Interaction with Classical Currents

structed by this procedure are different from the "in-states" which we introduced before. For example, if we take the state l0) whene there are no out-particles (the vacuum of the outgoing pariicles), then obviously we have (O I {/ito)(*), iLil (r')} lo) :

6p,Do)(r' - *).

(II.18)

Note that this state is different from the vacuum state of (11.7). From (1I.5) and (ll .i3) we obtain the relation between the inand out-fields:

dx' Ajf)(*) - Ayt 1x!: I D (x - x')i,,(x')

(11.1e)

We now go over to p-space and write

AgI \ ' : +Z4i+leihx P (x) f,t llv

tAl p\ gYt rt-* \ :

llza

| >ry lv

f,^ lr.-

(k)a e-in,o*(a(k)l, (rr.20) ao)

-' ) l (, 1. "1. 2 I ) ' rs\ - i n , f i * (\ r- -)l(rk Ll s ' i xvr' f i\(-t-)t!g

From these equations and (Ii.19) we have aa)(ri - 6at(k) :.-r*,r,

(k, o) i"^a.

(r1.22)

Here we have lntroduced the Fourier components of the current density by setting

(lc,AJf e-ih'i*(i)(rc, frJ];d,".[r.23) oor-,LZt,!)(tr)leih"iQ\ ir@):+ ' ' i ^ u[' o Vv t"'it

The complex conjugate of. i@ (fc,fto) is always |'*lt)(k, fto), and the factor id,tr'ensures the correct reality properties f.or the i,@). Simtlarly, for the particle number operators we find

- N(D (rg) fr(i)(rc) +

(k,r) o* tt)(k)* 5#in ^)l, (rl + f^ vo (rc, + HV# i* @(k,a) at^\

l

gL.z,)

with

N(4 (k) : a*Q')(g att)1ts1 ,

(il. 2s)

(rc): A* 0)(k) a0)(k) . fr/(,r)

( l r .26 )

and

As an example, we shall consider the state which has no inparticles. Thls is the state which we denote as the in-vacuum A question of physical interest is the calculation of the l0). number of outgoing particles of given momentum vector k and given polarization i. We shall confine ourselves to a single momentum

50

G. Kiill6n, Quantum Electrodynamics

lc and a single polarization f,, for simplicity. In our notation an in-state of z particles is la), and one of ,c out-particies is lZ). The probabilitythat n such particlesare emitted by our system is clearly

wf,i(tt): l(tl0)1,.

(rr.27)

li,>:+y n l @\"t6> .

0 1 .28 )

From the equation

we find

: (6ra' (##)"o>: i* A(,\ r0)

:rt(r116;)"

:l

I

\rL.zJ)

From (11.24) the expectatlon value of .& js

n : ( o t f rl o ): j l i l , .

(11.30)

I n t h i s n o t a t i o n a n d u s i n g ( 1 1. 2 7 ) a n d ( l l . 2 9 ) , w e h a v e

rs?@): i, lrr,(rc)], |(0I o) 1,.

(ll.3l)

The last factor in (II.3I) does not depend on a and is determined from the normalization condition

ZwP (k): I

( r . 32 )

l(o loSlz: ,-n .

(r1.33)

n-O

to be

Ine emllteo

photons therefore have a Poisson distribution: u,:

)@)ns-a

(11.34)

This says that the emission of a single photon is statisticatly independent of the emission of any other. From (11.8) through (II.t0), the total energy of the emitted photons is found to be

t i m ( 0 l r 1 l o ) : zt -J Jf f d * , d * , , I a " * n' g - x ' ) D ( x - x , , ) x

ro*o''

J

*" f a i0*'o ry) L

a i 4 , . " ) a i 4 : ' ta i , g l l : >I Ar'i +' ArL Ar'i I

T F;

II

(11.3s) a)n(ltUt).i ,

This was expected because of (I1.30). The continuity equation for the current 1u(r), ( i l . 36 )

Sec. lI

51

Interaction with Classical Currents

has been employed in obtaining (II.35). ( 1 I . 2 3 )w e h a v e

That is, from (I1.36)and

(Ie,a)12: li
/r r a 7\

n@(It):n@(tq .

( 1 1 3. 8 )

and from (1I.3 0) ,

T'hereforethe term in (11.35) with scalar photons and a minus sign is just cancelled by the term involving the longitudinal photons. In a slmilar way the distribution of emitted photons can be found for an arbitrary initial state. Beoause this result does not contain anything essentially different, we shall omit the explicit calculation. An interesting special case of this general problern is provided by a current distribution which is time independent in some special coordinate system. It appears that the results would be uninteresting because it is quite obvious that no photons are emitted. The probabilities zo, vanish by (11.34), since the factor D must vanish for time-independent cr.urents. This follows from (11.30) for all ar different from zero. In this case the interesting quantity i s t h e H a m i l t o n i a n , w h i c h i s g i v e n b y ( l l . 8 ) t h r o u g h ( 1 1. I 0 ) :

H:H(o)(AP)+E,

0 r .3 e )

- x')D*(x- x')ILEL4# Do@ E: + II or'dx"Idsx -

-]t,,.

Do(x- x')i,@)i,@').) I I o'*d.x'

not

By the use of Do@ - *'):

4*;6?"* rrt:lu-fr'\

-

x o * x ' o ),

,

( n.4r ) (1r.42)

the term E in (lI .39) can be written

- !# i,@) i,@') " : +I +#I I f## W W + il : - +I f# I I 4#: i,@) i,(n") #6*) :-*ll

!#!i,(n')iu@")'

- i' II '"i'ii'" 'i'@)iu(n") 0r.43)

This quantity corresponds to the electromagnetic self-energy of the stationary current distribution. The total energy 0I.39) is the

52

G . f d t l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec. ll

sum ofthe term (I1.43) and the energy of the free particles. This is intuitively the correct result, although the static current distribution obviously contradicts our assumption of a vanishing current for lrol ->@. D e s p i t e t h i s , t h e i n t e g r a l ( 1 1. 4 ) i s c o n vergent, since the infinite time integral over the function (I1.41) certainly converges. In a theory with a non-vanishing particle mass, this would not hold because the integration would be over the inside of the retarded light cone with a non-vanishing density function Ao@-x') . With a view toward later applications, we shall give another proof for Eqs. (1I.39) through 0I .43) . We consider an almost stationary current distribution

ir@):e-dt""l'1"(n).

(II

AA\

This vanishes for lrol-+oo if o( is non-zero, but for finite times it c a n b e r e g a r d e da s " a l m o s t c o n s t a n t " i f a i s t a k e n s u f f i c i e n t l y smaIl. \A/ewill describe this by saying that the current is "switched off" for lrol-+oo, and in the limit a->0, the switching-off is done adiabatically. In this process .E is not constant in time as ln b u t . 4 3 ) , v a n i s hes for xo-->- @ t as is obvious from (tl.t0). 01 R a t h e rt h a n ( 1 1 , 4 3 ) , E n o w b e c o m e s

E: L I I !#( H#l* x Jr@') I,@")-

t}x + ala-*,;]e-"{tn-' ""'|+t'.-'r.."

II +#

r-a{trnt*t'o-t,,'t) Ir@)Ir@').)u'.,0

The first term in brackets is of the order of magnltude or (and not oc2!) and vanishes if the switching-on is adiabatic. The other t e r m s g o o v e r t o ( 1 1. 4 3 ) . I n t a k i n g t h e l i m i t , , 0 i s t o b e h e l d constant as o( goes to zero. For the time-dependent current (II.44), the total energy (Il. 8) contains one more term of the following f orm:

t II 0,..r_i-#lrh?6(,..,- x.oix,o) t't'): + A l | ) @f). 6 @ , " ' - x o * t ; ' o ) ] u e - "It,' @

: + I I !#!lW

(ll. 4 6)

t qA@ @)]'t",-r".'t I,@')

The integrals appearing here converge for a=0;therefore the whole term vanishes in the adiabatic limit. By this explicit calculation, we have verified a special case of the adiabatic theorem of quantum mechanics. We can summarize the moSt important result of this example as follows: If a state is an eigenstate of the Hamiltonian and if a parameter in the Hamiltonian is adiabatically changed (in this case the current), then the same state is also an

Interaction with Classical Currents

Sec. l1

53

eigenstate, after the Hamiltonian is changed, but with a different eigenvalue. If the currents vanish, then the Hamiltonian is just gtol 1,4jPr)with well known eigenstates which can be characterized by the particle numbers. From Eq. (tl .46) it follows that the nondiagonal terms in the Hamiltonian vanish in this representation, and therefore that these same states are also eigenstates of the complete energy operator H(Ar). If the eigenvalue of the operator 1 1 t 011A f ) ) i s Z n r o f o r a s t a t e , t h e n , b y ( 1 1. 3 9 ) , t h e e i g e n v a l u e o f the operator H(A*) is lna*E for the same state. Here E is s i v e n b y ( 1 1 . 4 3 ). we should note that there are other ways to In conclusion, introduce free particles into this problem. For example, we can introduce a free field ,4jft (x, T) which coincides with the complete field A,,(z) at some arbitrary time I. The field AI?)@,I) is defined by A I P ( x , T ') :

-

[ a t * ' l n ( 'x J I ,l: T

This field satisfies

x ' '\

aAp\x') bD(x -:'t * A (,,\l ht a7\ "tt\-ll'\rf,'='/ A*6 ' Axn

the homogeneous wave equation

aAf)(,{,r) : o.

rir 1A'l

ln this way particles have been constructed which would appear if the current iu@) were suddenly (not adiabatically) switched off at xo:T In general, they are not of much interest. If we were to use a Schroedinger picture which coincided with the Heisenberg picture for xo: I then, in a sense, these particles would be the "natural" ones. For simplicity, we take T = 0 and write the a!t*!,o) in thuirFourier represchroedinger operators Ar(r,o) ana "'o sentations A , ( r , o') : + > W [ b ( 4 Vvo"^ll,.

( 1 0 )e i l . r+ 6 * t t )( r t' ) e - . i krt lf

(rr.49)

aer@ : =f ell)UdllT lba)1d eil"..- b*t^\0c\ -iro | \ - - l e-iktl r , 1 tt. 11.50) 1,"_o-FhLi'\tu) V 2lu"\tu)o F o r t h e o p e r a t o r s b ( / () t c )i n ( 1 1. 4 9 ) a n d ( I l . 5 0 ) , w e c a n u s e t h e m a t r i c e s ( 6 . 1 3 )a n d ( 9 . 2 ) . f t r e s e s a t i s f y t h e c o m m u t a t i o nr e l a t r o n s in the Schroedinger pictwe. The Hamiltonian is not diagonal. By means of a simple calculation, we find that the state of lowest energy (the vacuum) has the expansion

t0):c['o''',l],7rlW)" r'^4tn@)(^)(r,))1 , (r1.sr) where

lntb)>: fil4"to@s.

ur. c 2,,

trA

G . K i i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

From the condition (010):1, to be

Sec' 1l

the normalization constant c is found _ 1 ,

trt-a'i^

li(i)(k)l' 203

(ll.s3)

The physical vacuum is therefore a mixture of "particles at time zero" given by (11.51). Particularly in the older literature, this result is often described by saying that the physical vacuum With the conventions we contains a mixture of "free" particles. employ, we must be more cautious with this term because we will often make use of both the incoming and outgoing free fields. Acare identical cording to the results found above, the in-particles with the physical ones, yet because they are described by a free field, they can well be called "free" particles. In particular, if the current is a point source, then i('z)('c) is In independent of lc, and therefore the sum in (1I.53) diverges' this case the physical states cannot bg expanded in terms of the Nevertheless, the two states of free particles at time zero.' but cannot be expanded in kinds of states exist simultaneously In a strictly mathematical sense, they do terms of one another. This shows only that no not belong to the same Hilbert space. free particles at time to the be attached to physical meaning is p o i n t s o urces. w i t h m o d e l i n o u r l e a s t n o t a t zero, given here of the adiabatic theorem and the The discussion for introducing free particles has been spevarious possibilities field interacting with an electromagnetic with cific to a model The adiabatic theorem is very general given classical currents. and holds under quite weak conditions in the ordinary quantum A general proof for a system with mechanics of point particles.z For has not been given yet. freedom of degrees many infinitely calcuour model we have verified the hypothesis by an explicit lation. Later we shall often make use of the adiabatic hypothesis. Although the various free fields have been constructed only for a special example here, the discussion can be taken over, almost unchanged, to the general case of two interacting fields.

particularly emphasized by van Hove, @en Physica, Haag 18, 145 (1952). In this connection, he has introduced the word "orthogonality". (If the constant c is equal to zero, then every term in (ll.5l) vanishes and, according to van Hove, the vector l0) is therefore "orthogonal"to every vector lnlt\S !) See a l s o R . H a a g , D a n . M a t . F y s . N 4 e d d . ? 9 , N o . 1 2 ( 1 9 5 5 )a s w e l l a s A. S. Wightman and S. S. Schweber, Phys. Rev. 98, 812 0955)' 2. M. Born and V. Fock, Z. Physik 51, 165 (1928).

CHAPTER THE

FREE

]]1

DIRAC

FIELDX

1 2 . E q u a t i o n s o f M o t i o n , L a g r a n g eF u n c t i o n , a n d a n A t t e m p t a t a Canonical Quantization The Dirac equation for a free electron is

:0. ? * + m)v@)

0z.r)

Here tp designates a quantity with four components Vr,@)...tpt@), and the terms / are matrices (T)rB, which obey the anticommutation relations {y, y,J : 2 6p,. (lZ ,2) lf we write out explicitly a1I the matrix multiplicatlons in (12.1) and (I2.2), r,r,'eobtain

$ 1 $ ( n , \ ^ a , .+" ^m)d"Pj : o' vP(x) A\ A\ril"na;; i, lW) "u 0) po* Q) "BQ) Bu1: 2 6o,6na.

p-r

\tt.J)

(r2,4)

Usually the short notation of 02.1) and (12.2)is most convenient and leads'to no confusion. An expliclt representation for the matrices y is not necessary for most calculations; however, one can be constructed in the following manner:

I o - i o " "\ Tu:\ioo o ), tI

(rz . D/

0 \

zn:\o _r)'

( r 2. 6 )

Here.0 is the two-row zero matrjx, .I is the two-row unlt matrjx, and the quantities d' dre the Pauli spin matrices: o, , : t / 0 { \_ 1 , o . , : l / 0 \lo/'"'-\;

-r\ I o )'

t,t "o'"- :\ lo

0\ -r)' I

r ^z-' ./ ) \r L

* See also Chap. B of the article byW. Pauli in the Handbuch der Physik, edited by S. Fliigge, Springer-Verlag, Heidelberg, VoI. V, part I.

G. Kdll6n, Quantum Electrodynamics

56

Sec. 12

The matrices 7 are therefore four by four, and one readily verlfies that they satisfy Eq. (12,2). With the use of this representation for the T t wa can attempt to flnd the plane wave solutions of (12.1). It is stralghtforward to show that for every wave lt,(x)

:

(t2. 8)

uo(q) ei(an-eo't

of given spatial momentume , there are two possible values of. qoz

(12. e)

Qo:*E-*Wr+?nr,

. and that for each 4o there are two independent solutions We shall enumerate these solutlons in the following table:"9\@)

x

4

12'

! -

1

0

-

O

1

_q,+,r!r_ m*E

d"

-q,li8y

4'= mlE

mlE

*:" 4t\

d--x0^,

rr+E

mlE

o.*io^.

a,

m* E

t/t+ E

J!__:__:_:,L

\ \

r/

l"

m*E

I

The two solutions with z:l and r:2 go wlth the value {o:E,and These solutions are the two others go wtrth the value llo:-E. normallzed so that

lwlvt (q)uf\(q): d,,.

a:1

02.ro)

Moreover, one can show from this table that lu{t,t (q) "tir (q) :

6oF ,

(r2.1r)

I:T

,

uf)(a): - ftttrd.,-*)pn , zay,@)

(1,2,r2)

r_L 4

Za!, @)utp(q):

/:z

tiT rr, - m)po. "',

( 1 2. 1 3 )

Here we have used the notation

a@): w*(8)'yE ,

(r2.r4)

and qt+t :

(q, i E) ,

qt-) :

(q, -;' E) .

( 1 2. 1 5 )

Equation (12.10) is the orthogonalityrelation for the solutions of the wave equation, and (12.11)shows the completeness of the set of solutions. Physically the two different solutions for each value

Sec. 12

E q u a t i o n s o f M o t i o n , L a g r a n g eF u n c t i o n , E t c .

ST

of qo correspond to the two possible orientations of the elecrron spin. Obviously the total spin is then one-half. The transformation properties of the Dirac theory have been discussed in various textbooks.t We shail not enter into a discussion of them here except to note that V @) : tt* @)yE ,

,l @)',p(x); is an invariant and that

i rp(x) yrtp (x) has the transformation properties of a four-vector. In particular, ,p*(x)rp(x) is not an invariant but is the time component of a vector. Formally the equation of motion can be obtained from the Laqranglan

s: -v,@) (r* * *)v@)

( 1 2. 1 6 )

by allowing variations of rp(*)and rp(x)as if they were independent fields. The canonical momenta can be obtained directlyfrom (12.16): n r ( x ) : i r l @ ) y n : i r p * ( x )'

(r2.r7)

nq(x) : s.

/rt

ro\

The momentum conjugate to rp(r)vanishes identically, and the time d e r i v a t i v e s o f t h e s e f u n c t i o n s d o n o t e n t e r ( 1 2 . 1 7 )a n d 0 2 . 1 8 ) . Thus it is impossible to express the time derivatives as functrons of the momenta. Despite this, we can construct a Hamiltonlan which is a function of. tp(x), its spatial derivatives, and the momentum nr(x). This Hamiltonian is

tr:i,'@)v'w .::, o, ,nr ,y:^:"-1,,;. ) av{.) In this form the function g(r) has been eliminated, and the Hamiltonian contains only the independent field tp(x). The CUrrent denS jt.'

fnr fha

Triran olan+rOn

iS well

knOwn:

i,(*):ietp(x)TpV@).

(r2.20)

These quantities transform Like a vector underLorentz transformations and, as a consequence ot (12. I), satisfy the continuity equation

+#: t'fo{irr\

+ e-Wy,,t'@)l:

: i e l- *rp (x)Ip@)+ m'p(x)rp(x)l : o.

(12.2r)

l. See P. A. M. Dirac, The Principles of Quantum Mechanjcs, T h i r d E d . , O x f o r d , 1 9 4 7, p . 2 5 7 .

G. K51l6n, Quantum Electrodynamics

58

Sec. 13

Previously we regarded the electromagnetic potentials as operNow ators which satisfied the canonical commutation relations. This we shalt attempt to interpret the field tp(x) as an operator. procedure is often called the "second quantization" of the electron In this language, "the ordinary Dirac equation" should be field. the first quantization, but we shaII refer instead to the "classical Dirac theory',. ln this theory the field rp(x) is a classical quantity After the second which is also the state vector of the theory. dynamical varithe but is quantization, v@) is not a state vector longer be interrp(r) no particular,lt)*Pc) can theory. In able of the preted as a probabilitY densitY. Previously we expanded the electromagnetic field in plane Now we do the same for the field 9(r) : waves.

u| (q) nt't(g)+ ,'rr" FE'o)>luy (q)e\')@)l .(tZ'ZZ'l ] ,i ta*-r'.15' Z- *' ) -_

tt^z41

(" t uY a \ x\:+, 1 ,1

Y'q

/)|

\

t:r

a(')(q) are therefore operators which we now regard The quantities The Hamiltonian can be expanded in as the dynamical variables. t h e m : terms of

H : l d' , x . i ( ("x \ : I E I i V

l':-r

Q2.23) (q)- f a*ot(q)o(,r a*?)(q)a(,) @)f j,

as can the charge: 4

a : * i I ir@)d}x: eIq ) o*t't(S)o?)(q). /:7

(r2.24)

Here we could attempt to use the representation (6. 13) for the matrices a(') . This is equivalent to the requirement

lnrQ\, tP(x')),,:r" :

- i 6 (n - n') .

(r2.25)

This does bring the energy into dl.agonal form; however, this apAccordinq to (12.23) the eigenproach is scarcely satisfactory. as well as positive values. take negatirre can energty the of values unlike the electromagnetic field, there is no subsidiary condition Furthermore, here which excludes the states of negative energy. whereas limited, is not in same state the the number of electrons e x clusion t h e o b e y e l e c t r o n s t h a t e x p e r i m e n t a l . l y k n o w n i s it principle. Hence the orthodox method of canonical quantization cannot be employed, and we have to develop another method in Sec. 13 for quantizing the Dirac field. Quantization of the Dirac Field by Anticommutators modified quantization method, we sha11 require that the 1" - i "* commutator of H and an arbitrary field operator be equal to we if holds clearly times the time derivative of the operator. This other any since rp(x), property operators for the only require this operator in the theory can be expressed in terms of tp\x) ' From ,J2.22 ) ancl (12.23) it follows that a sufficient prescription for 13.

Sec. 13

of the Dirac Field by Anticommutators

Quantization

quantization

59

is

la*o:1n,o(,)(q), a\')(q,)l : - ao (q) 6,, 6ro,, (') la*o)(q\ o\,)(q),o* ts)(q')l : a* (g)d".dnq,.

( 1 3. 1 ) 03.2)

of which the T h e E q s . ( 1 3. I ) a n d ( 1 3. 2 ) h a r r e m a n y s o l u t i o n s , canonical quantization is only one. They are obviously also sat'i c fiad

i f

razo ronrr

ira

t

{a* r,t1Ol, oG)(q,)} : 6,, 6
0 3. 3 ) 0 3. 4 )

These relations are completely symmetrical in a and a*, and we have even more freedom here in the definition of particle number operators than we had for the electromagnetic field. Since a and A* now anticommute , we can rewrite (I2.23) as

H : Z " { i " . v ) ( q ) d / ) ( q ) + i o ? ) ( q ) a * ( ,-)2( q} .) q

tz:1

r'-3

(i3.s)

)

We now define two new operators b('l by b ( l ) ( q ):

(r3. 6) ( 1 3. 7 )

a*@(_ q) ,

at,(s) : a*(r)(- g) .

T h u s b ( / ) u n 6 6 ' * ( 4a l s o s a t i s f y t h e a n t i c o m m u t a t i o n r e l a t i o n s( 1 3 . 3 ) and(13.4).The total energy can be expressed in terms of "particle numbers " N+ and -l{- :

H:ZE q

+ l)' r - t ' t 1 n , , I (N.(")(A

( r 3. 8 )

/:L

Lr* (')(g) :

(r3. e)

6*{.r)(q) ot ) (q) ,

Lr-t"r(q): b*v)(e)at'r1n,.

(r3. l0)

T h e t a s t t e r m i n ( 1 3 . 5 ) h a s b e e n o m i t t e d f r o m ( 1 3 . 8 )s i n c e i t i s o n l y a c - n u m b e ra n d o b v i o u s l y c o r r e s p o n d s t o t h e z e r o - p o i n t e n e r g y o f t h e f i e l d . T h e q u a n t i t i e s J V +a n d / { - i n ( 1 3 . 9 ) a n d( I 3 . 1 0 )s a t i s f y

I{*(1 _ N*) : t/-(1 _ N-) : o,

(13.il)

as can readily be shown from (I3.4). They therefore have only the eiqenvalues 0 and I and no others. We shall interpret them as the number of particles of given momentum and given spin. By the choice of the anticommutators in (13.3) and (13.4) we have ensured that there can be only one electron in a given state. This is the exclusion principle which we find here as a consequence of the quantization with anticommutators. It is one of the most important results of the theory of quantized fields that quantization with urr I

p r

.

Tnrrlan

J v r v s . r

and

F

1vAv r/ yi n nar r l vl

t

7 . u.

D hL rv! c iuLr r :

L

A,7 =:,

A ?l vwr

/lqrAl \rr!v/.

60

G. K51l6n, Quantum Electrodynamics

Sec. 13

bounded eigenvalues for the particle numbers lead.s to unphysical In particular, the consequences for particles with spin one-half. In a similar way it can be shown energy is not positive definite. spin canno_t be carried of particles of integral that quantization out with the incorporation of the exclusion principle.r term" in (12.24), we obtain forthe If we drop a "zero-polnt charge,

(g)]. 1O;- rv--t') Q : t Z [-l/+t"t

/r? lr\

q'r

This quantity is not positive definite. Rather than being troublesome, this is a most satisfactory feature of the theory. We have now found that the solutions with "negative enerqy" in (12.8) and (12.9)appear in the quantized theory with positive energy but wlth reversed charge. This corresponds to the "hole theory" of Dirac; however, it is obtained here as a consequence of our quantization, without any additional assumptions. If we wlsh to have the particles N* represent electrons, then we must take a negative value f . o r e i n 0 3 . 1 2 ) . T h e p a r t i c l e s N - t h e n c o r r e s p o n dt o p o s i t r o n s . Dropping the zero-point charge in (13.12) can be formulated quite simply in ,-space. Instead of defining the current by(I2.20), we takeZ

(r3. r3) Here the notation 1s (r) rle@)- rte@)rt @)). (t3 . 14) W @)yr, rt @)l:lrt @),yr rp(x)):2^ 0 r)" B(rt' " " If we introducethe series irr.rrrinto (13.13), it is a straightforward calculation to obtain the expression(13.i2)for the charge Q. In a.similar way, the vacuumexpectationvalues of all the componentsof the currentvanish:

( o l i , @ I)o ) : o .

/r2

|R\

It is not possible to give an analogous simple prescription in .r-space for subtracting the zero-point energy. For the spatial components of the momentum Pu,by using (3.12) we obtain

(- q))+ ,}= 4# :>.0,{; t".0,(s)- n-t"r I 0r.,, :Zou(lr*t't10,*N-t'r10,1.

ru = - ;[ asxtp1x)yn

I

Q,f

The particles

N-,

which

were taken to represent positrons,

have

1 . W . P a u t i , P r o g r . T h e o rP . hys.5,526 (1950). References to the older literature are given here. 2 . W . H e i s e n b e r g ,Z . P h y s i k , 9 0 , 2 0 9 ( 1 9 3 4 ) .

Sec. I3

Quantization

of the Dirac Field byAnticommutators

6I

a m o m e n t u m q a c c o r d i n g t o t h e d e f i n i t i o n s ( 1 3 . 6 ) , ( 1 3 . 7 ), a n d (13.10). This is in complete agreement with the general result ( 3 . 2 1 ) , s i n c e t h e l a s t t e r m i n ( 1 2. 2 2 ) i s a c r e a t i o n o p e r a t o r f o r positrons according to (13.6) and (13.7). Because of this , the xdependence of thisterm must be s-'p' if p is the energy.momentum of the one-positron state. As a final topic, we compute the angular momentum for these particles. The symmetrical energy-momentum tensor can be constructed by the method given in Sec. 4. The result is

7,,(x):v(x)r,"ff+ i fi O'Al(y^ly,, y,)t T,ly,, y^l+ I trr.,rl f - y , U p , y i l r l @ D).

' o r . L ^ , ,d Lfo 1 c o mp l oner F rr th, r t 1nt( i tegri t i o n l Jrat:

r. 1. . : trt _ -

Ii'{9): 1

{l) I(l)

J 7 '1

(9)J- 7.(1 )

7(0j ! - l J 1 1i l

'r.[o s x l l u (*) l yn d3 (*),

1,

/l

2,

\L' '

i t^ -1" 8,

"x \ty)\ X ) , T E

with

I n ( 1 3 . 1 9 )a n d ( 1 3 . 2 0 ) t h e p r o d u c t o t t p ( x ) a n d g ( r ) h a s b e e n r e p l a c e d by the commutator Llrl@),rl@)l so that the vacuum expectation value of the angular momentum will vanish. In momentum space t h e t e r m j ( , 9 )c o n t a i n s a f a c t o r and is conseu t r ? ) @ )u f , ) ( q ) : 6 , , quently independent of the state of polarization of the particles. We shall therefore interpret this as the orbital angular momentum. We now consider the component of the other term Jf(l)in the direction of propagation of a particle. For a single electron state, if we take the direction of propagation as the z-axis, from 0g.ZO) and (I2 .22) we readlly obtain

lll)lq): +(- r)'+'lq). In a similar

fashion,

for a single positron

( 1 3. 2 r )

state we have

l|',)lq'): j (- r)/-rtto',

\rJ.

LL)

The complete similarity of (I3.2I) and (13.22) has been obtained by the choice of index in (13.6) and (13.7) . In working out (13.21) and (13.22) we have used the matrix representation for o;; which is given in (I2.5) and (12.6):

OL

G . K A l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

o'i:

/o* 0\ , und cyclic permutations. \O o^,)

/re

t?\

The Charqe Symmetrv of the Theory of the in the interpretation There is a certain arbitrariness theory, as has already been noted in connection with Eq. (13.12). Either we take the particles -l{+ as electrons , .Iy'- as positrons, and give a negative value tothe quantity e of (I3.12) and (13.13), or we interchange the roles of electrons and positrons and then a More precisely, the theory is invariant must be taken positive. 14.

under the transformations A/+ -

(r4. l)

AI_ I

(r4.2)

e<---e

This invariance property can also be expressedin x-space. We recall that the operator rp(x)contains annihilation operators for the electrons and creation operators for the positrons, while the reverse is true for the operator tp(x) or rl*@). It is clear that the transformation -Iy'+?/y'- must correspond essentially to an interc h a n g e o f r p ( x )a n d y , ( r ) . I n o r d e r t o o b t a i n t h e c o m p l e t e s y m metry, it is not sufficient just to interchange rp(r) and tp(") ' but rather we must investigate the slightly more complicated transformationsl (14.3) (*). y"(x) - tp*(x): C "prpp If it is possible to find a matrix C which has

(C-'yrC),p:

n r nvny va. r t i a qr * v

v

Pl

C B o,

( 1 4. 4 )

(CB*)* ,

(14. s)

-

0 4 .6 )

CoB: (C-t)"p:

tuhr r av

(yr)p",

then the operator tp'(x)wiII satisfy the same equation of motion as Ittlx) :

: (x) mc')v o):, lr- v c* + m]'t' 0 * * *) v'@) (r, * a :c[-

g#ramtlt(x))-0.

lu.',

In these transformations, C is a matrjx of the same type as the y/ -- tm lts^ f r i ^ c s and dnes nnl- nffocf iha "narljgle number indiCes" of u the operators g;(x). If we introduce an expansion for rp'(z) analogous to (12.22) and interpret the states with positive qo as electrons, we see that the Hilbert space of the electrons has simply

l . w . P a u l i , A n n . I n s t . H . P o i n c a r 6 6 , 1 0 9 ( 1 9 3 6.) T h e m a t r i x C . o f P a u l i is a little different R e v . 7 4 , 1 4 3 9( r 9 4 8 ) .

from our C, S e e J . S c h w i n g e r , P h Y s .

Sec. 14

The Charge Symmetry of the Theory

63

been interchanged with that of the positrons. Moreover, we can show that the entire theory is actually invariant under the simultaneous transformations (14.2) and (i4.3). Thus the new current becomes 'l * : : ) iL @ ) :- , f O 'V l ,y,,p'(x)l +l c'v@),y * c' ,p( *) I : # n 'W l,c-rypcv @)l:, fOOf y,v@D: i u@),l

ttn.rt

Here use has been made of the following result, which is obtained f r o m ( 1 4 . 3 )t h r o u s h ( 1 4 . 6 ) : , y L @: r t ' ! p * ( x ) ( y n ) e " : - ( c - ' y n v @ ) ) e Q n ) B * : :

L , , \ r = ' Jol, - (C-r T E C ) p a ( C u t p ( * ) ) o ( y n-) B( C * - ' ) " pr p p ( i , f

The Lagrangian (12.I6) is not invariant under this "charge conjugation", even if it is "symmetrized" and written as

e,: - Ilp t.t,(,-h* | v{,)].

( 1 4r.o )

We find that 1l-,,,t a \,,,1 .l -^ - r l -a5a; ( x \y + m y v z=\ x ) ,v l x ) y t . 0 4 . 1 1 ) 2 l l p l x ) , \ y a * i m ) , , p\ x )) : z l l# Because both of these functions give the correct equations of motion, we can take the completely symmetricexpression

-, :?Z- 1\ z rl t( ?r z- L)( :?-\ -

ll-

t..\ l..A

\

o l r l ( x ) , \ay * + m 1 V @ l - | - r f -o Q , \ Y + r n V \ x )I '| \ x. 1) l | I l- a"

$4.12)

as the Lagrangian. In (14.10) through (I4.I2), we have given up our original assumption that the fields in the Lagrangian are to be regarded as classical functions for which the commutators are identically zero. The quantitieswhich are varied, g@) and tp(x) , are to be regarded as operatorsalthough the variations themselves may be regarded as independent c-numbers. Then the varied fietd operators will not obey the correct anticommutation relations; however, this does not cause trouble since the correct equations of motion are obtained. The energy-momentum density Q, obtained from the Lagrangian (I4.I2) is a priori charge symmetric. We must now show that there actually exists a matrix C which satisfies Eqs. (14.4) through (ia.6) . This is most simply done giving the matrjx. Using the representation of (12.5) by explicitly and (12. 6) , it is clear that

(Yr)"p:(Y'u)p" for P:4ot :1 o' 0 p ) " 8 : * ( T r ) B *f o r P '

2,

( 1 4. l 3 )

3.

( 1 4 .r 4 )

64

G . K i i 1 1 5 n ,Q u a n t u mE l e c t r o d y n a m i c s

Sec.15

If we choose C : lzl+ ,

04.15)

then the matrix C is unitary and antisymmetric, i.€., ( 1 4 . 5 )a r e s a t i s f i e d . F u r t h e r m o r e , C - t y u c: T E T z T p T z T a- : T p f o r p : 2 , 4 , C - ly p C: l t T z T p T z T a : T p for p:1,3.

(14.4) and ( 1 4. 1 6 ) ( 1 4, 1 7 \

As a final topic, we shall give a new proof of (I3.15) in which only the charge symmetry of the theory is used and not the explicit form of the expansion (I2.22). Under the interchange of the two Hilbert spaces, the vacuum is clearly invariant. Therefore as a consequence of charge symmetry, we have

( o l [ p ' ( r )T, r r ! ' ( x ) l l 0: ) ( 0 l L , , t @ , y u ' t U ) ] l o > . ( r 4 . 1 8 ) But from (14.8) we have the operator identity that

(r4.1e)

(*)l : - lrp(x),y,rt @)) lrp'(*),y,,rp'

) n d ( 1 4 . 1 9 )w, e o b t a i n t h e v a c u u m e x p e c t a t i o n B y c o m b i n i n g( 1 a . 1 8a -':

I r ra

nf

l-' h"a-

r 'r -a - n- l-' n' r

,'

tp\x)i

(ot,i,@ l o) ) : 0 .

(14.20)

In a similar way, we can show that the vacuum expectation of any odd number of current operators must vanish:

r )) : 0 . < o l i , , Q t ) i , " ('x. .ri), , , * , ( x r , +| o These proofs are not completely valid since they lation with unbounded quantities. Physically, at (I4.20) is certainly correct and we shall see later and (14.2I) can be obtained by a suitable limiting

value

Q 4' 2 r )

involve manipuleast, the result thatEqs. (14.20) process.

15. Anticommutators and Commutators in r-Space, The S-Functions In analogy with the treatment of the electromagnetic field, we can work out the commutator and anticommutator of tp(x)and tp(x'). From (13.3) and (13.4), we can expect a simple expression for the a n t i c o m m u t a t o r , w h i c h i s n o w a c - n u m b e r . B y ( 1 2. 2 2 ) a n d ( I 3 . 3 ) , ( 1 3. 4 ) , w e h a v e

{rt"@),rte@')}: : + >, >, l"y)(q) @')e-iqe)'+iq't+),' a@(q')}+ . I {a*o 1ny, ' "P

/l q r\

q--q'G

: +v 4 tI4 * " {' -tu-u r* ' l) @)u g ,@) u$) @) r i,?' - r. { ,' - ,r [ ei qt-)u- '-,t !n o t 1 q1 \/:l q f-3 With the use of (12.12) and (12.13) we can perform the summation over / in (15.I):

Sec. 15

Anticommutators, Commutators, S-Functions

65

{rt'"@),rpp(*')): : :

+ # #

Zq #

(,'{Uy q'*'-' m)p, eid*\ x)- (i y q'-'- m)uneioHt,' *)}-->

(ls.2)

-(iy q'-'-m)pneiqt-'@'-x)) q'r'-m)uneioitt"'-'t I +#l(iy -r) (i y - m)f (q" dq eio@' + mr)u(q). S I "6

The right slde of (15.2) will appear quite frequently in what follows, and it will be convenient to have a special notation for it. Accordingly, we shall write { ' , p " ( * ) , r p p ( x ' ) }-: i S p n ( x -' x ) ,

/ r( a )

wnere S ^r'r't- =+

"qq\"t-

enlt

l ' d q e i e \,"(/ i' y1 .a - r"u' t\d,F^ 6 ( q 2 * m 2 \ e ( q \ .

1*2"

/r( /r

\rJ.r,t

Thls function S(r) is obviously quite closely related to the functions / (r) which we studied earlier. From the remark following e q . ( 7 . 3 7 )a n d f r o m ( 1 5 . 4 ) , ( 7 . 6 ) , w e h a v e

s(,):0+-*)t1,1 In particular, for ro:9,

(rc. ci

S(r) becomes S(r)1,.:o:iyn6(n).

(rs. 6)

Equation (15.6) can be shown directly from the integral represent a t i o n ( I 5 . 4 ) . A l t e r n a t i v e l y , i t f o l l o w s f r o m ( 1 5 . 5 ) b yu s i n g ( 7 . 3 5 ) . Just as the function A(x) can be used to solve the wave equatlon (Z - m')u(x) : 6 (r5. 7) wlth given lnitial conditions , the S-function can be u s e d t o s o l v e the Dirac equation /2\-

with the inltial

\ yz ; + m ) v @ ) :o

(rs. 8)

y(r) : u(u)

05 .9)

condition for xo: T.

From (15.4)or (15.5)it follows that the S-function satisfies the equation

0***)s1,1:0, and thus that

( r s. 1 0 )

66

Sec. 15

G . K d t l 5 n , Q u a n t u mE l e c t r o d y n a m i c s I d z x 'S ( r -

,t(x):-i

x')ynu(n')

/iR

ll\

., _T

is always a solution of the Dirac equation (15.8). Furthermore, according to (15.6) this solution satisfies the initial conditions ( 1 5 . 9 )f o r x o : T , and hence is the solution of the specified probIem. In a similar way, we can also solve the inhomogeneous equation l - . a ' . . . \. . . , . . r r t - \ t \y ar+n)v\x):l\x) with the same initial

{\ rrFJ . rr .z\, ,

as above by the use of

condition

- i t dsx'S(x - x')ynu(n'), (15.13) v@):"1 3@ x')f(x')dx' ro-T ,'o:T

As in Sec. 7 we can go over to the limit I+sumption that the last term tends to the limit )s w r i t e ( 1 5 . 1 3a rP(x) : tProt (x) - / s " ( ' -

oo. With the as,l\o)(x), we can re-

Here the retarded S-function

S" (r) is

rlaf inad

I- tt'l

for

sa(r):

,o) 0rl

for

to (

Io

(tS. 14)

x') | (x')d.x'. hrr

)

/'r( 1(l

0.

Just as for the functions D(r) and A(x) ,we can also introduce an advanced S-function S,a(z)anda function S(*) by f

O

s'(t):is1ry S(r):

for xo>Q, I

f o rx o < o , i

- g e ( x )s ( x ) .

05.16) / ' rR 1 r \

These functions are used in a manner simllar to the corresponding They satisfy the differentlal equations /-functlons.

-d(x). (rs.iB) s*(x\:(, b4 + *\s(*):(, ! + m\ ! + *\I so@): ^'', "', Y O1 ) " \'. dx I \', Ox and have the integral representationsl

s-(,): i*n I a.per'ffi ' 5 6 ( r ) :O + I o Oei p'(i yf-*l l rV l a

(1s. re) t int( p)6( p' + *' ) \,Qs.zo\

l. As in Eq. (7. 24) , the symbol P refers to the principal va1ue.

Sec. 15

Anticommutators, Commutators, S-Functions

S n @ ) : T r a roJ p r ' o (' i y f - m ) p I

f

.f-

,,;Ay,.

67

- i n t ( p ) 6 ( P ' + * ' ) )I . 0 s . z t )

I

na*

The proof of these equations is not trivial, but requires some care because of the differentiation of the function e(x). As an example, we give the calculation for the integral representation (I5.19):

s ( ' ) : - ) z, @ )"( ra+x - * )I t 6 1 + - * \I 2 1 r 12+' - l y n ab(^ ioa.l((rAs . z z ) ' :\ '6 ax " The time derivative of the function e(x) is just 2.6(xo), and so the last term in (15,22) contains the function A(x\on the surface xo- 0, From (7.35) we therefore find /

2

\-

S \( 'r' /) : [ ^ t J - - m l / h \ \t ax

I

/ 1(

,'

??\

F r o m t h i s , ( 1 5 . 1 9 f) o l l o w s b y a s i m p l e c a l c u l a t i o n , u s i n g t h e i n t e g r a l r e p r e s e n t a t i o n o f . Z 1 x ) . T h e o t h e r e q u a t i o n s ( 1 5 . 2 0 ) a n d( I 5 . 2 1 ) can either be obtained in a similar way or can be proved from the relation

(rs.24)

So,o@):S(')T is(r) .

T h e d i f f e r e n t i a l e q u a t i o n s ( 1 5 . 1 8 )f o l l o w m o s t e a s i l y f r o m t h e i n tegral repre sentations . W e n o w r e t u r n t o t h e a n t i c o m m u t a t o r( 1 5 . 3 ) . F r o m ( 1 5 . 6 )w e s e e that if the two times are equal, the anticommutator takes the value

{rp"(x),tpB(x,)},,:,6: en)8"6(n_ r,) . This equation can be written .

in the following

{r*(x), p (x')} "":4:

i 6 (n -

rlR

,q\

form: n') .

(rs.26)

The above equation has a certain formal similarity to the canonical commutation relations, the sole difference being the sign. I n a s i m i l a r w a y , f r o m ( I 2 , 2 2 ) , ( I 3 . 3 ) , a n d ( 1 3 . 4 ), w e h a v e

{rp(x), ,p(*')\ : {rp(r) , ,p(x')} : o .

( r s. 2 7 )

In (15.27) it is not assumed that the two points r and r'have s p a c e- l i k e s e p a r a t i o n . The commutator lrl"@),,pp(*')l

(rs.28)

is not a c-number but we can take the vacuum expectation value of (I5.28), just as we did previously for the anticommutatorof the electromagnetic potentials. After a straightforward calculation we obtain

(oll,y"1x1,,t'p(x')lj o) : sll(2,- r)

(15.2e)

68

C . f a i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec. 15

s ( , ) ( r ) :( , i - * ) 1 , , 1 "r 1 : , " 1 , ,I A p r , r , ( i y f- * ) 6 ( p r f z z r )(.l s . 3 o ) \,a.r.r J l2n)".1 *, since the The function S(l)(t) does not vanish for space-like function 7tr)(x) does not vanish there. Ifwe attempted a discussion of the measurement of the Dirac field in a manner similar to that which we previously gave for the electromagnetic field, we would find that disturbances propagate with velocity greater than that of liqht. This would be in contradiction to the basic postulates of There are no experirelativity and therefore is not admissible. ments , however-- even gedanken experiments--by means of which Consequently the field rp(x) we can study the field g(x) directly. cannot be used to transmit a signal between two observers . By We do i t s e l f , E q . ( 1 5. 2 9 ) i s n o t i n c o n t r a d i c t i o n t o r e l a t i v i t y . have to check that the commutator of two components of the current, which are observables, vanishes for space'Iike separations. For these quantities we have

liu@),i, @')) : - t; ll,p(r),vr,tt(x)1,lrp(x'),v,v (x')l) : : - e2lrl @)y (*) , rp(x') y,tp (x')) : ,rp :e2 (ttt(x')y,v@') rp(*)y,,vQ)-rp (r)yr',t@)rp(r')y,rtt@'))'|

,,,. ',,

The first transformation in 05.31) is verified by noting that the difference between Elry"(*),rye@')land y"(x) yp(x') is a c-number which vanishes in the commutator. By means of the equation , t t " @ ) r l t p @ ' ) -: i S * p ( x-

x ' ) - ' , l t p @ ' ) r t " @,)

(r5.32)

we can rewrite 0S.St) in the following way:

(x'-x)vrrl,@)) lir@),i,@')l:i e'z(r1t(x)Trs(x-x'1y,v@')-',t@')v,s ! Ior.,rt ' (x\) (y"rl tp"(*) (y 1x'1) 1 ez(y,"(x) vo@') rtp "(y,rtt(x')) r(yrrp(x)),) . | B-vB@') B e c a u s eo f ( 1 S . 2 7 ) , t h e l a s t t w o t e r m s i n 0 5 . 3 3 ) c a n c e l , a n d u p o n us lng

: ') v*Q)rt'p@ * l ' p * ( r ) , r p a @ ' )|l s r " { r ' x ) ,

0s.sa)

we obtain the result

(*),r,t(x- x')y;p (x')l-lrp (*'),y,s t*'- 4yrrp li"@),i,@')l='t{1, @)l}l * t {t, ly,s (*- *')y,s (*'-r)l - sply,s (*'- x)v,"s (x-"')ll O,.rO I : * {f, Ol,Tus (* - x')v,rt(x')l- W@'), v,s (x'- x)v,rt'@)l}.J appears in each term in 05.35), the comSinie a factor S("'-r) mutator vanishes for all non-zero, space-Iike separations.

Sec. 16

The Dirac Equation wjth an External Field

69

16. The Dirac Esration *ith a Ti*u-Irdependent, E*ternal E1e..romaqnetic FieId The classical Dirac equation for an erectron in an external electromagnetic field is I

tA

\

r

(16.r)

ly\;;-ieA)*m]v?):o.

If the external field is time-independent, we can expect solutrons of the form ,1"@) : The function

u(n) satisfies

uo(n) e*ia,'" .

the eigenvalue

(16.2)

equation

(ro. s)

,f u(a): ?ou(r), with the Hermitian operator ff, f f : a * ( -.

J

' a I eA * ( t ) ) ' f m Y n l eA o @ ) ai ,

06.4)

and the Hermitian matrices ap , (X.b:

1'Y'at\

:

o / ",). \or o/

T h e r e f o r e w e c a n a s s u m e t h a t t h e s o l utions orthonormal and complete :

0 6 .s ) up(n) of. (16.3) are

! dsxulr"t (u) u!'t (u) : 6nn,, lulot (n) (n'): 6op6(n - n) . "?

( 1 6 .6 ) (167 . )

The eigenvalue po in (16.3) can be either positive or negative. From the charge symmetry of the theory, we see immediately that if fo=E , E) 0 , is an eigenvalue, and with the corr e s p o n d i n g e i g e n f u n c t l o n d e n o t e d b y u , ( n , E )' o, f . t, h t re n t h e f u n c t i o n u'"(n,E):C"pk@,-E) is an eigenfun.iior,l withthe eigen_ value ps:-8. Here the matrix C is the one used in Sec. 14. In the quantized theory we write ttt"(x):

Z {"!\ (n) e-iE","at")* u'J"t(n) eiE"'"6*@)y ,

(16. B)

E"> o

and, as we did previously, we require {rl"k), tp,(x')},,:,t,: 0r) p (n - n,), "6

( 1 6e. )

1. (Translator's note) In general, the functionu, is an eigenfunction (with negative energy) of a Hamiltonian equal to that of Eq. (I6.4), except that the sign of the four-vector potential must be reversed. consideration of the probrem of an electron :.n a coulomb field makes this clear. only if there are additional svmmetries can the change of sign of A be ignored.

Sec' 16

G . K i i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

70

(16.10)

{rl"@),tpp(x')}"":,t:{rl"@),pp(x')},":q:o. F r o m E q s . ( 1 6 . 6 ) t h r o u g h ( 1 6 . 1 0 )i t f o l l o w s d i r e c t l v t h a t Ie*

(n) , e@')j :

: {a@, a\"')tt

{b*

(n) b\,t:l} : ,

:... {a\"), b(tu')}

(r6. tI)

6tutu, I :

(16.12)

0.

Instead of (16.9), we obtain for two arbitrary points x and - - i S p o Q ' ,x ) , { r p " ( * )r, t p \ ' ) }

06.13)

(r') eiEn\';-'dl'06'14) sp*(x',t)=i Z ln!) @)up (u') ,-iEo@i-rtaa'{") (n) u'u@) E"> o

Clearly

the

function

S(x, x,)

l, (* with the initial

satisfies

the

equation

differential

- i ed @))n *) t (x'x'): s'

(rb.rci

condition

(16' 16) s (x, x'): i Ynd(n - n') for xo: xL ' ( 16 ' 1 5 ) In fact, the function S(x, x') can be taken as defined by and (16.I6). This S-function differs from the corresponding funcdetion for the free field in several respects. In particular, it just of a function is not and quantities and *' r pends upon two xx' difference the In analogy to (IS.5), let us try to express S(r, r') as

- i eA(r)l s (x,x') ' x'). . , r *)) A (x, , : {rl , ( t a"q=

(16.17)

The function A(x, r') then satisfies the differential equation (r a :. t, \'2 jlaT-teA\n))

o

e - ,r)I /(x,x'\:g, 26rurF,\.,

(16.Ig)

with the initial conditions for xo:

Ylo.

( r 6. r e) (16.20)

Actually, these transformations do not really help too much, since the function A(x, x') still contains two spin indices, owing to (16'i8) is the last term in 0O.fg). In general the solution of Eq' ( 1 6 .15). As E q . o r i g i n a l o f t h e s o l u t i o n almost as involved as the constant of a case the first consider us J.et example, a simple *ugr,"ti" fi"ld H . In this problem the transformation (16'I7)does turn out to be useful . With no loss of generalitY, we can assume that H is in the direction of the z-axis and that eH> 0' Then we may take the vector potential ,4(r) as A(n) : (0, H r, 0) .

(ro. zU

Sec. 16

The Dirac Equation with an External Field This brings the differential equation (l6.Ig) into the form (

1o

-

22

,;T

- 2 i e H r ; -a

7I

-) e z H z x- 2 m zI e o r r U l Z @ x, , )- o , ( 1 6 . 2 2 )

First, let us considerthe simpler equation tAzAl L A T -s- 2 i x 6 , *'- mzl cl x,x' ): o,

and the corresponding IA1

( 16. 23)

eigenvalue problem

L-t +zix,

* xz*m,lu@):nzu@).

(16.24)

The eigenfunctions for (16,24) are obviously u n t , ( u ) : J - d t U t + a " I 1 g - , ( x- l . ) ,

(16.2s)

Eznn:m2+k2+2n+1.

06.26)

Here the function H,(x) in (16.2s)is the normalized eigenfunctlon for a harmonic oscillator with frequency l. For this function, we shall use the integrat representation

:w -# #_fi?,,,-#,,. H*(x) "+ Thus the function G(x, x,) in (16.23)

\ro.zt)

can be wrltten

o.!Gk.x,\=v IJP-! .,u - u-t')th(z-z')ly1,('- l)H,'(x'-:;sin lEnr'Q'o-*o)l ' en1z ,\-')-----E;*.(16.28) ,?-uJ

Both the integration over r and the summation over n canbe done explicltly. It is then possible to transform the integration on & so that 06.28) becomesl G " \ ("x' .. x" t, -\ -

t(x'- z) ] i - , +-4' q ' , l : _ _r #_(_2_" -_*_t *"J : l t 1 o o r c n , " . - | t ' - r t t , ' + * fJf 6; tnr t ,(16.29)

). : (x' - n), + (y' - y), * (r, - z), _ (xL_ xo)r, o:il@,-x)z+(y,_y)rl.

(16.30) (ro. Jti

F r o m t h i s s o l u t i , o no f E q . ( 1 6 . 2 3 ) , w e c a n immediately find the corresponding solutionof Eq. IA.ZZ1. We introduce the notation M : eorzH, tro.JzJ so that M2: szp1z. JJ,f uo.

l . I . c 6 h 6 n i a u ,p h y s i e a ,H a a g l 6 , g 2 2 0 9 5 0 ) . S e ea l s o J . G 6 h 6 n r a u and M. Demeur, physica, Haag ILTI(1951); M" Demeur, physica V,933 (lesl),

Mem.Acad. Roy.Bets.l[rvo. s (irjssl; t. K.;;;ulFronr. T h e o rP. h v s .6 , 3 0 9( l g s r )i;. s d r r . * s " r , t n u . . R e v .8 2 , 6 6 4( r 9 5 1 ) .

Sec' 16

G. Kaitl5n, Quantum Electrodynamics

72 We then find

:!L - 1t * "']*ao rlrl-" ^* t;a)' r- 1v'v1

A (x,x,) :'#

i6

.,1 2d

2e

eH

t-) =*;4 ot.,li; "ftl-"^+ v,*- # ""*l

^^. ,H \

" ;-|:::7'

,i6.-|2d

^^.,H\

06.34)

=-eH\ZT-""'-ti) 2d. -slneH

eH 2d.

llere we have defined (D(x,x'):s

-J!L

2 6" , ' - ! ) ( x ' +

r)

(16.3s)

introduce a retarded' an advanced' Just as we did inSec. 7, we can These are obtained directly from /-function. and an "barred " or (16.34) by multiplication bv -; t{ + e(x x')), lrlt e(x x')) , -$e(x- x') we obtaln' for exif *e tet H tend lo zero in these functlons ample for Z1x, x') ' _ ;,,_ " ; + L ) . , r (16.36)

Z ( x , x ' 1 - i-F

J

dae2t

7' since from This is the same result which we obtained in Sec' +o

PFia

(16.37)

: * et'rP'+*') lY,i Io*

it follows that

-# a'P *iaf ,"^' : o' ao I + ;d,I ft r, u;e I ffi "'V I ou. u4)'l ^') : ;" I #'t- s *" : -!- [ an'j'-l-"

In this expression, we have used the integral

'i"it'fA:l)]' ' "

:l_t\r,(cos(lzol ,'*0" f',)+'r'6 ! d.p .

I f:*

'

'l

iq2

(tutfi)- t #[sin (lzol eil)): ffi dpo(cos | _t |

^.

.

u

L, l(to'sg) l'

]

(x, x,) bv In a similar way we can define a function 7{D

: (o|[v,(*),pp(x'))to> {rl*

- ; ee @')]- *\ t(o'J{*"') . uo

Obviously this function can be written

(16'40)

Sec. 16

The Dirac Equation with an External Field

73

@

ntt)@,x,) :

-1 s.Lt(t-t)).h(z-2,)t L f f Y:! n l)n\2

n:o"

"

coslEhL\.r'ox H*(x - t) H,(x' - l) -&fr

and the subsequent calculation that for A(x, x') . The result is

proceeds along lines

I

,I

xo\)

(r6.41)

similar

to

xtr)@, x,): @@, x,)Ji*o*+ rtl--^*#), =+ , 8nz lql

x fcos *-#'r#] At this point it is possible

e

loeH l2q 2 \aH

_rH\ ---2dJ

( 1 6. 4 2 )

2" . -"' "H-' eH 2a

to define a function

Ao(x, x') : - 2i tr(x, x')4 /{1)(x, x,) , and so forth. factor @(x,x') in (16.34)and (16.42)which was definedby . T,h". (16.35) can also be written in the followingway: (r6.43) T h e l i n e i n t e g r a l i n ( 1 6 . 4 3 )i s n o t i n d e p e n d e n t o f t h e p a t h , a t r e a s t not if the electromagnetic field is not identically zero, but is defined as the integral along the straight line between x and, x,. In our case, with a constant fierd, the other factor in (16.34) and (16.42) is a function only of x-x, . In general, this is not the case: it is possible to show that after splitting off the factor @(x, x') the remainder is independent of the gauge which is used. If the external fierd is not constant, but depends upon the spatial coordinates, the calculation is very much more complicated than that of the previous example. As yet, no explicit examples for the singular functions have been given in a spatiaily varying external field. Recently, however, wichmann and Kroill have evaluated the expression

p ( r ): ( o l l , t @ ) , y n v @ ) l l o- ) [ d . x(,o l l , p ( * , ) , y o , p @o, ;) .] ld ( r ) ( 1 6 . 4 4 ) for a Coulomb field. This expression is quite important for the so-called "vacuum polarization,, (c.f . Sec. 29) . Their result is €1

Q(p) 2vt'_:e:s _l a-,trr 1laPltl+u")l' 1 . E . H . W i c h m a n na n d N . M . K r o l l, p h y s . R e v . 9 6 , 2 3 2 ( 1 9 5 4 ) ; 1 0 1 , 8 4 3( 1 9 s 6 ) .

AA

G. Kiill5n, Quantum Electrodynamics

, the charge of the external Here Q:Ze breviations have the following definitions:

14 yt

and the other ab-

field,

( 1 + u z \ ( 1 - z ( 1 - 1 t- \ )

,

t

, -. (1-z)(t+uz)

c:7rr=LIog

Sec. 16

(16. aGa) (r6.46b)

1-20-o-'

V,r,

(16.46c)

u:--3:,

2yr+t2

s\R):VR"-7" f

-

Zez 4n'

,

0 6. 4 6 d ) (16. 46e)

summation over The summatlon over A in (16.45) corresponds to a integratlons are The eigenfunctions of varlous angular momenta' of the representations integral the of obiained by transformations . '--llrutio" radial eigenfunctions (16.45) represents an analvtic functlon of / in the the use of ,"gionlZl< I . A powei series in 7, which results from ordirruryperturbationtheoryintheexternalfj'eldproblem'therefore gives a series which is convergent for lyl< I '

CHAPTER THE

DIRAC

FIELD

IN

FlELD

AND

INTERACTION.

THE

IV ELECTROMAGNETIC

PERTURBATION

THEORY

1 7 . L a g r a n g i a na n d E q u a t i o n s o f M o t i o n we are now ready to attack our major problem: the interaction of a quantized electron field and a quantized erectromagnetic field. As equations of motion, we will continue to use the Dirac equation (16.1), except that Ar(r) is now the operator of the quantized electromagnetic field, and not an externat field. In c e r t a i n p r o b l e m si t i s n e c e s s a r y t o u s e n o t o n i y t h e q u a n t i z e d field, but also to add an external field. Then Ar(z) becomes the sum of both these terms. In order not to complicate the formalism too much, we shall initiatty confine ourselves to the quantized field only. As a second system of operatorequations, we wilt use Eq. (11.2) with the current given by the Djrac operators of Eq. (13.I3j. We obtain all of these equations if we choose the Lagrangian to be t h e f o l l o w i n g e h a r g e - s y m m e t r i ce x p r e s s l o n : 9:9*l9e*

9w ,

(17.I)

e, : - i[o ot,(,* * *) v{4]- - !Y, + mrp(x), Qz.z) e 1x11, +t ee: '-

aAt,@)\ - !(!4!-L - aAr'@\(a4,@- -6i-l _ - 1 j4!!L !4@\ n- ^, 4 \ 'axp \t/.rl 0ro 7ru / \

T--dr"--;t

s* : + A*@)lrp(x),yoy@)l .

g7.4)

terms g* and ga are formally identicat with (14.12) and Tl"-iyo (5.10) and the term g* can be written as

9w:

A,(x)iu@)-

(17. s)

In the usual way, we obtain the desired equations of motion from this Lagrangian:

(, *

* * ) v @ ) : i e y A ( x )y tQ c ) ,

D Ar(x) : -

- ir@). + l',p(x),y,rt @D-

(r7.6) (17 7\

The canonical quantization is considerably simplified because the "new" term -9* does not contain any time derivatives of the

G . K 6 1 1 6 n ,Q u a n t u m E l e c t r o d y n a m i c s

76

Sec. 17

same The canonical momenta are therefore the field operators. can we and previously, functions of the dynamical variables as recommutation canonical immediately write down the equal time lations:

:lW, A,(x')1,":"i, iA,(x),

34#),":",":0, (17'B)

laA l v \ , A , ( 4 )) z o : x o , : - i 6 r , 6 ( r - n ' ) OxO

,

(r7. e)

I

-tp(x')}, :,'" : o, {ry(r), rp(*' )}, ":, r, {rp(x), " - n') , l e 6 ( n {rp(*),',1@')}"0:xL:

: o, y (r')f,,:";: lY#?,'P@)),,:,' SA u@), yA tp(x')f ,":,":lY#!,,t (r)f,,:,r:o' r(x),

(17. Io) (17. II) (r7.r2) ( 1 7. 1 3 )

the commutators for For free fields we were able to construct is now imposarbitrary times from those for equal times' This equations of simple satisfy not sible, since the commutators do are not even they separations' time-Iike for motion. In particular, expressions for give explicit to able not are we and c-numbers them. InSec.Ilwewereabletosolvetheequationofmotion(II.2) (ll'5)' In the present by means of a retarded singular function in motion by this case, we are not able to solve the equation of to an a transformation make to method; however, we can use it integralequation.Insteadofthedifferentialequations(I7.6)and (17.7) we can write (J/ . .t{, ,p(x) -- V(ot(x) - / So(" - .x')i e Y A (x') Y (x') d'x' , Ar(x) :

Aft (x) *

I

d' ' ) f o ^O- *')# l rp( *' ) ,yur t' @'

(I/.r-c/

I\\o)(x) and Here it is necessary to introduce two new operators soluobviously are Af)@) lnto the equations of motion' These tions of the free-field equations

(r!+*\rtot1v1:0, \'

0x

(.r/ . ro,

/'

f l , 4 1 9 ) ( r:) 0 .

(r7.r7)

(17'15)contain the reBecause the integral equations (17 '14) and the initial values formally are '4!o)(r) and ,ura"O functions ,ttt@) prinxo--.f'.In f o r a n d A r ( x ) o p e r a t o r s 9(r) of the Heisenberg to us allowing as motion ciple, we can regard these equations of c a l c u l a t e t h e f i e l d o p e r a t o r s a s f u n c t i o n s o f t h e i r i n i t i a l v a l uproblem es. to the At first sight this is quite a different approach the eigenvalues from that studied previously (where we looked for

Sec. 17

Lagrangian and Equations of Motion

77

of a Hamiltonian operator) . Here the analogous problem would be to find such a representation of the operators tp(x) and ,4,(r) that the Hamiltonian operatoris diagonal: /t

H (A ,',t) : H$) (A , ,p) * H\') (A , ,p)

I a\ "

0 7. r e )

H6(A,,i : H9 (A)* HIP@),

Hp@): ) I o'.14#Y#!+ !!#Y#?), 07.zo) : t I o".fot-t,(,,, HP@) , fa + *)y'@))

(17.2r)

: - -oi n'*o,(i l,t@),yuv@)l Htl)(A,y,) . I

(r7.22)

statement of the problem of quantum elecThis is the "classical" trodynamics. If we were to attempt such a calculation in detail , we would soon find difficulties. In fact, it would turn out that the theory formulated here is not well-defined mathematically and contains several infinite quantities. Great progress has been made in recent years with the realization that it is sufficient to regard these infinite quantities as a physically unobservable "reof the constants m and e. As we shall see later, normalization" it is quite important to have the formal relativistic covariance apparent at all stages, for otherwise the infinite quantities cannot be uniquely identified. The diagonalizatlon of the Hamiltonian 07.18) is formally not a covariant problem and it wouid be quite to arrive at a unique interpretation of the infinite parts difficult of the theory in this way. It is therefore preferable to attack the problem only by the use of the covariant equations of motion (I7.14) and (17.15), rather than by means of the Hamiltonian directly. A previous example has shown that it is possible to diagonalize the complete Hamiltonian (11.8) by solving the equations of motion for the field operators (11.5) with the use of "adiabatic switching". given of adiabatic switching Although the proof of the validity above does not hold here, it is straightforward to attempt a similar method. The justification of the method will presumably follow when we know the results of the calculation. We therefore write the equations of motion (I7.I4) and (17.15) in the form ,l @, u) :

Ar(x, a) :

yrot(x) - i t I So (ry-

el:r(*)+ *

[

x') e-al''ol y A(x' , a) \t (x' , u) dx' ,

(17 .23)

o o(, - x') e-ot'it lrp(r', o),T, v (x',a)l d.x', (r7.24)

and regard Tp and A: ^" functions ically interesting quantities are

of

g\ot, A(ol, and or .The phys-

,t@):hrytp(x,a) , A,(r) ::tylAt"@, d).

(17

' tr\

( r 7. 2 6 )

78

G . K i i 1 1 5 n ,Q u a n t u m E l e c t r o d y n a m i c s

Sec. t7

W e e x p e c t t h a t t h e t w o ] i m i t i n g v a l u e s ( L 7 . 2 5 )a n d ( 1 7 . 2 6 )e x i s t and diagonalize the Hamiltonian operator 07.18). At present, we cannot give a general proof of these conjectures but must test explicitly whether or not they hold when we find a particular solution. The operators rp(o)(r)and Af)@) are therefore to be chosen so that they diagonalize the Hamiltonian operator lor xo-->- oo; i.e. , H(A,rt)|""--€ : Il',(o)(1(0), ?(0)): Hf)(1o)) + HJo)(?(0))

(17.27)

is to be diagonal. Finding two such operators is just the problem which we have solved in Chap. II and IlI , so we can use those solutions here. From now on wb regard the operators gtot(r) and A , P@ ) a s k n o w n . T h e y s a t i s f y

(r), rrot1x,)}: - i s (*' - r) , {rp(ot (0 | [9(o) (r' - %) , (x), tptot (r')] lo) : S(1) IAIP@),Alot(x')): - i 6p,D(*'- ,) , (o 11/lot(x), trtot(r')) lo) : 6t,, Do)(*' - *) .

( r 7. 2 8 ) ( 1 7. 2 e ) (17.30) (r / . J.r/

In (17.31) it is implied either that we are evaluating only gaugeinvariant expressions or that we are using the method of the indefinite metric. Slnce it will later be useful to discuss quantities which are not formally gauge-invariant, we shall require from now on that the longitudinal and scalar photons be treated by means of the indefinite metric. The metric operator 4 is prescribed as commuting with ?(o)and 'y(o)'

(*), ,i : o. w\ot(r), "tl: [?(o) For the complete operator

(r7.32)

Ar(x) , we have

l A u @ ) , r t l{:A n @ ) , n \o: ,

/r7 ??\

exactly as for the incoming fie1d. For the complete Dirac field we cannot prescribe the relations correspondingto (I7.32), because the coupled equations of motion do not leave these quantities independent of the degrees of freedom of the incoming electromagnetic field. In order that the quantities present in the theory have the correct reality properties, we must define ,p(x) bV

,t @) : q tp*(x) q y+ . In the theory with coupled fields, tha

( r 7. 3 4 )

the continuity equation for

cr rrrant

a i p @ )- i t 2* . ".p."-l|

o t^t.\ A. ,rr^,'TPV(X)]:0,

/ 1z

? c'l

Sec. 18 Perturbation Calculations in the Heisenberg picture implies for Ar(x) z

aA,:jL: aAwL Atu

(17.36)

0r,

Both (17.35) and 07.36) are operator identities. condition for the incoming field holds in the form

79

If the Lorenrz

+t,p):o,

( r 7. 3 7 )

then it also holds for the complete field. 18. Perturbation Calculations in the Heisenberg Picture For the coupled fields, we cannot exhibit an exact solution ro the equations of motion. Rather, we must attempt to find useful methods of approximation. Because of the small value of the ,e2

1

N , it is straightforward to consider the right * * ( 1 7 . 1 4 ) ( sides of a n d 1 7 . 1 5 )o r ( 1 7 . 2 3 )a n d ( 1 7 . 2 4 ) t o b e " s m a l l , , a n d to attempt a solution in the formof a power seriesin e.We therefore write charge ,

V @ ) : V ( d ( x ) + e t p t t t ( x! ) e z y z \ ( x1) . . .

,

A u @:) A ' f )( r )+ eA t i \@ )+ t ' A e ,@ ) )* " ' ,

08.1)

0s.2)

and, upon introducing these series into the equations of motion, we obtain recursion relations for the different orders of aoproximation:

- + [ to@* x,)y, i toy, \(x,), (x,)\ (x): (jl B . 3 ) y ,(n+t) " / ' y p(,-^) \ " / ) " d.x, ", , 1 \') 2I *Zo

- ' - 't1 i y r t .\t-(- xt ' '.)p, y u r r , - - t ( x ' ) ) d x,' . - to A Y + t\ )' - (t x ) :2+ f D < t(' -x * LOLy (19.4) J '^p

In (18.3)we have symmetrized the right side. Although this is not necessary, it wiII turn out to be convenient. The symmetrization i s c e r t a i n l y a l l o w e d , b e c a u s e b y ( I 7 . 1 2 ) a n d ( 1 7 . 1 3 )t h e o p e r a t o r s Ar(x) and y(x) lor rp(r)] commute for equal times. In the lowest order, we have -i !)(1)(x): I S p ( x - x ' ) y t r Q ) ( x ' ) r t o ) Q t ' ) dt x ' t l t t r t ( x ) :- i ! t t o t ( x ' ) y A t o ) @ 'S) a @ ' - x ) d . x , , Afi(*): D o ( x - t c ' )l l t ) t @ o ) ) , y u r p t o r ( x ' ) l d ,x , + I @"), y,ytot@")l} x ,tz) (x)= 1// So @ ,') T,{!tto)(x'), ftp{ot t x")y*x x Dp(x'- x") d.x'il*" I I S ^ ( r - x ' ) y ,' ,"'S * ( ; (x") (x'), Alo) 6*' tr*" xyot @")} ,' {Atot

08.5) 08.6) (r8.7) I l(18'B) )

G . K i i l l 5 n , Q u a n t u mE l e c t r o d y n a m i c s

80

Sec. lB

, l t ] ) ( x )- + I I D * ( * - x ' ) ( [ r p ( o ) $ ' ) , y u S * ( t ' -x " ) y , p ( o ) ( * " ) l - l ] Or.nl + [ ? ( o ) @ " ) y , S n ( * -" * ' ) , y u , , p t o J ( * 'A) )t )$ I @ " ) d x ' d x " . ) In principle this process can be continued to arbitrary order withThus we can obtain the general equations out any difficulties. for the a-th order approximation to the field operators. The physically observable quantities, for example, the components of the crrrrenf- ca n a I so be constructed in a similar fashion. Because v g r r v r r u ,

we shall later study the current proximations here:

operator,

we give the lowest ap-

(*)), ilP@)-- i t *' (*),v,',p(d

0 B r. o )

(r')] + il]t(") : t I dx' (ptot@),yuSo@- *') y"rtt(o) I - *),ynV(o) -l- [9(0) (r)])A!,0) (*,),J @,)y, Sn@,

(r8.1r)

(r")]}] x (r"),y,rltto) (x'),lEtd dx" lrto)(x),yrSp(x**')y,{y,(o) tfXO:i[ tP\ ' 6 J J[ax' xDp(x'-x")-

O, dx" lt/0)@),y u Sp(x - x') y,, Sp(x' - x") y,og$\(*")l x a r[r[ x {Atl\ (*') ' Al! (*")} -:; (x') y Ato)(x') Sn(*'- x), y, Sp(x - x") x o ,'0,', SEto\ [[ (x")l* XY Alo)(x") tYOt

-,

08.r2)

(")f x " 11,rrd1x')\t, fi (x'- x)yu,V(o) (x"),yst$)(x +i [[a*'ar" [{[v,0, -

-

+ II o'

x") xDo(x' (x")v,,sn(x" x')1,, So(x' x) , yrtp\o)(x)) x dx"lrrrot

x {.4t?(x'),Af)(x")}.

A s i s a l r e a d y a p p a r e n t f r o m ( l B . l 2 ) , t h e h i g h e r a p p r o x i m a t i o n st o the operators rapidly become very complicated. We shall refrain from giving any further explicit examples,l and instead turn to a detailed discussion of the properties of the lowest-order approximations . InEq. (18.6)an advanced singular function appears for the first time. This does not mean that the operators f (r) become tptot(x) as r0->f oo , but rather it is only a consequence of the "reversed" . The same order of writing the difference of coordinates x'-x holds for the advancedfunctions appearing in (18.9), 08.II), and 08.i2). As can be seen from (18.5)through (I8.12), the solutions found in this way contain products of operators ,rptor(x) , ,lt!(x) andsingular functions. This means, for example, that the complete operator rp(z) has a non-zero matrjx element between the vacuum and a state wlth an incoming electron and incoming photon. Even the which allows order rp(r)(z)contains the product Af\(x').rto)(x'), @ysik

2 , I 8 z ( I g s o ) ;2 , 3 7 r ( 1 9 s 0 ) .

Sec. 18 Perturbation Calculations in the Heisenberg Picture

81

this transition: ( 0 1 ? ( r ) lq , k ) = - i e [ S * ( x - x ' ) v , < o l A t o \ ( x ' ) l k ) ( 0 1 ? ( 0 ) ( t ' ) l q ) d x ' - 1 " ' . 0 8 . 1 3 ) Here I q,h) is the state mentioned above; lq) is a state with oniy the electron and l,t) a state with only the photon. The same term also gives a contribution to the matrix element (hlrp(x)lq) z ( k l r p ( x ) l q ) : - i e I S o @ x ' ) y , ( k l A l o t ( x 'l )0 ) ( 0 l V ( 0 ) ( r ' ) l q ) d x ' * . ' . . 1 ( 1 8 . i 4 ) T h e r i g h t s i d e s o f ( 1 8 . 1 3 )a n d ( I 8 . 1 4 ) c o n t a i n i n t e g r a l s w h i c h a r e not actually convergent. We recall that these integrals are to be t a k e n a s t i m i t i n g v a l u e s , a c c o r d i n g t o ( 1 7 . 2 3 )a n d ( I 7 . 2 4 ) , s o t h a t f o r ( 1 8 . 1 3 ,) f o r e x a m p l e,

xl Uyf - m)y et^t

(olylxyiq, k)=-t enm ax, I I rh';H /

eih X --,lfzvu

u- (: q \ : ^ " " ,-' - , " : , , ' ' " '

I

, i Q t ' e - d l x ' +0 1" ' :

l/lR l5) l| \ r v ' r w /

Vv

, e i ( q + h )r r /,\ : - 1' u(q) e (q h)+,nzliy(q* k)- *ly e\^) + fO;

+

I

we have used the followingexpressions: in (18.15) '(^) (o1,llot (*)lP) : ;i? eih', V

( I 8. l 6 a )

( 0 l v ( o ) (, ro)r :

( r 8. l 6 b )

V2

ct)

I c . f . ( s . 2 8 ) ]a n a

){r'o'

,

is not l c . f . ( I 2 . 2 2 ) l , M o r e o v e r , t h e f a c t t h a t ( q+ A ) ' + m 2 : 2 q k zero for all photons (withro$0) has been used. In the theory of free fields, we have seen that the field operators create states with only one particle if they operate on the vacuum. From the calculation above it follows that this is no For example, if we Ionger true for the theory with coupled fields. we obtain, among other states, operate with I on the vacuum, states with two incoming particles (one electron and one photon), according to (18.15). On the other hand, we also obtain the onenarficle sf^tes in this manner. The matrix elementS fOr these ys!

Lrvrv

transitions are different from the corresponding matrjx elements for the free field.

From (lB. B) we have

: (olp(o)(,) (ot,t@)tq) |u>+ + II so('- x')y,X (x")l\ | q) D^(*' - x") d,x'dx"x (0 I {?(o(r'), lrt'@ @"), y,V(o)

- t,e2

f f ^ So(' - x')Y',So(x'- x") Y', x JJ (x")}l q) dx'd.x"{ . . . . (r") {,t!1,(*'), Alo\ x (01?(0)

( I 8. 1 7 )

82

G. fiill5n,

Sec. I8

Quantum Electrodynamics

The expressions appearing in 08.17) can be simplified ably. Beginning with the second term, we have

consider-

( o 1rptor (x'), Ali,Q")} | q) : 1r"; {A!o,t

l

I

frR 1R)

= -I ( o l v ( o ) ( x " ) l z > < z l { A t o ) ( x ' ) , A ltqo)).( ix "")"}" " ' lz>

The second factor of (18.I8) can only give transitions in which the number of photons is changedl Either two photons are produced or the same photon is first produced and then destroyed. The sum over intermediate states in (IB. IB) therefore contains states of either an electron, or an electron and two photons. .For the last class, however, the first factor is zero, and we have

(0 I ?(o)(r") {Al,o,) (*'), Alot(x")} lq) : : (0 I ?(0) (r'') | q) (ql {Al,o,) (*'), A[ot(x")] | q',: : (sl gtto) (x") I q) (o I {Al,| (r'), Alot(x")} loS : : 6,,,"Dn (x,_ *,,) (Ol rpto> 1rS. 1x,,1 In a similar way,

( r 8r.e )

we obtain

(o l {?(o) (x"),y,,,p(o\ (*")l\ | q) : (r'), 7r{o) : - 2 5

I

oa.zor

In working out (18.20), we have taken (ql I'p(o)(x"), rtt@) (*')) | q):(o

I ]tpa,(""), rto) (x')) | Q): Strr{x' r

"), (I8.2oa)

atthough the state 1q; is already occupied and should not occurin the sum because of the exclusion principle. However, the contribution of this state is proportional to V-r and it can therefore be neglected if. V becomes infinite. fv lv n l l a c f i vnL cr r r v rrv

nrrr

raqrrlfc

rrra

( o l v ( " ) l s ): ( o l r t o r ( xI)q >- ' :

harra

. . ' .[t l[

t-t.- x')y,x

]

. , .[ . S ( r\ .)_1 , . y. r' -' , \D r ( r ' - .x" " )I *L *SK n\ . (. x ' - -x- " )| _D \ t t( l - x " )t Jl x }

x 7 " ( O rl < o l ( x " ) l q ) d * ' d*x.". . .

(18.2I)

I

J

The integrals in (IB.2l) can be done and we shall work them out in Sec. 3I. For the moment, Iet us avoid this and consider (IB.2l) only as an example showing how matrix elements for simple transitions can be developed from complicated expressions like (I8.8). It remainsto establish that the series obtained here satisfy the postulates (17.25), (I7 .26), and that they diagonalize the Hamiltonian. Regarding the diagonalization of the Hamiltonian, it is an immediate result that the conservation 1aw (4.22) for the energymomentum tensor is no longer valid, because the Lagrangian con-

Sec. 18

Perturbation Calculations

in the Heisenberg Picture

83

xu explicitly. They are contained in the tains the coordinates charge. By the equations of motion or by means of considerations similar to those of (4.26), it follows that arpr _ ag a,r-4'

(r8.22)

The right side of (I8.22) is to be understood rather than (4.22), as the derivative with respect to the coordinates with the field operators held fixed. In the present case of a variable charge, we have

orr, _ o9 oe -oe - . ox, oap We consider the time derivative we have

gL:olo

[ a ' r ! oxo }":-;

J

(r8.23)

of the Hamiltonian.

[ 4 s , aoxp? n: -

J

0 8 .2 3 )

From

! . !oxo ' t a'*a oe

08.24)

J

The change in the Hamiltonian during a time interval (ro, ri) can therefore be calculated from the implicit equation u

: H (A(x'),,p (*'))- [ a,"L{ffi,tl)L H (A1x),,t @)) x'

\ff

.
Up to now, the detailed nature of the dependence of the Lagrangian upon , has not been used, and Eq. (18.25) is completely general. F o r o u r L a g r a n g i a n ( 1 7 . 1 ) t h r o u g h ( L 7. 4 ) w e o b t a i n

(r8.26) and therefore we have r i.f

(x"\

0e ,, ^ ^-, H (A(x),y(x))=H (A(x'),tp(x'\)- V J dx" W (*" ),yrrlt(x")l Ar (x")- a 4 - ' \ L a ' L r l r'

In Eq. (18.27) we can aliow *[ to go to the time derivative of the charge explicitly,

oodnd, if we introduce we find

- *i: j or r-al'61 x (*),,p(ur(.)) H(A(x),,,t@)):H(0)(1(0)

I ur.r0 y,,t@')lAu(4 . I * lE@l)7

In (18.28) it must be pointed out that the operator H(d(A(d, ?(0))is time-independent. The last term in (18.2S) contains a factor c(, but wlthout further calculation it cannot be concluded from this that this term vanishes for a->0. If the matrix element under conoo, the integral has orsideration tends to a finite limit for xo)der of magnitude

84

G. Kiill5n, Quantum Electrodynamics : eu'o: Q (1), a i e""'"d.x'o

/la

?q\

-@

and therefore does not vanish for a->0. With theuse of our solution, we can verify that each term in the series development of the integral has a time dependence of the form to f-

tro(nra*dfo)

aIxn"'-re@'d+ipo)''odx'^:--- l' ." " J lnra*tPol", -@ lHoro,h

iq

fhc

diffprpncp

of

thc

| J

2",-rdd,z.

08.30)

-@

cv r iynv rcr nv r r a l r r c s

nf

iLhr rcv

o v Fnv tr r f a t o r

H@ (Alo\, ?(0)) for the two states being considered and nr and nz are positive integers. For the terms in which ps is different from zero, i-ha inteoral (18-30) is therefore of order a. For the states for there are always which B'io) in (18.28) has the same eigenvalue, symmetry operators (parity, momentum, etc.) which commute with all three terms of (18.28) and which have different eigenvalues for These quantities are constants of the states under consideration. the motion, even during the switching, and matrix elements of the last term in (I8.28) between two such states must vanish. Therefore, in the limit or+ 0 the difference between the two operators H(A,rp) and l7(0)(Ao), p\o)) has been expressed in a power series j.n a in which each term has diagonal form. This showsthat our solution of the equations of motion (I7.23) and (17.24) actually "diagonalizes" the Hamiltonian and therefore that ttre search for these statement of the probsolutions is equivalent to the "classical" lem of quantum electrodynamics. It is more complicated to discuss the existence of the field operators for a->6. For example, it is clear that (18.30) does not since this expression is of order of exist in the limit if nrsI, magnitude d.-trLrr if y'o vanishes . A more careful investigation, which we avoid here, shows that the appearance of such terms is connected with the change of the eigenvalue of the energy, and that they are to be understood as series expansions of expressions of the tvpe ,-i6E

lt"-o(u-t)l

08.31)

If we had fixed the initial conditions for a finite time 7, we would have had an expression xo-O (I), rather than ro-O(a-l) . Such an infinite phase in the field operators is of no physical significance and can obviously be avoided by a very careful treatment of the boundary conditions. We do not pursue the matter further because we shall later remove the change in energy from our equations (c.f . Chap. VII) . Apart from these infinite phases, a is present in our solutions only in integrals iike +@

2rl f u _ > 2 n 6 ( 'b- . \ . t rfr,^ore- n a l x o l'-"; p" :" r " : -., p. "t J "ur+

(1g.32)

Sec. 19

S -Matrix

The

8s

or sometimes in inteqrals of the form

I

d x ' o e - * " ) , , - ,-i i,p o @ o - * a :

_@

]-_

: - i p- - | + n a e o ) . Po''""\rtrt.

1xd.+ipo

t( 1 1c9 . 3 3 )

I n t h e l i m i t a + 0 t h e r i g h t s i d e s o f ( I 8 . 3 2 ) a n d ( 1 8 . 3 3 )h a v e s i n g u l a r i t i e s n o s t r vor n r vn Uc r r tLh r ra a rnl d i O n S ri rnr l4y .o .. REpUcq a u ser Ll tq a - l -u ft rU u LnI C U t tl S D u Dr yr q o iul r l e S e expressions must Iater be integrated over p(actually, only spacetime averages of the operators have a physicai meaning), the existence of these expressions is certain. We shall content ourselves with these admittedly incomplete remarks about the adiabatic hypothesis. Actually, we ought to discuss whether or not the series used here are convergent and define a solution, and, if so, whether one is justified in operating termwise with them. We shall later return to these questions and shall even try to carry through the discussion without power series. For the present, we shall not go into this further, but rather study the applications of the theory more closely. 19.

The S-Matrix The field operators which we have considered up to now are nor especially suited to practical applications of the theory. In principle, the electromagnetic field strengths are measurable, as we have indicated in Sec. 10, but this is more a question of the method of interpretation of the theory than of the experiments which are actually done. In actual measurements, the determrnation of interaction cross sections is of great importance. Typically, in these experiments a number of particles of known momenta and energies meet each other. During a rather short time they interact with each other and then they continue again as independent particles with measurable energy-momentum vectors. In general, these new energy-momentum vectors differ from the original ones. Under certain conditions, new particles can be produced in the collision or some of the original ones annihilated. It. is clear that our formulation of quantum electrodynamics with incoming and outgoing particles is quite appropriate for the discussion of such collision problems. In order to relate the incoming and outgoing fields, we have to find-a generalization of the method u s e d i n S e c . 1 1. W e s h a l l d e n o t e r t h e i n c o m i n g f i e l d s , o r i n f i e l d s , b y 1 k r " ) ( x ) a n d , ( e i ' )( x ) a n d t h e o u t g o i n g f i e l d s , o r o u t fields by Af"') (r) and ?c"') (r) . First of all , we know that these quantities obey the same canonical commutation relations. Acl. (Translator's Note) The usual notation in English for the rnfield is /(in) and n61 7(ein). We have retained KAIl5n,s notation only in order to avoid resetting all the equations. Similar remarks apply to ,4(out)- 7(aus).

86

G . K 5 i I I 6 n ,Q u a n t u mE l e c t r o d y n a m i c s

cording to well-known theorem.,l with the following properties: : ttaus)(z)

th"ru must exist a matrjx -S

0e.r)

5-t1p(ei")(z) S ,

0e.2)

At"E-(x): 5-t 7f;i")(r) S ,

0 e. 3 )

SS*:S*S:1. By means of this xo-->+@.

matrix

Sec. 19

we can express

the

Hamiltonian

for

/l'(o)(1(aut,tpl^"")1, as a function

of the Hamiltonian (1(aut, ?("u9) : /1,(0)

for xr-->-

ai

S 1Ii(0)(/(ei'), ,yr("i")) S.

( l e. 4 )

and Introducing the eigenvectors of the operator .i7(o) 12("i"),rp("i")) denoting them simply by ln), we find from (19.4), l7(o)(1(ein), Vkin))ln>: Enln) , 11(0) (1(aus) , ,(aus) ) S-t ln) : E,St lzr).

(le. s) 0e.6)

Tha ctatac oiconcfafae nf fha anarnrr f' n- r- v e r y 'S - r l at \" / a r a t h a r o f o r a large times and consequently the probability that a state ln) makes the transition to a state ln') is given by

unn,:l(ra'lSl")1,

( r e. 7 )

The general problem of finding the outgoing particles, given thosq which are incoming, is therefere solved if the matrjx S is known." Equation (19.3) says nothing about the general structure of the matrjx S except for its unitarity. It is quite clear that S has to be and that even if. e is non-zero, the matrix the unit matrix if. e:0, elements of S can be non-zero only if the states lz) and lz')have the same total energy and total momentum. We therefore expect that S has the general form

(z'lSln):6n,nl

( n ' l R l n ) 6 ( p '- p ) .

( r e .8 )

l. See, for example, P.A. M. Dirac, The Principles of Quantum Mechanics, Third Ed., Oxford,1947 , p. 106. Here it is explicitly assumed that the states lz) form a complete system which is equivalent to the assumption that no bound states exist. See also A. S. Wightman and S. S. Schweber, Phys. Rev. 98,812 (1955), acnaairlltr

v e y v v r s r r l

n

y .

Atc,

v e v .

2. The S-matrix was originally introduced into quantum theory by w. Heisenberg, Z. Physik 120, 513(1943). Thetheory of the S - m a t r j x i n q u a n t u m e l e c t r o d y n a m i c sh a s b e e n d e v e l o p e d , f o r e x a m p l e ,b y F . J . D y s o n , P h y s . R e v. 7 5 , 4 8 6, 7 7 3 6( 1 9 4 9,) a s w e l l a s b y C . N . Y a n ga n d D . F e l d m a n ,P h y s . R e v . 7 9 , 9 7 2 ( 1 9 5 0 ) .

The S -Matrix

Sec.19

87

Here (a'lRla) is a non-singuiar function of the energy-momenrum vectors of the particles considered , and p'and p are the total enerqy-momentum vectors of the states In fact, lrz,) and ln) when we explicitly calculate the S-matrix below, we shall find that these conjectures, implicit in (I9. 8) , are confjrmed. If the expression (19.8) is squared in order to obtain the physically ir teresting transition probabilities (I9.7), the delta function enters quadratically: a completety meaningless result! We have to go back a step in our calculations momentarily and recall that with a (finite) periodic boundary condition, the spatial delta function is replaced by the symbol dpp. and this quantity can be squared without difficulty. If af0 we do not obtain an exact delta function for the energy in (19.8), but rather expressions of the form 1d,

;

{lq

q)

",+W;-p,Y

The expression (I9.9) can be squared without difficulty, but the integral ,.

1

f F(x\azdx

!t!r-", J 1o,a ,qz ,

0e.10)

where F(r) is a regular function, does not exist. erations show that the integral 1

Simple consid-

F(x)as

I r t'"tr f J l a r , r t y d x : 2 -r\0) ",

(19. lr)

does exist. We can thereforewrite symbolically v n n ,: l ( n ' l R l n ) 1 , 6 p n ,6 ( P ' o P o )J 2xt d.

(re. r2)

If a goes.to zero the transition probability becomes very large. In ordinary quantum mechanics, in treating collision problems, the quantity of interest is not the total transition probability but the transition rate. Recalling that the particles have been interacting with each other for a time of the order of. llu, we see from (19,12) that in quantum electrodynamics also, the transition probability per unit time has a finite limit if o( goes to zero. As we shall see in Sec. 20, we can write

( n ' l s l n ) : 6 , , n * ( n ' l R l n ) d o , p6 ( f t -

p o,)

0s.13)

rather than (I9.8), and obtain for these quantities ?won,

at

A , ^ ' o-

:l@'lRln)l'6p'o "o

,n-

po)

.

( r e l. 4 )

From the quantities (19.14), the interaction cross sections, etc., can be obtained in well-known ways. We shall not give general formulas here, because theycannot be written without complicated notation. In later sections we shall often compute particular inter-

G . K i i l l 5 n , Q u a n t u mE l e c t r o d y n a m i c s

88

Sec, 19

action cross sections from the S-matrix. In principle, we can determine the S-matrix from the equations :) S-trtot(r) S : ?0) (r) + I S (* 1rt"*r(z)

x')i e y A(xt) y (x') dx', 0s .$)

(Af*) (x): ) s-re tot(x)S: Af) {4-[ n @- *' )T l',t@'),yu,t'@')fd.x',le . 16) or [S,ytot@)l:

- S/ S(r -

(19.17)

x ' ) i e y A ( x ' ) y ( x ' ) d . x ',

[ S , , + ; o r ( r )S] :I o W - 4 + l r t , @ ' ) , y r r t ' @ ' ) ) d r ' .( l e . 1 8 ) tn (I9.15) through 09.18) we have again used the earlier notation We now develop the matrjx Al|)@) and rp(o)(r)for the in-fields. S in a power series in e , analogous to those for the field operators:

0 e .t s )

S:{f.5tt)-p... I n t h e f i r s t a p p r o x i m a t i o n , f r o m ( I 9 . 1 7 ) a n d ( 1 9 . 1 8,) w e f i n d -i [s(1),?(0)(*)]:

f dx's(x

x ' ) y t r t o t @ ' ) v @ @,' )

09.20)

(41: + [ a"' O (" - *') llt)@) (x'), yrrt(o\(x')). ,4(0) [S(1),

(I9 . 2I)

(r) and rlt\o)(r), it folSince the operator Af)(r) commutes with gJ(o) Iows from (I9 .21) that the first approximation $(r) f e the S -matrix must be of the form S 1 1 ) : _ t I d * , l r t $ ) ( * , ) , y r r l ( o ) ( * , ) ) A(txr ,o)t* s ( r t .

(l9.ZZ)

In (I9. 22J the lstm s(r) is independent of ,lf) (x) . From (19.20) we find that s(1)is also independent of the operators forthe Dirac field and consequentlyis a c-number. This c-number obviously can. not be determined from the commutation relations (I9.17)and (19.I8) S(1): - S(1)* ; From the unitarity of the S-matrix, it follows that that is, the number s(1)must be pure imaginary. Without altering the properties (19.1)through (19.3), we can always multiply the S - m a t r i x b y a f a c t o r e t d ,w h e r e t h e p h a s e d i s a r e a l n u m b e r . W e c a n t h e r e f o r e s e t 5 ( i l i n 1 1 9 . 2 2 )e q u a l t o z e r o b y d e f i n i t i o n ; t h i s only fixes the arbitrary quantity 6. Thus we have as the first approximation to the S-matrix the result

5rr): _ bJ dx, SEtt (*,),yurt'\,1x,)lAtf\(x'). In a similar way given by

gtzt:1

it

+@

_f

dx,

(re.23)

can be shown that the next approximation

(x'),y,,tp
is

I ( r e. 2 4 )

')) (x' ') , y,,y)@ (x' Alot(x"). ) x ltpto>

Sec. 20

T i m e - D e p e n d e n tC a n o n i c a l T r a n s f o r m a t i o n

Bg

In (I9 .24) an arbitrary c-number has been set equal to zero, as was done before. The method indicated here leads to rather complicated calculations if the higher orders in the S-matrjx are considered. Despite this, the results are remarkably simple, and it is to be expected that there is a more direct method to calculate this matrix. 20. Treatment of Quantum Electrodynamics by Means of a TimeI)enendent Can6pig6l TranSfOrmation In this section we shall develop another method of solving the differential equations of quantum electrodynamics. This "new" method isthe one which has played the greatest role in the development of the modern theories. It is closely connected with the interaction picture given in Sec. 2. From the fact that the in- and out-fields satisfied the same canonical commutation relations, we concluded in Sec. 19 that there must exist a matrix S with the properties (19.1) through 09.3). Actually, for an arbitrary time xo the field operators satisfy the same commutation relations as the in-fields, and from this we can conclude that there is a timedependent matrix U(ro) which has the following properties:

V @) :

(20.t)

U-, ("0)tp
A,(x): r-'!ro, Af,)(x) u(xo), U * ( x o )U ( r o ): To determine the matrix tions

(20.3)

U ( x o )U * ( x o ) : t .

U(xo) we have the two differential equa-

'lr1l: u -r u-'yn,t (,* * *)v@): - tl- r-#(r-tyny@ : i U-tl!# u-, y4't)@)lu : i e(J-1 y A@\ ,(o\ry, or [\ir'-'

(20.2)

?.o']: eY4YAto)!)b',

I

(20.4)

| ) ( 2 0 .s )

and

: r' {l+(f^r-1,Af,l* rAu@) rlffu-,!4u;' l*

'

or

I,ro.u, Lui,,-]lu : - 1]u, t p6,, * ILI|;u L,Agl, y,,t,o)t,,l'"

- u -(u . ug- - 't\' . -A- !( 2+, 1zl !-U -'." A!'ix o' l +[]!! U l''LA*o',

l7xotg^o

1, liar,

:] u-.1 oPl,!,+

r,o.a

) *lr*',v,v,(o)1. aU __, [/-1 must be a function From (20 .5) and e0 .7) it foltows that e o f / ( 0 )l r \ a n d Yt ' , @\ .)"(/'.) . M o r e o v e r , f r o m (\ -2"0. . 5. ,) w. 'e h a v e t h a t ? - U g - t axn

must contain a term of the form

90

G. Kill5n, Quantum Electrodynamics

_;

e t6:

I

(x'),Trrtt@ a'"' ;r
Sec.20 (20.8)

to

and we readily verify that this term is sufficient to satisfy (20.7) also. The matrix U(xo) therefore satisfies the differential equation t

a \!'ot

:

H$ (A@ (x), y.,$t(x)) u (xo),

( 2 0 .e )

with the boundary condition

U(-*):r.

(20.10)

As we did before for the S-matrix, we have set an arbitrary cn u m b e r e q u a l t o z e r o l n E q . ( 2 0 . 9 ) . I f x o b e c o m e sl a r g e , w e c l e a r l y have U(f

o o ):

S.

(20.1r)

In order that U tend to a well-defined limit for lrol+ oo , the charge must be adiabatically switched on and off, as was done p r e v i o u s l y . W i t h t h e s e a s s u m p t i o n s, ( 2 0 . 9 ) a n d ( 2 0 . 1 0 ) c a n b e incorporated into the integral equation

u(xo):

t-'f l-,

J

(x')) U (x'o). dx'oH! (A(0,(r') , ,lt(o)

Q0 .12)

*@

The canonical transformations (20.I) through (20.3) can be regarded as the transition from the Heisenberg picture, Ar(x) , y(x) to another picture , AII) @), ,lt@@) . In the new picture the operators are known, but the state vectors which are qiven by U(xo)ln) are not simple and must be found by solving the "Schroedinger equation" (20.I2) or (20.9), (20.10). Here we are using ln) for the state vector in the Heisenberg picture. This new picture is identical with the interaction picture discussed previously, and Eq. (20.9)asreeswith Eq. (2,11). The integral equation (20.12)can be solved formally by a power ''o

*o

lo

U(xo): | - i I H0 (*L)d"L- I dx'oI dx',iH(l)V;)H$)(x';)+ "' : -@ *m @

s

.1, n-O

ro

ti

th-l)

? l j*arL1*d,:i..._Lt.tr'ur')(xL) H @ ( x V ) ) .

10,,

Here we have writtet't gtr)(xo) as a shorthand for HQr(A(o)(x),rl@)@)). C l e a r l y , t h e s o l u t i o n ( 2 0 . 1 3 )c a n , i n p r i n c i p l e , b e u s e d t o c a l c u l a t e Afthe field operators A*(x) , rp(*) as functions of the in-fields. which we shall not go into ter so_mecomplicated transformations, here,r it can be shown that the Heisenberg operators are given by the expressions l.

See, for example, F. J. Dyson, Phys. Rev. 75, 486 (1949).

Sec. 20

Time-DependentCanonicalTransformation

A,(x\: AI!)(,) * + f, i, i o-t...'ff"o'y, il-L

-6

-@

9t

I

( (20.Laa)

x[u<'>1xYt1, l ,l l.. . lHl) (x;),A fi(x) ...11 -

-(&-r')

,p(x):,t'@@)+ T2 i ! I d r L . . . I d . x f , i *

(20 . 14b)

x [aru1r5,i;, p(o)(r)] ...1]. [...[H(1)(r;), These formulas are not especially suited to practical calculations and several rather complicated transformations are nece_ssarybef o r e t h e s i m p l e f o r m u l a s ( 1 8 . 5 ) t h r o u g h ( I 8 . 1 2 )r e s u l t . I F o r t h e calculation of the field operators, the method of the interaction picture is not especially suitable. The situation is quite different for the S -matrix. Here the series (20.13)essentially contains the result. It is -@

@

S:1 * Zeil" n:l

x[

,?'-1)

2 0f \..1 s ) I a r LI d r ' i . . . j d x , w t H < t t 1 x ' ) . . . H Q(@

_@

_@

_@

F o r w h a t f oal l o w s , i t i s u s e f u l t o t r a n s f o r m t h e e x p r e s s i o n ( 2 0 . 1 5 ) s o m e w h a t .' I f . F ( x r . . . x n ) i s s y m m e t r i c i n a l l v a r i a b l e s , i t i s evident that f

J

brtxn-rbb f -

f

-

O * r a x r . . . d x n F ( x r . . .,x n ) :1 f d, x r . . . f d- . x " F ( x r . . . x *( )2. 0 . 1 6 ) ,t J J J J

&A&aa

The product of the interaction operators in (20.15) is certainly nor symmetric in the time variables, because these do not commute with each other for different times. Despite this, we can transform the reglon of integration according to (20.16) if we require that an operator with a greater tirie always stands to the left of an operator with a smaller tirne. To do this, we introduce a ',time ordering" or simply a " P -symbol " in the fotlowing way:

: P(A(x) a@il) ;:: :t',,:,\ {1\;);"^lT,))

(20.r7)

The generalization ofthe P-symbot to several factors is obvious. The S-matrix (20.15) can now be written as an inteqral from - o to *oo: 91 r :\n f f p ( H | . I ) ( x ; ) . . . a ( , ) ( (r 2 [ ,0) ). )1.8 ) s: r + f -# Jdr;...Jdx? n:I

The expressions (19.23) and (19.24) found earlier for the first apl . S e e , f o r e x a m p l e , J . S c h w i n g e r ,P h y s . R e v . 7 4 , t 4 3 g ( 1 9 4 8 ) 7 5 , 6 5 1( 1 9 4 9 )o; r G . K e i l l 6 nH , elv. phys.Acta22,637 0949). 2 . S e e f o o t n o t ei , p . 9 0 .

gZ

G . K i i l l 5 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec. 20

proximations to the S-matrix are clearly special cases of (20.15). Obviously one can verifyl directly that the formulas (20.14) and ( 2 0 . 1 8 ) s a t i s f y t h e d i f f e r e n t i a l e q u a t i o n s ( t 7 . 1 4 ) , ( 1 7 . 1 5 ) ,a n d t h e r e l a t i o n s ( 1 9 . I 7 ), ( 1 9. I 8 ) . In conclusion, we shall give a proof of the relation (19.Ia) bv means of the methods developed here. We first remark that the s e r i e s ( 2 0 . 1 3 )c o n t a i n s a f a c t o r o f t h e f o r m ( n ' l U ( x o ) l n ): ( n ' l R l n ) d n o ' rt; [,'ot'-n"t";d''

(20'19)

shall therefore supin every order of approximation to U(rr) . ;" pose that all matrix elements of the complete operator [/ have the form (20.19). lf we let ro become very large in this equation, we find ( 2 0. 2 0 ) ( r c 'I S I n ) : ( n ' l R l 6 p p ,6 ( f L - ! o ) , M ' ) * l n ) ) , ") where the matrix R in (20.20) is to be identified with the matrix R in (I9.I3). For finite ro the probability that the system be in state lr?') at time ,o is ,l(n'l(J(xo)ln)l':

tl '(' n - -,-l' R - - l' -z' )' 1-2o^ p p ' J1 ,"r,r, " " d x ' de i ( r i ' - o t t ' t ' - r t : t('2 0 ' 2 r ) l

The time derivative of the quanti;z0.ri probability per unit time:

t, the desired transition

'2 I f" ' - ! l <' n ' l u ( x ^ \ l n, l 2 : l ( n ' t R t n ) 16oo'l + I*drLei(Pi-Po)G6-d dxot - ro)vo-,r,): )\/!)Dl * j' or, ei(p'o l ( n ' t R t n ) 1 2 oPP' ^ t] oa\PL

- ro)@i,-' o)ri, r' @6 aoo, _j*a

(20.22)

Po)'

The result (20.22) is identical with Eq. (I9.14) and is time-independent. At first sight the time independence of the quantity (20.22)seems surprising. Physically one might expect.that the transition probability would slowly fall to zero for increasing time, because of the decrease of the probability of the state In) due to the decay. Our failure to obtain such a result is clearly dependent on theuse of the ansatz (20.19)for the matrix U. Actually (20.19)is only correct up to terms which become very small if Z goes to infinity. In many applications of (20. 22), the sum of all transition probabilities from the Thus, a given state jz) is usually proportional to V-L . probability that the state lr?) is present, even after the collision, t.

C. Kiill6n, Ark Fysik 2, 187, 371(1950).

Sec.2l

The P-Symbol and the Norhal

Product

93

is practically equal to 1. Physically then, it is clear that the probability that a finite number of particles collide must be proportional to V-t if they are "released" in a very large volume Z. Only with this requirement does the concept of an interaction cross section make sense. In this way we can understand why (20.22) is formally time-independent. It is now clear that this method can encounter difficulties for very large times. In particular, if the transition rate is time-independent, then for very large times the total transition probability may be so large that we should not employ our formal solution (20.I9). In these circumstances, evidently we shall have to include those terms which were omitted in the perturbation series because of their dependence on V. In particular, only if the interaction time d*l can be chosen so brief that

1 -: tV

is

rrcrv

small

is

(lg-12)

norrAci,

The

iimit

a->

0

which

would give an infinite transition probability is not actually allowed in (19.12).For problems in which there are also bound states present it can happen that the simultaneous limit a->0 and V-> a is impossible. The method of calculating the S-matrjx developed here cannot be applied without some changes. In Sec. 28 we shall study a simple example of such a case and use a slightiy modified method of integration. In those cases where we obtain sensible i n t e r a c t i o n c r o s s s e c t i o n s b y m e a n s o f ( 2 0, 2 2 ) , w e s h a l l u s e t h e S -matrix and the concomitant restrictions without hesitation. 2I.

Calculation of the P-Symbol. The Normal Product O u r n e x t t a s k i s t o c a l c u l a t e t h e P - s y m b o l i n ( 2 0 . 1 8 ), o r t o caiculate matrix elements of the form

(/,'lS(')l ) A) l o L) " . . . ' \(1[,t ( 0 ) ( xy' ,) ,, , w , 0 , ( x ' )U " > : : :2) ,:,.. 1.1J I o r ' . .J. f a * r r l n'l P l(Zt.t) . . .11/(D y^rtrc\(*(")))Alot @(tu)), @o'>\lns .

)

Here the states lr) and ln') are ej.genstates of the Hamiltonian for the incoming particles, i.e. , of H o ) ( A @ ) ?, ( o ) . W e c a n c o n sider these states to be characterized by a given number of eleetrons and photons and write them as products of the form

(2r.2)

ln) : lnx)lnq).

H e r e l z o ) i s a n e i g e n v e c t o r o f 1 1 l o( ), 4 ! 0 )a) n d d e s c r i b e s t h e p h o t o n s rrrhinh

Ara

n raqanf v r euv r r L,

y vh r r jr lf e v w

lZ.\'i

r ,.q,/

s eat nL

Le

cincnrrer:for er yvr r vsuLvr

n v rf

H ^ ( 0 /)" 1 0 ) ) a n d w )

"2

d e s c r i b e st h e e l e c t r o n s f c . f . E q s . ( 1 7 . 2 0 )a n d ( I 7 . 2 1 ) ) . B e c a u s e the operators l(0) and ttot commute with each other, we can write t h e m a t r i x e l e m e n t ( 2 1 . 1 )a s a p r o d u c t o f t w o f a c t o r s :

: !;l, (*'),y,,,t@) ('')l... <";lp(l,t(') lA*' ...laa"> (*()),y,*V(o,@a))l)1")("Llp(,4t:) .. . [,t(o) @,)... A@,) @{,,\))ln).

(21.3)

94

G . K 5 r l 1 5 nQ , u a n t u mE l e c t r o d y n a m i c s

Sec. 2l

The individual P-symbols appearing in (21.3) can be worked out b y t h e s a m e m e t h o d w h i c h w a s u s e d e a r l i e r i n t h e p r o o f o f ( 1 8 . 2 I ). In order to make the procedure more systematic, we shall follow Wickl and introduce the so-called "normal product" of the free field operators. According to (5.28) , we can write the electrom a a n a f i cL { v

rrrvvrrv

fiold

e q

F

qlrm

nf

trrrn

h^rf Psr

c ..

Lr

Af,(*): Al:,@) + AI;,\x) . Here the operator

A';) (r) contains only the annihilation

: AIj)@) 4n,utt) (k), rt A#

(2r.4) operators: /rl (l

and the operator AI;)@) contains only the creation operators:

. aFl\,(trr ) - +

f +,

, 1 tot * t t(tk ) .

( 2 r .6 )

lvfilzo

These quantities

satisfy the commutation relations

(x),Al+)@',)l:lA? (x),tr-t @')f: o , lAl;,)

(2r.7)

(r')I 0) . (21.8) lAli)@),A? (x')l: - i 6,,D?)@'- x) : (o ;zlor@)Alo: In a similar way we write the Dirac field operators as

(x) : rl\ (t) + ,lt") (*) , 1p@)

( 2 re. )

1p@) (x) : rt't (x) * y,Q (r) ,

(21.10)

where the "+ operators" contain only annihilation operators and the "- operators" contain only creation operators. In place of ( 2 1 . 7 ) a n d ( 2 1 .8 ) , w e h a v e t h e c o m m u t a t i o nr e l a t i o n s { r p ( *( )* ) , r t - ) ( x ' ) } : - I S ( ) ( x ' - x ) :

( O l ? ( 0()r ) r t o )( x ' )10 ) , ( 2 1. 1 1 )

{ r p c ) ( * ) , r r +()x ' ) } : * i S ( + ) @-, * ) - ( o l r r t o(rx , ) t y t o t ( x ) l o, (>Z I . I Z ) { r p ( *( )t ) , 1 p F()x ' ) }- { V e )@ ) , r p ( (. r ' ) } : { r t t t '()* ) , , p c ) ( * ' ) }

O .( 2 1 . 1 3 )

Now we define the normal product of two or more factors as an operator product where the creation operators always stand to the Ieft of the annihilation operators . We denote the normal product b y : . . . : a n d w e a b b r e v i a l s 4 t o )( r { ' ) ) a s A ( i ) . W e t h e n h a v e t h e normal product of operators Alli (r) , 1. G. C. Wick, Phys. Rev. 80, 268 0950). Theseproducts had already been used by A. Houriet and A. Kind, Helv. Phys. A c t a , 2 2 , 3 l 9 0 9 4 9 ). T h e n a m e " n o r m a l p r o d u c t " w a s i n t r o d u c e d by F. I. Dyson, Phys. Rev. 82, 428(1951).

The P-Symboland the Normalproduct

Sec. 21

YJ

: A ( t ). . .A ( n ) : A @U ) . . .n t +@ ) )I n

+ I /(-) (1) 4t+\(1). . . AetQ- q a*t U+ D . . . ar+)(QI

(2r.r4)

* | tc | (i)A( ) (j)AF)U) . ..4t-)(n)-t i
In (2I.Ia) the normal product is clearly completely svmmetric in all of its variables. It is not necessary to state exactlythe sequence of the 7(+) (or /(-)) operators with each other since, accu n v r ud ri rnrav

fLov (\ 4? rl . 7 t \t t

lLhLaLr er ! n v vnrm r rm r rrrrurLae .

a ur ruar h u r a

statement

is

necessary

for the ?-operators, however, because in (2l.ll) through (21.13) the anticommutators are present. We now give this normal product a well-defined meaning by the conventions

({).. . rt+t : v U)... v @): : e(+) () ... et+t @)+j a. eH (i) eG) @).1 i:r ( 2 1l.s )

* l d . e t - ' 1v4G ) ( i ) r t + ) O . . . r r++. .) .(+e v e O . . . v c , ( n ) . 1<1

In (2I.15) g(l) stands for a factor which can be either Vn@U\ or ,1:to)@lttT ) .h e s y m b o l d p i s + l i f t h e n u m b e r s ( i , i , . . . ) a r e a n e v e n p e r m u t a t i o no f t h e o r i g i n a l n u m b e r s( I , 2 , 3 , . . . ) and -l if they are an odd permutation. With this definjtion, the following symmeffy condition obviously holds : (21.16) Thus in a normal product we can always calculate as if the Af)@) were commuting operators and the g(0)(*) and ,p$)(r) were anticommuting operators. To transform an ordinary product into a normal product, we have equations of the form

A ( t ) A ( 2 ) : : A ( t ) A ( 2 ) :(+0 1 / ( 1A) ( 2 ) l o > t (2r.rT) : : v O r p ( z ) : r p ( t ) t p ( z ) : y ( r ) y ) ( 2+) :( o l y ( t ) y ( z ) l o,) ( 2 1r.8 ) ,t,U)rp(2): : tp(t)tp(z): = : 9 (1) rp(z): + , (21.te) ,,p(l)rlQ)::tp(r)y(z):*(olrl(r)rp(2)lo>, (zt.zo) ' , t ' U ) r p ( 2 ) : : 1 t , U ) r p((ozj )r :p-(tl ) r p (oz)).f (21.2r) Equations (2I.17) through (2I.2I) can be proved directly from (21.4) through (21.16). As an example of the application of these equations, we shall write the free field current operator as a normal product: I

(x)'y, V@@): + ( 0 | ?(0)(/) y rp(o) (*) lo)-l lqo (t(), yr rp@(x)l : g {: ty
I

: 1rp
. 1l(o)(x) y,rtt@ @) : .

)

G . K 6 i 1 1 5 nQ, u a n t u m E l e c t r o d y n a m i c s

96

Sec.21

This result shows clearly the vanishing of the vacuum expectation i,alue of the current operator. It also illustrates the advantage of the normal product over the usual product: For a given transition, where a fixed set of particles is changed, there is one and only one kind of term, among all possible normal products, which conThese terms are just those having the correct annihilatributes. This statement does not hold for the tion and creation operators. product a and subsequent annihilation can where creation ordinary take place in the intermediate states--as we have seen inSec.18. It is important to require that no particle in the final state is iderr If this is not true, there are tical with one in the initial state. circumstances where several different normal products can give non-zero contributions to the result. the following matrjx element:

As an example,

consider

) B (q,kl:tpto)vt\\A(o)A@\:lq,k').(2r'23) ( s . h l F l q , l r ' ) - A ( k l : A t o t , 4 t o ) 1 l k '+ In all such cases it turns out that the "complicated" terms contain additional factors of V-'and therefore that they can be dropped if the volume Z is made very large. Therefore we need consider only the first non-vanishing term in expressions such as (21.23). A similar simplification has already been used in (18.20a). Our major problem is now the transformation of the P-symbol into a sum of terms each of which contains operators only as If we knew the solution of this problem, the normal products. calculation of matrjx elements in (21.3) would be straightforward. To make the algebraic manipulations as transparent as possible, we shall introduce two more symbols.

We define

r ( , p ( rv) Q ) . . v ( n ) ): 6 " , p ( i,)p ( i.). .

(21.24)

ordered, just Here the factors on the Ieft side are chronologically b y t h e factor 6r, as in the P-symbol . The two symbols differ ( 2 I . 1 5 ) . a l ways take w e s h a l l a s i n M o r e o v e r , d e f i n e d i s which With this this factor to be +1 if the operators g(x) are A(f)(x). convention, the following equations can be used for (21.I4) as well as (21.15). We also define

, 6 O d Q ) e 0 . . . E ( n ) : : ( 0 1E O q Q )l o ) : q ( 1.). . q ( n ),:

(2r.2s) (2r.26)

F r o m( 2 I . 1 7 ) t h r o u g h ( 2 I, 2 I ) , w e p r o v e b y i n d u c t i o n ( f r o m n t o n + I ) that

.. v @): : :e U)q Q).. . v @): + i, 6 ll v Q)...6Q)...v @):,(?r.27\ q (i) |q (z). and from this

(also by induction

from n to n+I)

ean

,1

The P-Svmbol and the Normal Product

97

v ( ) c p( z ). . . c p( n ): t v U ) . . . v ( n ): - f

* | : e O e e ). . . 6 t q. . . { ' O. . v @ )t:

(2r.28)

+ j,;I . t v U ) . . . i l @ , ) . . . U t ; s . . . i l U , ) . . . U* .U. .,.) . . . v @ ) : i.r< i,"
,

A c c o r d i n g t o E q s . ( 2 I. 2 6 ) a n d ( 2 1 . 1 6 ) , i n t h e " c o n t r a c t e d ' , n o r m a l products the sequence of the factors can be changed only if a factor dp is added. Care must be taken that two factors which are contracted with each other are never interchanqed. For example, we have

:q(1)... 6O v tn6Wl... E@): : - |q(1)... 6V.l6 Wlv 0 ... v @): (zt.ze) but

t v U ). . . i l @ i l O. . . q ( n )g: - : e ( r )

6 t i l $ 1 .t .1.q 1 " 1 , . ( 2 1 . 3 0 )

IFor the operators AjP@) the sign factor isnot present in(21.29).] Because two contracted factors on the right side of (2i.28)will then always have the same order as they do on the left side, it foliows f r o m ( 2 I . 2 4 ) , ( 2 1 . 2 8 )a n d t h e s y m m e t r y c o n d i t i o n ( 2 1 . 1 6 )t h a t

r ( q 0 . , . q ( " ) ) : : e U ) . . . p (*n ) : e ( t ) .. . 0 0 . . . v ( i ) . . . v @ * )) : i
t,i O q (z)v 0 ... q (n): : (0 | r (,p(t) e (2))| 0) :v $) . . . v @):,(21.32) : e U ). . . . e Q. . . e ( i l. . .e ( n ) :: 6 , , : E @ E ( i )v 0 . . .E @ ).: ( 2 i . 3 3 ) The vacuum expectation value of the I-product immediately. The non-vanishing terms are

can be calculated

(0 | r (,4t0) (x)A[ot(r'))I o) : (01P (AIP@)Atq) (x'))lo>: I - | 6 1 , , p {( tx)*' x ) - i t ( x - x ' ) D ( -* ' 4l:1,^. Qr'34) - 4 6 , , , l p r(tx\ '- x ) - 2 i D - " ) l : @, I : i6u,Dr,(x'- x) , Here the function Do(z)

I

)

is that defined bV (7.29) and

(01 7 (rt(o) (r) 1t\o) (x')) 10) : + So@,- x) ,

(21.3s)

(0 | r (?(0)(r) v@,(x')) | 0) = -+ S" (r - ,') :-(01 T (1p(Jl (x') y(o)(r)l0).(21.36) Because of the antisymmetry in (21.36)land the symmetryin (21.34) l,

98

G . K 5 1 I 6 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec.22

we can change the order of two operators which are contracted with each other in (2I.33). In particular, we find

: e ( { ). .. v 0 . . . v 0 . . .v @ ) :: 6 " : q ? ). . , i , U .). ' 8 0 . - . E ( s ) ' .( 2 1 ' 3 7 ) E q u a t i o n s ( 2 1 . 3 1 )t h r o u g h ( 2 1 . 3 6 ) c o n t a i n t h e d e s i r e d r e s u l t f o r a I-product; it can now be writtenas a sum of normal products. The coefficients of the normal products can be constructed from the Admittedly, in applicatlons of (21.3) we singular F-functions. shall have P-products, but always with the same number of operators rpt0)(z)and gtol(r) . It is clear that P (: rp(r ) rp(t) : ... : rt @)v @):) :

r (: rp( r ) v U) : . . . : rp(n) rp(") :) . (2l' 3 8)

For the mixed T-product in (2i.38) one can readily show that there is an expansion analogous to (21.3I). It must be rememberedthat two factors with the same time are not to be contracted with each other. Therefore we have, for example,

( z :)?: ( t )' , t ' 0 ' p Q ) , p ( -zf) : P ( : r (pr ) r(pr ): : t y ( 2v)Q ) : ) = ' r(pr ) . p ( tr)p ( z ) ' , p + * : tp(r)tp(1),p(2)v Q): + :rp(r)ii'0 ,t'Q),t,Q)' : : : t p ( 1 )r t t O ' ! ( 2v)Q ) t+ + S t s Q 1 : t)1 t (ttp) ( z )-:

/r1 eql

- i S r ( 12 ) : t p ( t )y ( 2 )| - + S l . ( 2{ ) S F ( l2 ) . With these rules of calculation, it is possible to write down immedlately the expansion of an arbitrary P-symbol into normal products. All that is lacking 1s a general discussion of the omen h i d d e n i n ( 2 1 . 3 3 )a n d ( 2 1 . 3 8 ) . W e s h a l l f o r e g o t h i s n o w , s i n c e we can continue the discussion much more simply in Sec. 22 by means of a method developed there. 22 . A Graphical Representatlon of the S -Matrix We now have all necessary means to compute any desired element of the S-matrix to arbltrary order. Even in rather simple problems the expressionswhich are obtained are very complicated and not easy to understand. There is an extraordinary method for arranging the results systematically which was originally discovered by Feynman.r In this method each normal product is represented by means of a graphical figure and there are rules according to which one can immediately draw these "graphs" and convert them into analytical terms. In his paper, Feynman originally introduced these graphs and rules of calculation quite intuitively, and only later were they established by means of the usual formalism.2 We shall not explore these intuitive considerations further, but rather shall show that our analfiical expressions can be represented by means of the graphs. Essentially, we follow the work I . R . P . F e y n m a n ,P h y s . R e v . 7 6 , 7 4 9 , 7 6 9 ( 1 9 4 9 ) . 2 . F . J . D y s o n , P h y s . R e v . 7 5 , 4 8 6 , 1 7 3 6 ( 1 9 4 9 ) . S e ea l s o R . P . F e y n m a n ,P h y s . R e v . 8 0 , 4 4 0 ( 1 9 5 0 ) .

Sec. 22

A Graphical Representation of the S-Matrix

99

OI IJVSON.I

In order to represent a matrjx element such as

. ! a x "Q ' o l P( t r t U y) , , . , t ) O. .:. : V @y) , , " y , ( 1lxn)r:S)x l

i1a'''

x ( r ' p l P ( . 4 ( 1.)A ( n ) ) l n )

,^^ ". JQz'r)

graphically, we draw ,?points on the paper and associate each variable xi with one point. Each variable xt appears three times in each term in the expansion lnto normal products. From the operator,4(l) there results either a factor in a normal product or a contraction Dr.(ii) with another operator A(i). ff two variables are present in the same function Dn, we connect the corresponding p o i n t s i n o u r q r a p h b y a d a s h e d l i n e . I f a n o p e r a t o rA ( i ) e n r e r s a normal product, we draw a dashed line from the point *t to the edge of the figure. We proceed in a corresponding way for the operators y(i)and yt(,j), If the two operators are contracted into a functlon Sn,we draw a solid, directed line from point xl to point xd . If these operators are in a normal product, we again draw the line from the point x'or xj ro the edge. In orderto differentiate among the various kinds of operators, the "electron lines" wiII be solid and the "photon lines" dashed. Moreover, the electron lines are provided with an arrow which is always to show the direction from the polnt of the operator tp(i)to the point of the operator rp(a). As an example, we show the diagram of the matrix element

#

II

o- d,x'(qlP(: (x)::pto) (x')y,V@ v@(x)yty@) @')') Is')x

: - +[[ a*a*,r-( Y::r?^ q|: rp,,, ; ::,1: i,{irllr'',' r?lI ! ( q l : t p < o t ( x ' ) y ^ S o ( * 'x- ) y ^ t p ( o ) ( xl )q:' ) ] D p ( x ' - x ) :

: -

* II

(22.2)

ox dx' (ql rp(o> (x)lo)y7Sp(x- x') y^(o 1y{o) (x,)| q,) y. x D o @-' x ) .

in the corresponding figure. Obviously the diagram (Fig. 2) gives an intuitive picture of the arrival of an electrofl g, , the emission

Fig. 2.

The graph for the matrix element (22.2).

of a virtual photon, then its absorption, and finally the further propagation of the electron q. One should be cautioned against l. A classification of the various terms in the S -matrix without the use of graphs has been given by E. R. Caianiello, Nuovo Cim. 1 0 , 1 6 3 4 0 9 s 4 ).

r00

G. Kil]6n, Quantum Electrodynamics

Sec. 22

too literal an interpretation of the diagrams, however, since this can easily lead to false results. In general a matrjx element is not represented bya single term a s i n ( 2 2 . 2 ) , b u t r a t h e r t h e r e i s a s u m o f t e r m s , b e c a u s ea l l n o n vanishing normal products must be considered. We can obviously obtain all these terms if we draw a1I the diagrams which can arise by connecting a points with the required number of "external" and "internal" lines. In all practical cases, these diagrams can be drawn immediately. The diagrams which correspond to the n -th approximation (22.1) to the S-matrix can be divided immediately into several groups. Each group contains exactly m! diagramswhich are distinguished from each other only by a different labeting of the variables r' . All these terms are most simply taken into account by dropping the factor a! in the denominator of. (22.I) and actually evaluating only one of the diagrams in each group. A further simplification results if we neglect a1I "disconnected" graphs. A graph is called disconnected if it can be broken into two separate partswhich arp not connected with each other by either electron or photon lines.'

Fig. 3. A disconnected graph.

Fig. 4. A connected graph containing a closed loop.

Thus, for example, the graph shown in Fig. 3 is disconnected, while that ofFig.4 is connected. The basis forthe neglect ofthe disconnected graphs is that their inclusion serves only to multiply all matrtx elements by the same numerical factor. This factor is clearly just the matrix element (0 lsl0), i.e., of the form eid, and therefore without physical significance. The following rules are to be used for transforming a dlagram into an analytical result: l. Each internal photon line corresponds to a factor t 6,,, ?o(ii). 2. Each internal electron line from a pointx'toapoint Nicorrespondsto a factor -* S"(ir). 3. Each external line corresponds to a creation operator if the particle leaves the graph and to an annihilation operator if the I _

(\ rTr qrr ar r nr usL lv ar f o r l sr

p r r vnLive/ )

external lines either! "vacuum bubbles " .

Tho diqcnnnccted

Some authors refer

part is to have no tn

ihaco

narlq

aq

Sec. 22

A Graphical Representation of the S-Matrix l0l particle enters. Note that with our conventions about the ar* rows in the diagrams, a positron is to be considered as running "backwards ". The photons are taken as entering if they are present in the initial state, otherwise they are leaving. 4. Each point rtcorresponds to a factor y,t. 5 . The whole expression is muttiplied by the factor e, (- 7)t+", where a is the number of points and I is the number of closed (electron) loops in the graph. 6. The whole term is multiplied by an overall factor (_1), , where P is determined by the permutation of the factors v@(x\ in the normal product. With the exception of the sign (- {)r in rule 5, these rules require no further discussion since they follow immediately from the considerations of Sec. 2I. The expression ,'a closed loop" of electron lines is quite evident and requires no further elaboratiorL Thus, for example, the graph of Fig. 4 contains just one such ioop. In order to prove the overall sign given in rule 5, we write all the factors :tlt(i)y,,rp(l): which enter the loop next to each other in the P-symbol. This does not affect the sign of the matrix eremenr. We must then evaluate the following expressLon: ,ttUt) y,,,rtt(ir)ii Qr)y,,,,rlt Qr)y (is)yq". . . y,h+ (in).

(22.3)

The contractions can be carried out immediately and give the result * * Sr(i, i,*r) , with the exception of the contraction rp(;r)ri Q,,). The contraction y (r;) rj;(i,1 sives the factor + i SF(i"zl) on account of(21.35). Fromthis the rule for the overall sign in rule 5 follows immediately. In conclusion, we shall discuss one further simplification which can be easily stated in this graphical representation. we can omit ail graphs which contain a closed ioop with an odd number of electron lines. As an example, consider the diagram of Fig. 5.

Fig. 5 . A closed loop having three electron lines.

Fig. 6 . The contribution of this Sraph exactly cancels that of the graph of Fig. 5 .

The assertion is that it may be dropped. for every diagram with a closed loop, which must be added, and in which the is just reversed. To the contribution of

?3486

The basis for this is that there is another diagram direction around the loop Fig. 5 we must therefore

G. Kd116n, Quantum Electrodynamics

IO2

Sec. 22

add the contribution of the graph inFig. 6. For simplicity we shall designate the points which are vertices of the loop by x7,... , x* ' of the diagram with the loop contains a factor The contribution

Sp[7,,sp('lz)y,,Sr(27)...y,,,sr(n1)7'

Q2 '4)

The contribution of the diagram with the reversed direction around the loop differs from the first expression only in the replacement of the spur by (22'5) S p [ 2 , ,S r ( l n ) 7 , * S r ( n , n- 1 ) . . . y , " 5 p ( 2 1 ) ] . In the final result, the sum S p [ 7 , , S p (z1) y , , S o ( 2 7 ) . . . y , , 9 p ( n r ) f *

t\ )- ,- . ^- /\

p S p [ 7 ,S , . ( t n ) y , ^ Sp ' ( n , n - ' t ) . . . y , , s F ( 2 1 ) ] will occur. In order to evaluate this expression, we need a few theorems about the traces of 7-matrices . In what follows, we shall often have to evaluate such traces, so we shall review their ^-nnarJ-ia PI vyvr

c

hara

From the fundamental property of the Trl, * l,Tp:

7-matrices,

2 6 p u,

QZ '7)

we obtain the expression - S p l y u , ) ' , , T h" ' y h ) : S p l y , , y , , l , " " . T ^ f : 2 d , , , , , S PI y , , " ' y , * f n-

I

ttzz.a)

=2Z 6,,,,(-1)' Splv,". . . T,;,1,,*,"' l^f * (-1 )"-l Sply',''' T'^l',1'I i:2 Using the elementarypropertyof a trace, Sp[,aB]: Sp[B /],

(22,9)

and (22 .8) , for n even, we have n

.T^]. Sph", . ..yd :2,6,,"u(- t)' Sp[2,,. .. 7,,-,7,0+,.. Equation(22.I0)

(22'r0)

us to calculate a trace with ?' 7-matrices

"tI.;, 7-matrices are known. In principle, this if all traces with tt-2 result can be used to calculate all traces of an even number of ^fricAc. v/ - m rrrsLrfvvr

For odd r

we write

S p[ ] r " ,... y , , ":f S Ph , , . . . y , , , y ? ) :S PL Z u , .r ., ., y , ^ T s, f

Q2'II)

where 75 is defined bY Tt:

TrTzTsTa,

(22 'I2)

and has the Properties

Tl:r,

(22.13)

Sec.23

The Physical Interpretation of the F-Functions

TsTplTpTs:0,

P:1,2,r,

103

4.

(22.r4)

E r o m ( 2 2 . 1 4 )a n d ( 2 2 . I I ) , i t f o l l o w s d i r e c t l y t h a t Sp hr,,. . . Tol : Sp ly uTn. . . Tat i : e 1)" Sp ly?y,,. . . yul : : ( - 1 ) ' S p [ ) r , ,. . . 7 ^ ] . Therefore, if n is odd, the spur must vanish:

S Ph , , . . . T , , n a :, f o .

\zz. tr) ]

(zz.IG)

Anothersymmetrycondition for traces is often useful: Sp [)r,,. .. y,^]: sp ly"^y,^_,...7,,T,,f.

(22.r7)

Equation (22.I7) is non-trivial only if n is even. For n:2, it is identical with (22.9). Using the expansion (22.I0), the proof for n+2 is readily obtained from the result for n. The general expansion (22.I0) gives the spur of n y_matrlces as a sum of n(ro-z) (n-4)...4.2 t e r m s . B e c a u s et h e r e a r e o n l y four different 7-matrices, related by (22.7), it is to be expected that considerable simplifications are possible in the results for Iarge n. Actually, it can be shownl that such simplifications occur for n>r2. we shall not pursue the matter further because ln our applications there witl not be traces complicated enough to make these simplifications important. F r o m ( 2 2 . 1 7 ) , ( 2 2 . 1 6 ) , a n d f r o m t h e s y m m e t r yp r o p e r t i e s o f t h e functions /n@),

/o@ - /) : /r@' - x) ,

ii /"@ x') : *. Or(*' x),(22.r8)

one can readily show Sp[y"So(12)y,,SoQ3).../,,5p(nt)):

] : ( - 1 ) ' S p [ 7 , ,S o ( 1n ) y , , S p ( n n, - 1 ) . . . 7 * S e ( 2 t n . l Q 2 . I g )

If n is odd, the sum in (22.6) therefore vanishes. We have now proved that all diagrams with closed loops having an odd number of electron lines may be dropped. This result was originally proved by Furry,2 using a completely different method. 23. The Phvsical Jnterpretation of the F-Functions A characteristic feature of the above calculations is the appearance of the functions Dn@- r') and So("*2,) in the matrix elements of the S-matrix. In the original integration of the equations for the field operators, we worked only with the retarded functions. The latter have the obvious interpretation that the value of a field operator at a point r can depend only on quantities at points a' for which the times x[ are smaller than the time xo , IFD

Caianiello and S. Fubini, NuovoCim.9,12ig l2S (1937).

2, W. H. Furry, Phys. Rev.5l,

(igS2).

G . K i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

I04

Sec' 23

phenomena, This is a natural consequence of the causality of all in complete correspondence with the classical treatment of a field theory by means of integrals over the retarded light cone ' In our expression for the s-matrix this causality has apparently been tost. The integrals over the variables r extend over the entire spaceandtheF-functionswhicharepresentdifferfromzerofor bothsignsofthetimedifference.Despitethiswecanreadily convince ourselves by means of a simple example that the F functions have a "causal." interpretationunder certain conditions. Of course it is not expected that the S-matrix will have as simple a structure as the fietd operators, since the time sequence of two integration points is not fixed from the beginning,' We choose the following term in the S-matrjx' as an example for the analysis of the F-functions:

(r')y^,td (x')| q") x a- ax"(ql rtt\o\ (q,q,| ',il |q,,,q"') : i [[ I O, . r, q"' "'' (x") (q.' (x") y * ) ^'!)o | ) x Dp(x' x") l rtd clearly this integral is obtained in calculating the cross section that two electrons with energy-momentum vectors q" and g"' collide and emerge with new energy-momentum vectors q and q' ' It is a photon straightforward interpretation that the electron q" emits a point t" the g' At momenturn at the point x' and as a result has its consequently and electron other the by the photon is absorbed momentum is changed from q"'to q' , The total probability of this transition contains the sum over all possible combinations of x,' and x,,. serious objections can be made to this naive interpretatiorL ThefunctionDo@)doesnotvanishforspatialseparations;not propeven approximaiely. The emitted photons must therefore be agatedwithspeedsgreaterthanthevelocityoflight.Followlng Fierz,l we shall show that a similar interpretation of the integral ( 2 3 . 1 )i s s t i l l p o s s i b l e i f w e c o n s i d e r n o t e v e n t s a t s l n g l e p o i n t s ' but rather the probability that the photon is absorbed or emitted in a finlte region of space-time R. In order to pursue this idea, we break up the whole four-dimensional space into regions R and rewri-te the integral (23.I) as

( q ' q ' ls " ' l q "' q " ' ) : r / n "a "I " ' '

(23.2)

R"R"

In, p,,:l

)tl

I a*' I dr"(qlrtot@')r^rto)(x')lq")x J

R'

R''

,,, ,,

x D,(x' - x") (q'lrro)@")r^rto)(x") lQ' >' ]

For the following discussion it is not essential that the operator yldlx)be developed in terms of solutions of the Dirac equation field. The only property which with an external, time-independent l.

M . F i e r z , H e l v . P h y s . A c t a 2 3 , 7 3 1( 1 9 5 0 ) .

sec. 23

The Physical Interpretation of the F-Funotions

105

we shall use is the well-defined energy of each state lq). Eram +!r^ ih+^^--l

rru* Lr'elnregrar representations of the functions Do@\,D*(x), and D1(x) ,

D p ( x ) : ? : . " [ 6- 'p- -r ; n ' { p l ^ +' i 'n- "6" (t "kt )z \ \ i (2n)aJ ,r. t

(23.4)

Do@)::-

(23.s)

[^

t d k -e i h , { pA! ., +r -i rnv e v \ (' 'k l ")t d ' ' t(, lA t ' ) .}

QnY1"-

DnQc)::=, \2n)-J

[-

f d h e i h , { p * _ t n e ( h \\6, -(\ h. r z1 ,\ \

(re ^\

\LJ.u,,

|.h.

we find the following relatlons upon taking the positive and negative frequency parts of these three functions:

DF)6):iia,

- }o , n 6 ( n ) l : 1 o f t , ( ,()2, 3 . 7 ) I a n ' , 0 , { n+

n F ( * ) :? t r * r " ' io u , , u , {}o; * i n d ( A , :) } 1 o k , ( , )(. 2 3 . 8 ) io(0 We have, therefore,

t on@):DkI @)+ D? (x).

(23.e)

Introducing the relation (23.9) into (23.3), we find . , tr n'n" : -r t'" ( " ,dx', ..l' dx" (ql V$)(x')yxy)tot 1,1 @,)| q,,) DP @,_ x,,,) \l?, , x . R"

l " ) y ^ y 6 t( x ' , ) x ( q ' l 1 p t i@ 1 q , , , t) -f d,x' I [ , d . x " < q l r p h l p c , ) y ^ y ) @ @ ) 1 q , , ) D- t ) x( x, , ) x x (q'l tptd(x") y^rrot@") lq"'>) '))

I

I l(23.10) |

T h e t l m e d e p e n d e n c eo f t h e f a c t o r < q l l p @ Q t , ) y ^ V h ) @ , ) 1 qi,s, ) g l v e n by e-i"i@l'-qi . A similar expression results for ir," btir", *atri* element. In the tlme integrations in (23.10) the limits of integration are not * oo, but ?,and 7,, , the limlts of the regions R, and R". If we require that the two times I,and T,,are veiy large com_ pared to the times lq'i - qol-, and lq(' - qi l-1, then the time inte_ gratlon is "almost" a delta function. According to (23.7), the first term of. (23.10) gives a contribution only if

ao- ali: q:;'- q;> o.

(23. u)

With these assumptlons we have Io',,p" av-i e2[.dx' d""
x
(23.r2)

106

u.

1'!11'NoIIell,

vuollLuril

E q ! r lsounuf vr nu dy rr zr qn rarm r rivcD

Sec.

23

/t'l

l?\

The terms which are neglected in (23.12) are very small if

T(qo- S;)>>t, Thus the integral

and

(23.I2) is non-zero

T'(qo-q;);>1. if

x'o> x'i -

(23.14)

T h e c o n d i t i o n ( 2 3 . 1 3 ) f o r t h e v a l i d i t y o f ( 2 3 . 1 2 )i s i n f u l l a c c o r d In order for the energy tobe with the usual uncertainty relation. defined with sufficient precision so that the differences (23.11) have a physical meaning, the system must be observedfcra time w h i c h s a t i s f i e s t h e i n e q u a l i t v ( 2 3 . I 3 ) . T h e c o n d i t i o n ( 2 3 . 1 1 )t h e n says that the energy of the electron in the region R'has been increased by the collision while the energy of the other electron has fallen by the same amount. This means that the electron in R" and that must have emitted a photon with the energy a:qo-I'i t h i s p h o t o n h a s b e e n a b s o r b e db y t h e o t h e r e l e c t r o n i n R ' . E q u a tion (23. 14) therefore says that the emission of a photon must always take place prior to its absorption. From Eqs. (7.19) and (7.38) it follows that the photon is propagated with exactly the velocity of light. In a similar way we can show that the secondterm in (23.10) gives the contribution - i e 2I d x ' J d x " ( q l r t o \ ( x ' )y ^ t p r c( *t ' ) l q " ) D n @ ' - x " ) x I R'R"I x ( q ' l 1 P t(dx " ) Y ^ Y t t@ ) l q " ' ) o \" )

rr. ,.,

if the following conditionsare satisfied: q',i-ao:q|-q:;'>0, T'(q{-qo)}}1, T ( q ' ,-iq o ) ) 1 , x:J> xL.

( 2 3. r 6 ) (23.r7) /r?

la\

H e r e t h e p h o t o n i s e m i t t e d b y t h e e l e c t r o n i n R ' a n d a b s o r b e db y the other electron, and the inequality (23.I8) again says that the emission must take place before the absorption. In this way the function Do(r) actually gives a "causaI" description of the collision. It is often referred to in the literature as the "causal p r o p a g a t o ro f t h e p h o t o n " , o r s i m p l y a s t h e " p h o t o n p r o p a g a t o r " . It is clear that the properties of the F-function discussed here do not enter only in this particular example, but that a similar discussion ought to be possible in other cases. We shall not carry the discussion further except to mention that these matters have been fully discussed by Stirckelberg and his coworkers.l t. See, for example, E. C. G. Stiickelberg and D. Riviel ruhcre references to the older H e l v . P h y s . A c t a 2 3 , 2 I 5 (1q50) \rvvvr, literature will be f ound.

CHAPTER ELEMENTARY

V

APPLiCATIONS

24. scattering of Electrons and Pair production by an External Field In these next sections we shall give a few elementary applications of the theory developed so far. Since quite complete textbooks exist in this arear (see also footnotes 2, 3) and also because the details belong in Vol. )()C(I of this handbook,4 we shall not attempt to be complete. The following calculations are to be considered only as examples of results which are obtaineid rather easily with modern quantum electrodynamics, although they were usually discovered by other methods, Iong before the beginning of the modern theory. It seems to us that by the calculation of a few examples which have already been satisfactorily worked out, we demonstrate the great simplicity of the new methods compared to the old. As the first example, we choose the scattering of an elecrron by. an external field. The quantized-?tectromagnetic fietd does not enter this very simple example, which can clearly be treated using the Dirac equation without the "second quantization". We work out this problem here because in the next chapter we are going to compute the radiative corrections to the interaction cross section, and then we shall want to compare with this simple calculation. Moreover, we can use this calculation to introduce a simple method for summing over the two directions of spin of the electron in the initial or final states.

Denoting the initial state of the electron by lq) and the final state by lg'), the quantity of interest is

( a ' t s l a ) - t ( - ' ) " [ . . . I a * ' . . .d x ' x

)

1,,o,,

?N:JJ

x (q'lP (:tpal1x'1y,,v@(x'): A,,(x').. . :1pb) (x,) y,,rl,\o) (xn):A,^1x,))| l) . Because there is an external field equation" is not (20.9), but rather

present,

the

l'-"-'

"Schroedinger

1. W. Heitler, Quantum Theory of Radiation, Third Ed., Oxford, loE/

2. A. I. Akhiezer and V. B. Berestetski, euantum Electrodynamics, translated fromthe secondRussianedition, Intersciencq 1965. 3. I. M. Jauch and F. Rohrlich, The Thqory of photons and E l e c t r o n s, A d d i s o n - W e s l e y , C a m b r i d g e ; N 4 a s s l.T 9 5 S . Phrrqik a.lif6.l h.' *S.FlUSge, Springer- VerIaq, 4. Handbuch =:: der :::j-:ji:' -rL'J -' ":' H"iJLrffi

G. KtiII5n, Quantum Electrodynamics

108

i

uuuYd: - i yrrtb,(tc): (Afi (x) 1 z;"' (r)) d.sxU (xr). (24.2) e [ : tptot 1x1 ot(o

J

In (24.2) A(f) @) is the quantized fteld and A?i"'@) is the external field.i The term A,(x)in (2a.1) is thereforethe sum of these two quantities. If we impose the further restrictlon that the external fieid need be considered only to first order (the Born approximation) and that the radiatlon field may be neglected ' (24,I) is then sim-

plified to (q'l S I q>: - eJ d,x(q' lptot(r) | o) 7r(0 I ?(0)(r) I s) A^;""(x).

(24.3)

The matrix elementspresentLn (24.3) are given by (o | lpb)(x) | q): + u?)(q)edc" , vv

(24.4)

where the functions w(')(q)are given explicitly in Sec. 12. Since we are now studying the scatterinq of electrons (and not positrons), we must take the lndex z in (24.4) to be I or 2. These two possibilities correspond to the two possible orientations of electron' spin as was discussed lnSec. 13. For a time-independent, external field

A|i*(*): At""(n),

(24.s)

we now obtain

(q - q'). 2n 6(qn- ql,),Qa. 6) (C,lS Iq>: - t att 1n,yr u?t(q)A7|"" wherethe notation (24.2)

At*e) : ! dsx Af;""(a)eiea

has been lntroduced. Accordlng to (20 .22) , the transitlon probabiltty per unlt tlme is then

: (q.- q,)lr. 2n 6(qn- cL) rn,Touv)(q)A?,""" + : + lav't

(24.8)

: - zn$ao (q)yuuv)@)av) (q)y,un(q,)Aii*@- q,) x x Ai"""(q' - C) 6 (So- qi).

To obtainthe last form in (24.8), we have made use of the reality properties of the potentials A?|"'(*). Since AT""@) are real and Ai*'@) is pure imaginarY, we have 'l (avt 1O', yru?t (q),4i'* (q - g'))* -- ux(r)l1l rrrnu{"t (q')x x,4i""(g' - q) - u*t') (q)ua')(q')1i'o(g' - q) : fi (2a.0)

: _uo)@fuuv,r(q,)Afi""(q,_q).

)

To avold resetting the equations, we @") r4ass for the external field. shall follow the German and use the out-field. from 7(aus), This is to be distinguished

Scattering of Electrons and pair production

Sec. 24

I09

From (24.8) we obtain the cross section for the scattering of the state lq) with polarization / into the state lq,)with polarization r' after dividing by the number of incident particles per unit time

and per unit surface area

d,o : -

#

t

or; wn(q')y, u?)(q)no@)y,ot't(q1) x if x A?,"* _ s,r) (q- q,),Ai""" (q,- q)6(qo da. ] , , n . , 0 , W)*

I

tn (24.10) we have required only that the direction of propagation of the outgoing particle lie in a given solid angle and we have s u m m e do v e r a l l e n e r g i e s p o s s i b l e . T h e l a s t f a c t o r i n ( 2 4 . 1 0 ) i s t h e n u m b e ro f s t a t e s p e r e n e r g y i n t e r v a l d i v i d e d b y V l c . t . ( 1 . 4 ) ] . Because of the delta function, the integral can be evaluated imm e d i a t e l ya n d i f w e d e f i n e q , o : A L : E a n d thenwe lq):lq,l:p, find do e 2E 2 - 1 " , , , , , -ai, : - .t671, ao''(o')r,uo,@)n?) (q')x @)r,,wa't X 1i*'(q - q')Ai"""(q'- q). |

,'n'",

If we wished, we could introduce the explicit form for the funslrons uvtQ) and work out the cross section for polarized electrons. In most experlments, however, the incident electrons are unpolarlzed. That is, we must take the average over both directions of polarization for the incident electrons. In general, the polarization of the outgoing electron will not be measured either, and therefore the sum of the cross sectlons for the two polarizations must be calculated. This sum and average can be worked out directly in Eq. (24.II) without havlng to use the expliclt forms of the functrons ot\')(q), which depend on the particular representation employed. Qtr m a as n. . vc rrrv

nf vt

uf

oo

ao

:- --

Pn

uY

.

11, \r!

''L-"-

.

L2),

we

have

1

t,t | ' -t,\, ' S \ - - - r , )' r, - ' , lq')ltuv) (q)u\1) 1n1 r,,wr'')(q')x A Ar"" (q - q') A?,""" x A?,""" (q,* q) :

(2dr T

(24.r2)

: -

TAd, sp ly, (i y q /n)y,(i y q' m)l x x Ai"" (q - q') Al"'"(q'- q) .

The spur in (24.12) can be calculated immediatety by the use of (22.I0) and the relation

sp[I]:a.

(?a 1"\

We obtain

!r; : -

$

(q.-q') A?" (q'-q).(za.ra) La,,{tq'* m2)- q,q;- qLq) A?,""'

Sec. 24

G. Kall5n, Quantum Electrodynamics

It0

Equation (24.I4) is the desired result. The general result (24.14) can be specialized for different Perhaps most important is the case forms of the external field. Here we have of an electrostatic Coulomb field.

A7"'(n): - fj; na,,r.

(24.15)

We obtain the form in f-space from (24.7):

A 1 " \' Y, (t o \ :- i" 6 " p.4. ^ 7 1 .

Q4.16)

e2

In this case p we have

q'and letting

Q'= q'+q'2-

2qq'-

angle,

@ denote the scattering t)

2 p ' ( l - c o s @:)4 p 2 s i n 2 " .

(24.r7)

After substituting in (24.14) and simplifying, there results

H : |^

'""?" * P*1, l'(' ** i7

(24.r8)

Idxrp"sn- 2l

Tr r hr firc,

fL hr reu rr e v r fvo r er ,e

-

ie

r u

tha

ralaiirzicfin

-onefaliZatiOn

known Rutherford scattering cross section.r then it follows from (24.18) that do dQ

I I

Zt' ^

| 16?IlltSrtj*

Of

In fact, if

WelI

the

f'11*'

,

-12

(24.te)

^Al'

I

where *D&_ r

h2

2m

(24.20)

kinetic energy of the electron. Equation is the non-felativistic (24.L9) is the familiar Rutherford cross section in the Heaviside units employed here. A similar problem, whose solution is to be found from the previous formula, is the probability of pair production b:f gUg-ak, external field. Again if the field can be treated in Born approximation, the S-matrix element of interest is given by equations similar to (24.3) and QA.6)z

Q , s ' l Sl 0 ) : - eI d x ' ( q l r t o t ( x , ) l o ) y , ( q , l v {lo0@ ) . 4, )f i " " ( r , )I: : _

_ 7A0 (q)yrut't(_ S;)A";'"(_ q q,) €

-r..t

,

,

I

In (24.2t) lq')is a positronand lq) is an electron. The index r is therefore equal to I or 2 and the index z' is equal to 3 or 4. Furthermore, the external field has been assumed to be time dependent and we have introduced the notation

1. Equation (24.18) was first derived by N. F. Mott., R o y . S o c . L o n d . , A 1 2 6 , 2 s 9( r 9 3 0 ) .

Proc.

Sec. 24

Scattering of Electrons and Pair Production

,4i*'(8): I d* Ai"" 1x1 e;a'.

I11

e4 .zz)

In (24.7) the integration was only over the three spatial coordinates; here we integrate over the time coordinate as well. According to (i9 .7) the probability for the production of a pair lg, g')

)i aau,s'ls I0)1, d8 d,q' -,r,r-!-ar,) (s)y, uv') (- q') x {rnye

(24.23)

x w?')(- Q')y, ,r, (q) Aii,, ? q _ q,) A,i""'(q * q,) dsq dBq,. The total probability

that a pair is produced at ali is therefore

q'- m)y,fx &, .[I ^r*#sp [(tzs m)y,(- i y

- :

x -4i*'(- S - S')A?""(q * q'):

:

#

(24.24)

d.Q Q)Ai*,e) , I r,,,(Q)-4i*'(-

where

r* " (Q ) : t t +: t'::q ' I": s p[(i zq-m)yp(-i yq,-*)y,]6( Q- s- q,) : :q'' J J : -

) - q , ) , 1 m r ) O ( s , ) O ( Qq-' ) x I d , q ' 6 1 q ' z + ?6n(2( Q x Sp [(i 7 (Q- q')- m) yr(i y q' -t nt)y,f ,

@(q): I ft + u{t)t.

(24.25)

(24.25a)

In (24.24),Eq. (I2.I2) has been employed in the summationover r a n d E q . ( 1 2 . 1 3 ) i n t h e s u m m a t i o no v e r / ' . T h e t e n s o r 7 ) " o b v i o u s l y has the transformation properties specified by the indices p and I and depends only on the vector 0. It must therefore be of the following' general form:

Tr" : A(Q\ 6r,+ B (Q2)Q*Q,: 7,,.

(24.26)

Now from

Qr SpLUy Q - q')-n)TuUr r' I m)y,f: +qil(Q-,r' )z+ *\-\ 1zt. zz) - 4(Q,- q!,)lq'' + *'l ) follows

T,Qu:O:Q"lA+BQ'1,

(24.28)

B (Q\ IQ, Q,- 6r,Q'],

(24.2e)

and hence

7,,: with r

Tuu^:1-3s. r e , . l d a '6 ( q ' ' +* ' ) 6 ( ( Q q ' )+' / n ' ) x x@(q')O(Q- q')sp t(ty (Q- q')- *) yuUy q'+ m)y,l.

8 3,2\:

It,n.,or

Sec' 25

G. Kiill6n, Quantum Electrodynamics

lI2

is the The spur in"(24.30) can easily be worked out and the result following integral tor B(Q2) z

B e\=#

o(q'I@Q- s')8lQq'-*'l=\ , *216(Qz-2Qq') Id.q'6(q'21! (za'st)

- q ' ) J- ' # " ' l ) : - .l d q ' b ( q ' ' m + ' )6 ( Q ' - z Q s ) O ( q ' ) O ( Q tf

WecomputetheinvariantfunctionB(Q2)inthatcoordinatesystem is where the spatial components Qs vanish. This assumes that Q immedifollows it time-like, but from the delta function in(24.31) ' ately that the integral must vanish identically for space-Iike Q find we In this waY

'[]l[a't'4v*Iffi* B(-Q,i=1 [,*

']i.I I

x6(zQrll(,1*'-ai):

i

(24.32)

t7 - .I cw?l - 4m') o(Qo) : ,I f, t -lit * z o(QZ + !&"lz"lf

Therefore in an arbitrary coordinate system we have I n2 m 2rz11t .;,' o (q ol- "0-nz).Qa3z) r,,(Q):+2 n l -lr 6l + Q\ \Q Q,-6,, u v,

in By introducing the external current and using the symmetry probability as total space, we can write the

*:la

[oOrf

lg

Q)Iltotlg'1 ,

Q-

Q4'34)

: +"(, - #)[ +So ? + - *'), (24'3s) (Q,) n@) A1:*(Q), ii*(0) : (6p,Q',-QuQ") Q u l i " " ( q :)o '

(24.36) (24.37)

another In a later chapter we shall again find these equations in -*'') connection. Here we remark onlv that the factor O(- t the obviously expresses the condition that the energy given upby system' coordinate every in greater than 2m external field must be 25. Scatteri.ng of Light bv an Elqctron shall work out the cross section for As the next exampGlwe for the scattering of a photonby an the Compton effect, i.e., The first non-vanishing approximation to the S-maelectron. trjx is clearlY

Scattering of Light by an Electron

"'

( o ' . h ' l S I A .a ')

2 J [[ J

1I3

- r')x or'd*"(q'lrro)(x")lo)y,,S"(x" I I

(x')lq)[(oI Ato) xy,,(olrto) @')lk>
Eig. 7.

The graphs for Compton scattering.

representations of the various factors into Eq. (2S.I), the two rintegrations can be carried out. The result is two delta functions, one of which expresses the conservation of energy-momentum for the process and the other of which allows the y'-integration in the So -function to be done. We finally obtain

(q',k'tslk,q): :';, *

v@a

an@)lr'''lr\|{* ff r, + I L \: .', I ro.

i @ , , ! : )? u r ' )u v )( q \o n \ t 6 b * k - q '+ 1,, T "" lq-h'r+m"' I

rr

k ' \. [ " " ' ' ' )

As in Sec. 24, the two indices r and r'denote the polarization states of the electrons. The vectors e and e, are the polarization vectors of the two photons A and A'. In order not to complicate the calculation unnecessarily, we shall specify the initial state of the electron as being at rest; i.e., {:0, Qa:im. In addition, if the two photons are taken as "real", i.e,, transverse, then we have Qe :

Q e' :

k e:

k' e' :

O.

Using (25.3) and the Dirac equation (iyq!m)u?)(q):0, simplify (25.2) to

tzc . J/

we can

(q',k'lSlk,q):- #r 1)...., uv,(q,)lr r,,' , 2,*q h * y ,y' , 2'.r^1',1*1 ' l (,. ZS '^ ' qh ' I L ' .4) V..' x u v ) ( q ) ( 2 x i 4 6 ( q+ k _ q , _ h , ) .

l

The conditions q 2+ / n 2 : A 2: 0 have also been used in the denomlnators. With the assumption that the polarization of the electron is random prior to the collision

Sec. 25

G. Ktiil5n, Quantum Electrodynamics

II4

and that both polarization directions are observedl afterward, the transition probability per unit time for the process is

uJ:

spt"'l (27t)46t-+n-q'-k'), (2s.s)

o : ^ r " . t -,'| ,

according to (20.22) . Here s p L . . _. l:

i.vh

lt s -pLl {\ ./ e y' 2eqLh

itth'\, ) + y e y,e ' -2- q' ,n, ' l,l"t'y g - m ) x I I t

]

iYh ^ \ r i ' i , y e 'y e + ; ; ; y e ye ' l( i yq '- * ) ] . u

[iYh'

(25.6)

|

)

cross section for the outgoing photon:

This gives the differential

do - eL t f a'q. A a - - n 8 " , * r J E ' JI-r , a r , S p i -. . . ]: -6\(' k- - q , - k ) 6 ( m l a - E ' - a , ) . ( z s . z ) over'da4'and da'can

The indicated integrations the delta functions: do

e4

d!)n,

1287*

be done by using

(25. 8)

a'2 :n[...1 ' I m2 @2 "Y L

The spur (25.6) should therefore be evaluated under the restriction

(* + ^ - r')' : m2I (k - lt'')',

( 2 se. )

kk' : kk' - a a)': n (.' - a).

(2s.10)

From the conservation of energy and momentum, it follows that the energy ar'of the scattered photon is a single-valued function of the scattering angle. Therefore we can give (25.8) as a function of only ar and ar' . To work out the spur (25.6), we write

y o r t ' r I ' ^ -l y e yt

#

- y e ye 'i y a t z r t '

]?,

(2s.II)

with

* -

h'h

-

?sH

'sh

\LC.tL)

The vector a has the property that qa:O.

/tc

lll

Now we break the spur into four parts

sp t...1: sp [I] f sp [II] f sp [III] ] Sp[rv] ,

Qs.14)

where sp [t] : sp A t y e' i y a (i y q -m) i y a y e' y t (i y (q * k -h')-m)], (zs.ts) I. Explicit equations for the various possible polarizations of the electrons have been given by W'. Franz, Ann. Phys. 33, 689 ( 1 9 3 8 .) S e e a l s o F . W . L i p p s a n d H . A . T o l h o e k , P h y s i c a , H a a g 20, 85, 395 (1954).

Scattering of Light by an Electron

Sec.25

lls

'rrnl (q sp[Ir] : 4(ee,)zspl'rtruu Uy q - *) Uy + k k) _ 4],,(2s. 16) sp[Ill] : sp[IV]:

I

- r; e') splyey e'i y a(iy q- *)',Iou(iy (q+ h- k') - -lJ (zs.r7) ) n d ( 2 5 . 3 ) , ( 2 5 . 1 5 s) i m p l i f i e st o U s i n s ( 2 5 . 1 3a S p[ I ] : - S p [ ( i z s + l n ) y e y e ' i y a i y a y e ' y e ( i y ( S +k-k')-m)l=1 : a2Sp l(i y q-f *) (i y @+ n- k')- *)f : a2Spli y q i y (h-,t;)l= [ trr.,rt - - - 4 a 2 q ( t- k , ) : 2 ; j ! : 2 9 ! l ' q n l @k'I Likewise,

we obtain

s p[ r r ]: - 4 ( a a s, )p, f' # o r ( s - k ) ) : 8 @ e ) _ z (ot f i 1 ,tt. t n 1 Sp[III] * Sp[IV]=41ee')Sp r t + m)y ey e'i y aJlI i y k'f: ft; l . ''o' 'r'nnu, 'r,iI onu; :4(e e')zsolty q iy k'):8 (ee')2 .l"t n aanrAinnl.' vrrryry nvevl

,

S p t . . l: 8 ( e e ' ) 2 + r #

(rE, t1\

U s i n s ( 2 5 . 2 I ) , E q . ( 2 S . 8 )b e c o m e s

++k,),] (2s.22) :;_:(ftf : ;,e;fl@_a'), If we introduce "Klein-Nis.hina

the scattering angle @ , we obtain the so-called formula"r for the Compton effect:

do I lr'P Zou,:\ 1" ) 7[- +." (r=;onr

I

a2(r-cosOf

l;67

.1, _

I4(ee,\rf.e|.Z3)

"o"o11

I

In many applications, the directions of polarization of the incoming and outgoing photons are not known. Then we must sum and average over the directions of polarization of the photons in a manner similar to that used previously for electrons: If

(2s.24)

(ee'\2: * (l * cosz@)

^,)",

do I e2\2t' --:f ( ' , " o ' @ ] l u*,1- * c o s 2o l . e s . z s ) 707=\4n)z@1;n -;;6yL;,W= ' 1 +;G-""o)t l With a view toward later results, we shall. Iook case of (25 .25) when atlm(l . We find

at the limiting

1 . O . K l e i n a n d Y . N i s h i n a , Z . P h y s i k 5 2 , 8 5 3 0 9 2 9 ).

G . K a l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

116

#--GirY t(t * cos'z@)

Sec. 26

(2s.26)

is the so-called classical electron radius which *; we shall denote by ro . Equation (25 .26) agrees completely with the result that is obtained from classical electromagnetic theory. The quantum mechanical effects fjrst show up when the energy of the incident photon is of the same order as the rest mass of the Integrating (25.26) over the angles, we find electron. The quantitv

o,tot:

8n, r-o: OT t i

(r\

r7\

where oa is the so-called Thomson cross section. The complete expression (25.25) can also be integrated over As the The calculation is elementary but tedious. solid angles. ra cr r'lt

uro

nlrf

a i n

1+1. 4(t+1)2 ^t. ^.,t ( t + 2 ^ ) \ ' t ,- 2 . i " ' ) * t , i ; + i t t 6tot:rrfr ftog

2 ( 1+ 3 1 \ I 1 1 6 . . r

( , _ z - i 1) , \ 2 5 . z v 1

with

) . : 9 nt.

(2s.2s)

I f , 1 b e c o m e s v e r y 1 a r g e , t h e c r o s s section in this aPProximation goes to zero at a rate given bY

n4 * 6 to tN

l"*

( + 2) ,)+ :l

/r(

?n)

As we shall see later, the higher terms in the S-matrix have no This is to be expected because effect on the result (25.27). For very high effect. here we are dealing only with a classical to (25'30) corrections higher-order energies, on the other hand, the meaning' (25.30) its loses actually result the very large and become However, the energy at which these corrections are evident is so on the region of validity ot (25 '28) large that these limitations The Kteln*Nishina forhave no experimental interest at present. For up to now' by experiments all fully confirmed been mula has which has the theory, we refer the reader to the book by Heitler, already been mentioned,r and to Vol. )OC(W of this handbook'2 26. Bremsstrahlung and Pair Production bv Photons in an External Field we consider ar ttr. next, slightly more involved application, the Born approximation to the cross section for bremsstrahlung' Here the process is that an electron, passing through an external emits a photon. As before' we electromagnetic field Ai:"(r), field and the radiation the external of both account shall take I. See, for examPle, footnote l , p . 1 0 7 . 2. Handbuch der Physik, edited by S. Fliigge, SPringer-Verlag , Heidelberg.

Bremsstrahlung and Pair Production

Sec. 26

II7

field only to first order.l With this approximation, we obtain the S-matrix element of interest from (25.1) by replacing the photon which is absorbed by the external field:

Q ', kls l O > : - t l l O .'d x" (q 'l tp to \1 x" )l o)y,,Sp( x"' - x,)"y",( oLV@@,) lg l (26.r) (x')lo>Ai:'"(tc,,)f xl : 4 -yJg: u b l t L. t , _ 2 q h t/ ,uef r ry tr i v ( q ,' *qh, )h- - y "nl ,xl l ' lfzot | 06.2\ x uvt (q) A?"""(q - q' - lt) 2n 6 (qo- q'o- ko). l As before, we sum over the two possible polarizations of the final electron and average over the polarizations of the initial electron. After a few transformations, we can write the cross section for the process in the following form: ,

do :

eL

b'

s e nf ;

da

;

,, dQ e dQ o;To,A?,* (q- q' -lg) Ai"* (k* q, - q) . (ZA. Z)

In (26 .3), P and.trt'stand for the absolute value of the momenta g and g', while al is the frequency of the photon, a:lkl. The solid angles of the final electron and photon are d,[)0,and ll)u and 7), is used for the quantity

(26.4)

The spur (26.4) and the cross section (26.3) must be evaluated subject to the condition

E:l,pryorz: Vp,r*mrIa - E,*a.

(26.s)

Equation (26.5) obviously expresses energy conservation for the bremsstrahlung process. There is no momentum conservation because the external field can take up an arbitrary momentum. This is expressed in (26.2) by the fact that we have only a single delta function for the time component of the vector q-q'-k. The spatial component of this vector, which we shaLl calt p from now on, enters as the argument of the Fourier components of the exl. Por a more complete treatment of the external field, see H.A. ; . Davres, B e t h ea n d L . C . M a x i m o n , P h y s . R e v . 9 3 , 7 6 8 ( 1 9 5 a ) H H . A . B e t h e a n d L . C . M a x i m o n ,P h y s . R e v . 9 3 , 7 8 8 ( 1 9 5 4 ) .

IIB

Sec. 26

G. Ktill6n, Quantum Electrodynamics

ternal field. If we replace the polarization vector e by the vector A in one of the brackets of (26.4), the expression can be transformed as follows: ivb-h\-ut 1, F1. . # y *2qk

--'Y - pi -. qr t' / r- ,L /n", -- ^It p 7 ' "^ '

, i v h ' '* h \ - m . ' ;: R 2q'h

R-rY

.

tt b

il+

-

m

*

zqh

-Y ' F. .

-i nY qV' hl

-; 7_ ittTt a -

m

TRTp:

(zo.o,

i^ta'*m

ivo*m

L^tl2^t

b---:---:----:--

2q'h

2qh

t"IP'

The factor iyq+* In (26.6) we have used the fact that ft2:0. c i e a r l y g i v e s z e r o i f i t o p e r a t e s o n t h e f u n c t i o n u v ) ( q )i n ( 2 6 . 2 ) o r in (26.4). In a similar way, the factor on the factor iyq-m gives zero everywhere and we see that the tensor Q, in iyq'*m (26.4) vanishes if one of the factors e is replaced by A. This is actually only a consequence of the gauge invariance of the theory and can be shown in r-space by the use of the continuity equation for the current operator ilo)(x). If we are not interested in the polarization of the photon,'this In will allow us to simplify the evaluation of (26.4) somewhat. view of our previous remarks, we obtain the same value for the spur (26.4) if we take the polarization vector a as longitudinal or scalar. Therefore we have

s I

L'tr, transv.

...2 t. ;",h _h\ : - "Sr lo\l/lt yt . ' " _ 2 q h

n

2

i y ^I ^' .-| ^I ^, . "

( q ,l h ] ; _ m

2 q. i ' i "

" ' y "\)(, .i y

\(26.7)

Pbolons

I

y t yt l _ _i_^:t_(_o:-_h: _\ -__w_. '_ _ _ _ ^ , /\\r^ I' | /' -2qh The expression identitie

)

s - m ) x lI

(26.7)

i v' l s- ' + h \ - m 2q'h

can be further

\,. 7tl\xTq

simplified

-nL) ' 1

by

means

| | of the

s

TtTpTt:

-

(26. 8)

2Tp ,

( 2 6 .e )

TtTpT,Vt,:4dp,.

The Eqs. (26.8) and (26.9)can be proved immediately from (12.2). In principle, further reduction of. (26.7) does not involve any new n r nv vhr vl o m r r r vq .

yr

\A/c f hcrcforo

oirre

onl v f hc

rAsult

:

1. The cross section for arbitrary polarization of the photon has been worked out by M. M. May, Phys. Rev. 84,265 (1951) a n d b y R . L . G l u c k s t e r n ,M . H . H u l l a n d G . B r e i t , P h y s . R e v . 9 0 , 1030 (1953). For polarized electrons, the cross section has been given for a few specialized cases by K. W. McVoy, Phys. Rev. 1 0 6 , B 2 B ( 1 9 5 7 )a n d K . W . M c V o y a n d F . J . D y s o n , P h y s . R e v . 106, 1360 (I957). General results for polarized electrons and phoi6is witl be found in A. Claesson, tuk. Fysik 12,569 (1957).

Sec. 26

Bremsstrahlung and Pair Production

119

Bp, , Ct,, l 1q,n1rTqn.q'nl,

(26.r0)

.r \tp!_ L tr2-nsv.

u2 lAp,, 2 ltshf-

Photoos

A r,,,: 4 (*, * k q) lkpq; + k,ql - 6p,(k q' * mr)) - 4m2lsrq', + q,qL- 6r,(g * 2mr)l Q, , : B ro 4 (*' + k q') lk, g,l k, gp- 6r, (k g rn')l - 4m2lqrql,+ Q,Qp- 6r, (Qg' -f 2nt2)l ,

C r , : - 8 n ' rk' rzk , + 4 k q ' ( 2 q p q ,q+p S+: q , q ' - 2 6 t , , l l ' )-4kq(2qrqi,+qLq,*q'"qp-26p,SS')+ - q,)- 2quqi, - 2r,L+ * 4qS'lk,(qL- qp)+ kt,(qi, -t t 26u,@q' m"))

( 2 6. 1 1 )

(26.r2) 1 t (26. t3) [ )

T h e c r o s s s e c t i o n m u s t b e i n d e p e n d e n to f t h e c h o i c e o f g a u g e o f the external field. In 1-space this means that the tensor I, must satisfy the conditions

Tr,(q,- q: - h,): Tu,(q,,-s; - hu)- o.

(26.r4)

Equations (26.I4) can be verified either by means of (26.4) and a transformation similar to (26.6) or by the use of the explicit forms (26.I0) through (26.13). This last proof can serve as a check that no algebraic mistakes have been made in the rather tedious reduc.tion of the sprir (26,7). These considerations about the use of the gauge invariance of the theory could also have been used as a check in the previous sections: in the summation over polarizations of the photons in Compton scattering or in (24.14). In the Compron scattering it must be remembered thatEq. (25.3) isnot satisfied for longitudinal and scalar photons; consequently the transformation which led from (25.2) to (25.4) cannot be done if the summation over transverse photons is carried out as in (26.7). From (26.3) and (26.10) through (26.13), we can find the cross section for bremsstrahlung in an arbitrary external field if the field is weak enough so that the Born approximation is meaningful. In most applications the external field is the Coulomb field of the nucleus, and for this case the general equations specialize to

Ai,"*(g):-i6rn6,

d,:#ffif ++lffidehdeq,

tzo.rD,

( 26 . 1 6 )

T h e c o m p o n e n t Z n nc a n b e o b t a i n e d f r o m ( 2 6 . I 0 ) t h r o u g h ( 2 6 . 1 3 ) :

I20

Sec. 26

G . K a I I 5 n , Q u a n t u mE l e c t r o d y n a m i c s

- TEr:=T#+E

@'I E')llmz{ a (p'cos h##_E

x

x l E ' z - a p ' c o s @ ' *p f ' c o s $+ r / f ] +

+ E))- V#_W + =V:h-+ r lmzr a (pcoso

x

x lEz - rttp cos@+ p p' cosfl | m2) *

(26.17)

+ z!ffi;a@+E')- =#3(E +E')+ 4(EE' - P P'cosil) (ru t' a 6' -$) '

- o*','

( E - P c o s@ ) ( E ' - P ' c o s @ ' \

I n ( 2 6 . 1 7 )t h e a n g l e s @ , @ ' , a n d d a r e d e f i n e d b y qq':pp'cos$

(26.18)

,

qk:atPcos@,

(26.19)

q ' l c : a p ' c o s o '.

(26.20)

Rather than use tl , we introduce the angle @ between the and (g', lc) planes: cosd : cos@cos@' -l- sin @sin @'cos@.

(q, It)

(26.21)

Then (26. 17)can be written in the form given byBethe andHeitler,l -TEt:

' ffi 9N ,1E 'z- g z1 *

sin2 @

P'2 sinz@' (E - p' cos@')z

p p'sin@ sin@' cos @

(E - p cos@)(E' - p' cos@'\ b z s j n z OI b , 2 s i n 2 @ , (E - pcos@)(E'- p'cos

[) F2 -L ) F'2 \ 2 " | 4 "

(48 - 8')

-Q',)+

-l,,^,,, J'-"--

-k), has often been used, I n ( 2 6. 2 2 ) t h e q u a n t i t y Q , : ( q - q ' r a t h e r t h a n p p ' c o s 8 . F o r t h e c o m p a r i s o n o f .( 2 6 . 1 6 ) a n d ( 2 6 . 2 2 ) with experiments, and for integrations over the angles, etc. , we refer the reader to Heitlerrs book.l The cross section for another process can be obtained directly from the previous calculation: the production of an electron pair by a photon which goes through an external field. The corresponding element of the S-matrjx is . / ^ n ,t te
'zq (x") lk> A1,.i* ,41,1) (*')| o)r[(01 x(q' I ?tor @')+ (01,4,:)(*') lk) A?,i"@")1.'lrzb

l. H. A. Bethe and W. Heitler, Proc. Roy. Soc. Lond. AUg, 83 (1934). See also Heitler, Quantum Theory of Radiation, Third Ed. , Oxford, 1954.

Bremsstrahlung and Pair Production

Sec.26

I2I

In Eq. (26.23), lA) denotesthe incident photon, lq) the electron p r o d u c e d , a n d l g ' ) t h e p o s i t r o n . R a t h e rt h a n ( 2 6 . 2 ) , w e f i n d

k,q'IsID: # 6 r" otlv,w#fl

y,+y'!3=#

r,)*l

x u ? ' ) ( _q , )A l : * ( k _ q _ q , ) z n 6 ( a _ E _ E ' ) .

.,nt 1ru

)

As before, the index I ia (26.24) is 1 or 2 and the index /' musr now take the values 3 or 4. Inworking outthe cross section it must be rememberedthat the density of final states is now different, that in the sum over spins inthe final state, (12.13) rather than (I2.12) must be used, and that the possible polarizations of the photon must be averaged over. Carrying out all this, we obtain ?4

do : n 1r;,, "*i' '

bb'dE

dQodI q,Tll A";* (It- q - c') A?",,(q * q' - tt),(z6 . zs) ,n

'n

-/ir

/6!,cc,r

rl'): +rn[(, 2@'-,!)+ tnr^+ r^ffr,)

" Q}!ffiv

r,* y,tW

Qt,q,+ /n)x

Ur t - *)] r^)

If we compare(26.26) with (26.7), we see that rjlt{k,t.q'): -Tr,(- k,- q',q): -Tro(k,-

,u, ),,u

C,q'). (2G,zT)

W e c a n o b t a i n t h e n e w s p u r f r o m ( 2 6 . I 0 ) t h r o u g h ( 2 6 . 1 3 )i f w e r e place g by-geverywhere, or, what is the same thing, E by -E and @ bV n-O and change the overall sign. In particular, for the special case of pair production by a light quantum in a CouIomb field, we obtain the cross section

o": &(*)'

?#L deq dpq,[&],

e6.zs)

with

9:lc-q-q'.

(26.30)

Equations (26.28) throush (26.30) as well as (26.16)and (26;22) are well confirmed by experiment, at least in the energy region where the Born approximation is expected to be good.r t . W . H e i t l e r , Q u a n t u mT h e o r y o f R a d i a t i o n , T h i r d E d . , O x ford, 1954.

G. K:i115n,Quantum Electrodynamics

122

Sec.27

Scatterins of Two Electrons from Each Other We now consider the problem of two incident electrons with energy-momentum vectors p and g which collide with each other and emerge again with energy-momentum vectors p' and q' . For initial and final states we therefore have quantities of the type 27 .

( 2 7. r )

{q)lo) . lf , q) : a*?)(f) o,*t"'t

Since the operators a*(') (p)arld a*(/') (q) anticommute with each other, it is clear that

( 2 7. 2 )

lf,q):-lq,f)' Thc annronriaie r

rrv

syyr

vy!

+Yev

(q',b'lSlb,D:-'^

.S-matrix element is

v

e2

.+

f

f

, l d x ' d x , " lL(\q1 ' lI t' t t \ o \ ( xl 0 ' )) 2 " , ( 0 1y t o ) ( x ' ) l q x)

l

2JJ

I

x < p ' l \ @ ) ( x " ) I 0 ) 2 , , ( 0 1 V ( o ) ( x " ) l f ) - ( q ' l . r p ( o t ( x ' ) l o ) y , , ( o l V ( oL) (^x_' ^) l,p ) x 'r) (x") l1)f 6,,,,Dp(x'- x") : [\Lt x
A j-r- : - : l a u . + q - b-' _ x' II yea,1,y +' fW_ q,l,y+ 1paffin1- ,

) I

-n ' \). 1 ( " ' n )

Here we employ the abbreviations

-4 : Sp ly^(i y p - *) y, (i y P'- m)l' Sp [2,(i y s - @ y, (i y q'- *)), (27. s) B : S pl y ^ ( i yP - d y , ( i y q ' - m ) f . S p l y ^ ( i yq - m ) y , ( i y f ' - m ) l , Q 7 . 6 ) c : - 2 sp ly, (i y f - m)y, (i y q' - m)v^(i v s - m)v (i v F' - *)) . (27' 7) " The quantity

?."1 irr (27 .4) is the

"relative

velocity"

of the two

1. For longitudinal polarization of the particles the cross sect i o n h a s b e e n c a l c u l a t e d b y A . M . B i n c e r , P h y s . R e v . 1 0 7 ,1 4 3 4 .

0ss7).

Sec. 27

Scattering of Two Electrons from Each Other

I23

incident particles. That is, it is a quantity which, in a coordinate system where one particle is at rest, reduces to the velocity of the other particle. For our purposes, it is not appropriate to define this quantity as the velocity with which an observer in the rest frame of one particle seesthe other particle approach. Rather, we shall define it as a quantity such that a,"tEp Eqis an invariant. By this, we make the scattering cross section itself an invariant. In the rest system of the particle / we have

a , . 1 E o E om-E e t | t ": * l E 3 -

*r:l*rEtr-

*n.

(27.8)

We therefore define

? r e:r p 4 . 1 1 ' p q l z - * a . "P's

( 2 7. e )

In a coordinate system where the two particles are moving u, and u2 , we have same direction with velocities ater:.:lur-

in the

( 2 7. 1 0 )

a2l.

The expression (27.I0) can reach the value 2 under certain condifr ir nv n q t L p

tha+

r v r r r Y r r u .

iq

dnrrlrlo

lha


Because of the conservation we can, for example, express energy E4 ds functions of E' incident particles. In order to introduce the three parameters

nf

lich+

of energy and momentum in(27.4), the two scattering angles and the and the quantities describing the do this in an invariant fashion, we A, , ,1, , and 7 by

^,: t2- 4L:

- (r + @*n) ,

^,:l!_4L:_(r +?#), y--ff:t*htAz.

(27.rr) \z / . IZ)

\zI.rJ)

The quantity y is therefore given by the incident particles and (27.13) expresses i2 as a function of L . We now ciefine an invariant differential cross section by

ddtfi +) ,-1* 1 l Ao7l+Zn+4B | i + C A , 1 2 f . r . dl,:ffi

(27.r4)

The invariant integral 1 is given by

I:

If

JJ

d3:'d:s'6(b-rq-b'-s'ta(A,--@-l')'): Lp'Lq'

r

at-\-'r

2mz

)

I

I I r^- '-r

f f

= q l l a p ' a q ' 6 ( b ' z t n 2 ) 6 ( q ' , -t*- r\ r)t o ( f ' ) o1("q1 '' t) 6 ( p * s ?- ib-,\-" q )a(1,-J!:-P'rlrzr'ttt L -\1 JJ

r

z*Tl,l

As in (24.32), the integratlons present in (27.15) may be carried out easily in the coordinate frame in which F:0. The result is

124

G . K a l l e n , Q u a n t u mE l e c t r o d y n a m i c s

Sec. 27

z* mr)@(!') a ((p+ q- p'),-t mr)o (! + q-p') xl r : 4 I dp'61p',

- (p- p'Y) :'iri o( ) o(P;- t - ^,). x o(,r,

Consequently,

in an arbitrary

coordinate

system we have

t" o0,\o0,\. llY'- t

I:

Ie7'16) (17

17\

The three quantities A, B, and C in (27.5) through (27.7) can easily be expressed as functions of the three invariants .1, , A, , and 7 by methods given previously. After some algebra and using the following equations (which follow from the conservationlaws) :

Pt:P't',) Pq':f'q,l

lc'7

1a\

fh':qq',) the result may be written as

,*fe 64m"

'

1?r)-1r).r(zy-r- ArAr).(27.rs) 4 + n 1?+ c LAzl:yr(A?r+

For the differential cross section (27.14) we therefore obtain do -itr.-

2ntfi -y2 J

1

7

qlne

l,Ltilv'Q?'+

. t. - t - L,1,)) ' 1Z)- .1r)'r(2y

"^

Q7 '20)

In (27.20) the two ),i are positive and related by Eq. (27.I3). The length ro has been defined bv (25.26). The expression (27.20) was first derived by Mfllerr and therefore the problem discussed here is usually referred to as "Mfller scattering". Equation (27 .20) is thoroughly conflrmed by comparison with experiment. We shall not go into further detail* however, such a discussion may be found in various textbooks. z We shall make a comparison with the formula (24.I8) for the scattering in an external field because it illustrates some matters of principle. For this, we limit ourselves to the non-relativistic case. Here, in the center-of -momentum frame, Eo: Ep- Eo,: Eo,: * I

p + sr:p'+8':0. We introduce the scattering

, "nL

C.MdIIer, Ann. Phys. 14,53I (1932).

2..

For pwamnlc

F.

?1\

( 2 7. 2 2 )

angle @ by

I.

N.

(t7

MOtt and H.

S. \M. l\4asscv-

Thaor"

^r

A t o m i c C o l l i s i o n s , S e c o n dE d . , O x f o r d , 1 9 4 9 , p . 3 6 9 . S e e a l s o A . A s h k i n , L . A . P a g e a n d W . M . W o o d w a r d ,P h y s . R e v . 9 4 , 3 5 7

(1es4).

125

1\atural Lrne wloln

Sec.28

pp,:pzcos@

,

(r 7 )'1,\

so that ^ At:

2b2 SIn'" @ , mz 2

( 2 7. 2 4 )

A" z :

212 C o S" '@ -. rn,

( 2 7. 2 s )

In this approximation the cross section becomes lt7

Witlr Z:

)A\

l, and using the present notation, we can write (24.19)as -

7t

?/t4

o

t r o : - . /"i - s n o d , O , ti

.

(27.27)

"(v

tn (27.27) the polar angle @ has been integrated out, d Q : 2 n s i n @ d @.

lon ao\ \Lt..et

O n e m i g h t w o n d e r w h y t h e t w o e x p r e s s i o n s ( 2 7 . 2 6 ) a n d ( 2 7 . 2 7 )d o not agree. Formally, the two "extra" terms on the right side of (27.26) arise because of the second term on the right side of (27.3). If only the first term in (27.3) had been considered, the definition of the scattering angle (27.23) would have given exactly (27.27). We obtained two terms in (27.3) because the two states lf , S) and lS,D cannot be distinguished from each other. In other words, it is impossible to decide, for example, whether the electron p' was originally the electron p or the electron g . If we arbitrarily determine the scattering angle bv Q7.23), i.e., we consider 1 and 1' as the "same" electron, then there is a certain probability that the two electrons in the final state have been "er thanged". We can then denote the "extra" terms in (27.26) as the exchange terms. Obviously, it is equally valid to denote the first orthe second term in (27.3) as the exchange term, provided that the definition of the scattering angle is in terms of the appropriate electrons . The non-relativistic form (27,-26), including the exchange effects, was first derived by Mott.r 28. Natural Line Width2 -As the final example, we shall consider the emission of liqht by an electron in an external field which is strong enough so that the Born approximation cannot be used. Atypical case is a timet . t ' l. F . M o t t , P r o c . R o y . S o c . L o n d . .A 1 2 6 , 2 5 9 ( 1 9 3 0 ) . 2. See also the article written from the experimental point of view by R. G. Breene, Jr. in Vol .XXVII of this handbook(Handbuch d e r P h y s i k , e d i t e d b y S . F l i i g g e , S p r i n g e r - V e r l a g ,H e i d e l b e r g ) .

126

Sec.28

G. Kdll6n, Quantum Electrodynamics

independelt external field with one or more bound states. For this we shall employthe method used in Sec. 16 for treating the external field, i.e., we assume that the solutions of the eigenvalue equation

|

,

a ,\ e A, "u', @ ' l \ l' u ' l )*l m y o l e A n @ " ' " ( r- ) E , u , l r )

l a n l - t , o_x' .k. t "'

"'

(28.1)

are known. In general the functions uo(r)range over both bound states and scattering states for the electron. We take z,(r) as the in-fieIds, i.e. , the incoming g -field is developed in terms of the eigenstates zs(r) which are given by (I6. B) with the operators ( 1 6. 1 1 ) a n d ( 1 6. 1 2 ) : V @@ ) ):

2

Eo) o

1 " " ( u ) e - i E n t xa @ )|

u , ( : r ) e i E n r ob * Q ) f ,

{a*(n\,ab'ty:{b*(nt,bl"'\}:6n,..

(28.2) (28.3)

As before, we treat the electromagnetic radiation field as a perturbation which causes transitions between the states un(n).Using the procedure given in previous sections, we obtain the first nonvanishing S-matrjx element for a transition from un(n) to wr,(n\ with the emission of a light quantum A : ef

(n' ,klS ln): - ,| | d,3xu*, (r) y,aq(n) t-ika t(x)2n 6 (E,,I llzV ot J

o-8,) . (28.4)

per unit time , with the photon emitted The transition probability into solid angle dQo, is therefore

y : #l!

a,ra,,1n1y e@ u,(n),-tt"*12 den.

( 2 8 .s )

If the initial and final states of the electron in (28.5) are both bound states,the whole expression is independent of 7. That is, the transition probability per unit time cannot be made arbitrarily smaII if only V is sufficiently large. Consequently, the formalism developed in Sec.20 cannot be correct, at least in the higher orders, since the transition probability is not time-independent. In order to proceed further, we have to modify slightlythe inso that it is tegration procedure for our differential equations, not required that the lifetime of the initial state be arbitrarily Iarge. We shall restrict ourselves to an approximation which was originally developed by Weisskopf and Wigner.r A more general theory of these processes, in which higher orders can in principle be considered, has been developed by Heitler and coworkers.' Because the initial state is now"short-lived"we cannot prescribe

l . V . F . W e i s s k o p f a n d E . W i g n e r , Z . P h y s i k 6 3 , 5 4 0 9 3 0 ). 2. E. Arnous and W. Heitler, Proc. Roy. Soc. Lond.l.!29, 290 0953). This paper also has referencesto the older literature. Seealso F. Low, Phys. Rev. 88, 53 (1952).

Sec. 2B

Natural Line Width

127

the initial conditions of this problem for xo-- - oo for, if we did, the initial state would no longer be occupied at finite times. Following Weisskopf and Wigner, we shall require that our system be in the state lz) at time to:0 and we shall use an "interaction picture" to treat the problem. This picture is defined by the following equations:

lx) :

U ( x o )l n ) ,

(28.6)

t aurX"\: H,(!,,0)(x), l,f, (r)t)u(xo),

(28.7)

u(o): 1.

( 2 8 .B )

Here l*o) stands for the state vector at the time ro and ff, is the interaction energy as a function of the operators v(o) of (28.2) as well as AP(*) of, for example, (S.28). The essential idea in this approach is actually only the boundary condition (28.8). In principle, the system of equations (28.6) through (ZB. g) is exact. However, following Weisskopf and Wigner, we shall approximate these exact equations by considering only the states lz) and ln', k) in the matrix multiptication of (28.7). That is, we consider only those states which are either the electron in the initial state and no photon or the electron in the final state and one photon. The physical significance of this approximation js not easy ro grasp. Certainiy it is clear that the probability that several photons should be present in the final state must be small, since the coupling with the electromagnetic field is weak. It does not follow directly from this that such states can also be neglected in the matrjx multiplication of (28.7). At any rate, the suitability of this approximation is probably best decided by comparing the results obtained with those of experiment. We define

( n l U ( x o ) l n ) : a ( x o ),

(28.e)

(n', klU(xr)l n) : bn@o),

(2B.ro)

so that the equations to be solved in this approximation

,i r:l1t v^o

,+:1L: w^o

are

: Z ,t ,*i / ursbe@n),

(28.r)

chei/-*,a(xd.

(28.r2)

T

Here we have used the following

notation:

Aot:En, la-En, (n', klHrl n) :

c h e i / @ ' oI

ca: -; ela-,(r)ystt) u-{n) ffia"x.

T?R I?I

(28.14)

(28.ls)

I28

Sec. 28

G. KdII5n, Quantum Electrodynamics

The boundary conditions for (28.11) and (28.I2) are obtained from (28.8):

a(o):r,

( 2I . 1 6 )

b u ( 0:16 ' To solve these oliminafo

tho

differential

(28.r7)

equations we can,

:for example, first

h. . ,o

b e @ o:) - i c hI d x Le i z ' " 6o ( x t ) ,

(28.i8)

0

3!J1t : ?ro

-

a(x'). I 1ru1,f a*'oe-1a@\'o-r'u) u i,", 1

The equation for a(xo) can then be solved, We set of the Laplace transformation.

for example,

(28.rs) by means

@

a ( E ) : I e - E ' , a ( x o )d x o

llq

tn\

0

and transform (28.19) into an equation f.or a(E). we obtain

[ ,-u'"

u"uYt

Jolo"

For the Ieft side

d.xo:le-E"o a(xo)]tr | E a(E): E a(E)- 1.

QB.II)

ff,i int"grul on the right side can be transformed in a similar way:

axoI ax'or-i/a(ro-r'u1 a(xL): I t-,'" :

f

r rt

,,

J oti

f

(28.22)

- t\E + i / @ ) r o + d / @ t i , - _ _ ! J ! \

t

o*oa\xo) axoe J

E{ilttt'

S u b s t i t u t i o n o f ( 2 8 . 2 1 ) a n d ( 2 8 . 2 2 ) i n t o ( 2 8 . 1 9 )l e a d s t o t h e r e s u l t for a (E) , ,\p,-------------f_.

11) By

means

of

transformation,

the

well

known

__1j41" a E+iaa

formula

f6r

Q8.23) inrzorl-inn

rho

Tanlace

we have e*i o

I

dE f a\xo): *, 2 1 t x II o _ \ .t

eEto

Iril,

"-+E_tZ;

(28.24)

In Eq. (28.24) the integration is along a path immediately to the right of the imaginary axis in the complex E-plane. If we introduce a real variable of integration z by E: eliz, then the integral (28 .24) can be written in the following way:

Sec.2B

Natural Line Width f,2 siz rn pe16

,

a \ x o ) : r " o li*

-?J

t

'n'

_

z-Le-

129

:

i "+t--;'

lCa

OE,\

dzeizro

f

e)

J,-

-@

hl

oal z. l+' A

- i n l , l c e l z 6 ( z- / a ) a'''

T h e l a s t f o r m o n t h e r i g h t s i d e o f ( 2 8 . 2 5 )f o l l o w s i f w e l e t e q o t o zero and use the relation

(28.26) The evaluation of the integral (28.25) for arbitrary times is quite complicated and not very interesting physicatly. Because the electromagnetic coupling is "suddenly switched on,, at ,0:0, it is to be expected that there wilt be "transients" for small times which will depend strongly on the choice of boundary conditions. These are therefore uninteresting. Consequently, we shall estlmate the integral only for times so large that xoBn)l , and xoE*2 7. With the additional restriction that

X lrul,6(/co)a En- En,,

(28.27)

we can write

a(xl x

dz e1zto

2n " i ,,

[-

lt

, -, Z+: l/,

-.i,ll

celz 6(/a)

: ,-i/

Ero- t

y :2nzlchPA(a@ , j -

1",t2

A E : - P ) ' r " :-ir

o.

",, {zg .2g)

(28.2e) (28.30)

The probability that at time ,0 the system is still in the inltial state is therefore I a (*o)l' :

e-r to

( 2 8 .3 1 )

The lifetime of the state is therefore tfy, where 7 is just the total transition probability per unit time, calculated from (28.5). For times such that the result (29.28)is correct, we obtain from (28.i8), -ck

be@o): laFor small times (7ro({)

/n a t?-

-rf . lr1o'-no'-Lr)."

eB.3z)

we obtain from this the probability that a

r30

c . Kii115n,Quantum Electrodynamics

Sec. 28

photon has been emitted into solid angle d.Qp, @

d a , I : : ; t b e @ )d t zo t : J '-""' 0

'-' ' I'ol"tlt ")'""\ : dQ" " I I

(28.33)

I a'ot xo1

F

\2ilf

{

) ,... N o*

1l't1' ' /" \2)-16

| ^.||2.,,2 fl 2xt,x^ | "*,1:-'

"

\zfl)o

I

I dQ,.

| A'@:O

"

Here we have used

A'a:

Aa - AE.

(28.34)

Apart from a shift lE of. the emitted frequency, this is just the result (28.5)with a transition probability proportional to the time. According to (28.31), during this time the probability that the system remains in the original state is practically l. With these restrictions, we can use the simpler formalism which we developed earlier. However, for large times ( yxo)l ) we obtain II h . (\."0/ r ^ \It z : ",i

Itnl' . (/, a)z a trz

/ta

2(\

The frequency of the emitted photon does not have a sharp value, but the emitted spectral line has a width of order 7. In addition, the maximum of the line is shifled by an amount AE . The level shift is not of interest at the moment. We shall return to this problem in greater detail in Sec. 37 when we have further developed the formal techniques of the theory. Here we note that the.result (28.35) for the line breadth can be understood at least qualitatively in terms of the uncertainty principle. Ifthe state lz) has a finite lifetime /-1, then the state can be physically realized only during a time which is shorter than 7-r . Now, since it is impossible in principle to define the energy of the state with greater accuracy than 7, the emitted spectral line must also have this uncertainty. According to this, it would be expected that if both the Ievels lz) and In')have finite lifetimes, the width of the emitted line ought to be the sum of two quantities y . In the preceding calculation it has been implicitly assumed that the final state la') is the ground state of the system and that in the state I n) only transitions to the ground state of the system are allowed. If these restrictions are not made, then the other transitions have to be included in Eqs. (28.11)and (28.12). The calculation can be carried out following the method used above and the result is what would be expected from the uncertainty principte. We shall not explore the matter further here, but refer the reader to the work of Weisskopf and Wigner for the details.I The fact that the result (28.35) can be understood by means of l.

V . F . W e i s s k o p fa n d E . W i g n e r , Z . P h y s i k 6 3 , 5 4 0 9 3 0 ) .

Sec. 28

Natura] Line Width

t3t

the uncertainty relation at least makes it plausible that the approximate equations (28.11) and (28.12) have a physical meaning. Certainly this argument should not be overemphasized, since a similar result must come out of every approximation using a unitary matrix U(xo). Tn the usual perturbation calculations a power-Gries is assumed for [/ and then it is terminated at the n-th term. The matrix

u:1-tu0)+...+u@)

(28.36)

is not exactly unitary and such a matrix is not suitable for treatinq the problem of the line width. Accurate measurements of the natural tine widths can be made only with difficulty since several effects such as doppler shift, collisions between the excited atoms, etc. also broaden the spectral lines. At the usual densities and temperatures these effects are very much larger than the natural line widths.r At least the prevrous measurements of the natural line widths2 are not in contradiction tothe above equations, although they cannot be regarded as definitely confrming them. More detailed discussion of this will be found in the article of R. G. Breene, Jr. in VoI . )CffII of this handbook.

I. See, for example, H. Margenau and W. W. Watson, Rev. Mod. Phys. B, 22 (1936). 2. See W. Heitler, Quantum Theory of Radiation, Third Ed., Oxford, 1954, p. 188.

CHAPTER RADIATIVE IN

THE

VI

CORRECTIONS LOWEST

ORDER*

29.Vacuum Polariz ation in an External Field. charge Ren"o{{nqlization system containing an external field which @a the Born approximation can be used. In that so enough is weak this section we shall not only be concerned with the s-matrlx, but also with the field operators in the Heisenberg picture. We write the equations of motion for them as I a '* \ r t @ ) : i e v ( t ( * ) + A d " ' " ( x )v)( x ) , \l a*a r"

(2e.I)

: )-1i u @ ) ' g A u Q ): - o : l , t @ ) , y u , p @

(2e'2)

with the assumption that we can expand in powers of the quantized field z{u(r) as well as the external field z{i*(r) , *" obtain the lowest order approximation to the ?-operator: V@):',p{o)(r)-ie J

S 3 @ - r ' ) y l A t o ) ( r ' ) 1 , 4 n " * ( r ' ) l r { t o ) ( x ' ) d+r '' . . . ( 2 9 . 3 )

For the right side of (29.2), i.e., corre sponding exPansion is

for the current operator, the

ss(x-x')v"',.@ (*)l+';[ a*'(W@) (r),yurt@) @')f*,l,ro @),vu irpc)=+Ldo) o,, (Alot(x') +,4i"o (r')) * ... . + [?(o)(r,)y,se(r' - r) Tp,,p@)(r)])

i \at '=t

)

From (29.4) we see that the vacuum expectation value of the current operator does not vanish identicatly if an external field is present. With the aid of (15.29) we obtain in this approximation . (Tr*iiator's note) The detailed comparison of the predictions of quantum electrodynamics with experiment is an active area of continuing research. Although this chapter provides an excellent introduction, a discussion of the higher-order calculations and the more recent experiments must be sought in the current literature . At the present time, a very useful starting point is the article "The Present status of Quantum Electrodynamics", by Stanley J. Brodsky and Sidney D. Drell in Annual Review of Nuclear $cience, edited by Emilio Segrd, J. Robb Grover, and H' p,i"rr" I.foye., Vot. 20, Annual Reviews, Inc., Palo Alto, Calif ', 1970.

Sec. 29

Vacuum Polarization, Charge Renormalization

<0lir16)l0):

/ ilr'Ku"(r - tc')A^,"o(ri') ,

133 /ro (\

K o o (-x 4 : + ( s p l z r s o ( r -x ' ) y , S t r )- ( /x ) ]+ l , ^ (2e.6) I

+ Sp [ZoS(il(r - x')y, S,r@'- *)]) . ) In orderto study the structureof the integral (29.6)more closely, we introduce the Fourier representation

Kn,(x- 4 : ;*

I

4Peti@-/\Ko,(f).

( 2 e. 7 )

A f t e r u s i n g ( 1 5 . 2 0 ) , ( 1 5 . 2 1 ) , a n d ( 1 5 . 3 0 ), w e o b t a i . n rz

f f

Ko,(f):, uF d'p'dP" 6(p- p'+ p") SplyuQy f' -m)y"(iy p"-+n)lxl JJ I

* {td* ( p ' , +n )' ll P -;$ pz+n'

- i ne (p" 6(p" ) , * r '))l+ + '

[tr n.el

I

6(p',+ *n)l\. ! 6(rt"s{nrrllo#*, * ine(p')

J

The expression (29.8) differs from elements of the S-matrix obtained in the previous chapter in various respects, one of them being that it contains two four-dimensional 1-integrations but only one "conservation law", 6(f -P'+P"). For the first time we are summing over an infinite number of intermediate states, and Before therefore we must deal with convergence difficulties. working out the details of. (29.8), we sha1l study a few general properties of the kernel Kr,(f). From the conservation of charge, oip@)_ /?o q\ o -| 0*" we have

- x'\ :0. !-oxp x....tt P' \ FoKr,(p):O.

(2e.ro) (29.1I)

According to ( 2 9 . 8 ) , K o , ( P ) i s a t e n s o r w h i c h d e p e n d s o n l y u p o n t h e v e c t o r p . Consequently Ko,(f) must be symmetric and of the form

Ku"(f): G(p)pup,+ H(f) 6u,.

/ro

rt\

F r o m ( 2 9 . 1 1 )w e o b t a i n

.puKo"(P) : p,lc(p)p,+ H(p)l: o,

( 2 e. r 3 )

or

K,,(P): G(p)lf uf,- 6,,p,). Finally from (29.14) it follows that

(2s.r4)

I34

G. Kdll6n, QuantumElectrodynamics *oz" x,,(*-x'):0.

Sec. 29 (29.1s)

Obviously this expresses the independence of the expression (29.5) from the gauge of the externai field. Equation (29.11) can be verified formally by means of (29.8) . If we multiply the latter equation by ipu , we find

: i" 6(P- P'+ P")x i 'P t,K,,,(P) il rr' d'P"

x lS p l (i yp"-m)(i yf'*m)(i yp' - m ) y,l-

- Sp l(i y f" - m) (iy f" I m) (i y P' - m)y,lx { . } :

I t [

(2s.16)

--#lloo'op"o(p_ p'+p")lp:(p", * m")-fi'@'z + ,mz)l* t r .J By the use of x6(r):9, ;o2

f

we can simplify (29.i6) to

f

; puKu,(P)=ft | | ap'ap"6(f'f' + p")lp:6(p',+ ,")-pi 6(P"'+mz)).1Qe.r Each of tfr" r*o terms in (2g.I7) is zero because integrals such as

I:ldf,f'"6(f'r*mr)

( 2e . 1 8 )

must vanish by considerations of symmetry. Here it must be noted that the integral (29.18) is actually divergent. Consequentiy, by a change of the origin of coordinates, an integral which does not vanish can readily be obtained from an integral such as (29.18). Despite this, we must require on physical grounds that (29.11) hold and therefore (29.18) may be taken as a definition of the value of the integral. If we observe that a calculation similar to (29.16) and (29.17) can also be done in r-space, we shall obtain this same convention. Thus we have

,-L *u,pl*):-e,

p, S
As one can readily convince himself, the vacuum expectation value of the operator i[o)(z) contains exactly the integral .I of (29.18). It must vanish. as the preceding discussion shows in detail. Bythe use of (29.14)wecan now write (29.5)in the followingform: /^,' / \,^\, a ' z Z ? t" x* ' 1 1 - ] f I a-rO -.. d x ,e i p ( ' - t C ) fvtyJAii"( A f n l e u s s / x' r ,_\l\ u l l p \ t c ) l:u)@" yJJ fr ) |' l:l,r r .r o, : d'Pdx's;o@-r\ G@)i?,"*(x') , er- I I I

G(P):-#K,,(P): : -

d.p', 6(p-p,+p,,) splyu;yf,-m)yuQyp,,-m)lx II d.p, (2e.2r) * {u(/',+ *1le o,}14 - i ne(p',) 6(f,,,+ *\l + &

+ 6(p",+ *,1[e U]p

I i n e(p,)6(p,,+ *r)l\ .

Sec. 29

Vacuum Polarization, Charge Renormalization

135

Here the experimentally observable quantity is the sum of the original external current and the induced current (29.20). That is,

ri'*(r)* (oli*(x)lo) :,=

(r')eip(x-/) . (2s.22) [ f apa*' U G(P)]iius

From (29.22) or from (29.20), we see that if the function G were a constant independent of. f, the effect of the interaction of the external field and the electron field would be only a multiplication of all currents by a constant. This wouLd mean only a change in the unit of charge of the external current and therefore would be unmeasurable in principle. In fact, G is clearly not a constant; however, as this argument shows, we can add an arbitrary constant to C and this addition means only a change in the unit of charge. Observationally, the quantity of major interest is thus the change in C when 1 varies. For an unequivocal comparison of (29.22) with experiment, it is necessary to determine the arbitrary constant in (29.22) according to a certain convention. In principle, we can do this by specifying the value of 1-G for some arbitrary value of the "frequency" P. In particular, we shall require, for external fields which vary extremely slowly in time and space, that the external current and the observable current be identicai. This may also be expressed by saying that the function G in (29.22)shouldvanish for f :0. Certainly the definition (29.21)for G does not contain an arbitrary constant and it is not to be expected that C(0):0. We therefore define the observable currents as 1

t;beob lx.\ \--t

f

f

(Z"Jn JJ

Again we stress that the introduction of the constant G(0) in (29.23) signifies only an unequivocal determination of the unit of charge.l This procedure is known as "charge renormalization" in the literature. In working out the function C explicitly, we note that according to (29.21) the imaginary part of G can be calculated easilybecause of the delta function

occurring in it.

We get

Im c (p): - EFF dp' lp'(p'-p) | zrnzlx J x 6(p',* mr)6 ((p- p'Y-t mr)lu(f')* t(f - f,)) : e2 dsb' | " _ bzt f : _ -t2n2p2 \*" ;) I yo,u;

x

- 2pp'+zr,llp'} *l +,';"JffIJl* " {u(0, ir l

C i nsl&r rr r r J v vclr t a L

onc

cnrrld

h r resrvz ov

cv h r raunr nr ya r ' l

+ha

rrnil

nf

(29.24)

a rhrasrrn a u :,s

vl ryt r

multiplying by a constant rather than by adding a term. However, in the approximation considered here, both cf these procedures are completely equivalent.

136 The

G . K i i 1 i 5 n , Q u a n t u mE l e c t r o d y n a m i c s jntcoral

r | l v { ] I 9 v v ! g g r l g v r v l 4 r f l

( 2 . 9- 2 4 )

is invariant

and

clearl v rrani shes

Sec. 29 :f -452 ls

smaller than 4nt2. lust as in the last chapter, we shall also evaluate this integral in a special coordinate frame: the frame in which the components of p vanish. In this system, (29.24) can be written in the following way:

rmc(p): ,{,+n,- pilh : +u + 2.0)'i;## 6(ztp"tv

: *U +240)*VT - *"(ry- -):

(2e.2s)

:*A *#)1[;'8.#"(* -*,), @ ( , ) I: U+ , 1 , 1 1 .

(29 .2sa)

Thus in an arbitrary system of coordinates we have

,++"(-+-*').

-#) rmc(p):e(p)*u

(2e.26)

The imaginary part of G therefore contributes nothing to the term G ( 0 ). The direct evaluation of the real part of G(p) from the definition (29.2I) is quite tedious. We spare ourselves a considerable effort if we note that the function

G(, - 4 : ;*

I

rtOleit("-/\dP

(2e.27)

must vanish for x'o> rs according to (29.6). This is also necessary if the induced current is to depend on the value of the external current only within the retarded light cone. If this were not the case, the theory would clearly be "acausaI". From this property of the function (29.27), it follows that the integral G(p, Fo+ ili : I dxG(x) e-d(pa-Qo+i't,no, exists if 4 is greater than zero. The function is an analytic function of a with no singularities plane. By the well known theorems of analysisr tion, the real and imaginary values, on the real relation

(29.28)

G(p,z) in (29.28) in the upper half for such a funcaxis, satisfy the

+@ -R_ e - ,G \ r (, rov.b l ^\:1

n-

o f J

-€

rmc(P'x)dx x_po

(2e.2e)

Fromthe imaginarypart (29.26),we can thereforecomputethe real part by a simple integration.Using the notation n@)(Pr)introduced I. See, for example, B. A. Hurwitz and R. Courant, Funktionentheorie, Third Ed., Berlin, 1929,p. 335.

Sec. 29

Vacuum Polarization,

Charge Renormalization

137

in (24.35), we find

(2s.30)

ImG(p) : 7ue(p)n@)(pr),

R e G ( / ) : p ' ( - ' ' o ' ,xo-'P- "o t J lxl

t. d r : p i r r r , l o" r - " r '1l xl - P o + x *!P.o] la r = ) J

**

I

*o

- x 2 ), 1 a z \ : p - ' Jo- xf qnu@ r -' ,( -p' z. 1

ltzs.trt

IrQ\(-o\.do :]I6)(p2) , aapz-=

I .

I ,

a

n t i l 6 z 1: p

f

J 0

no)(-"1a"

(2e.32)

p2

a*

Hence we have

G ( p )- G ( 0 ) : f r @ ) ( p -, 7 ) or(0) -tine(f)n(o)(p2). (2e.33) From (24.35) we see directly

that the function II@) (- a) has the value

12 irz

does not converge and the functions n*o(pz) and G(1) actually do not exist. This is clearly aresult of the summation over an infinite number of states in (29.2I). Untit now, we have not paid attention to the convergence of the sum. One might therefore suspect that the theory is not capable of treating the problem of the induced current. If we ignore all this for the moment and just write down a formal expression for the difference II
: eIna 64o,l;F - )-]:-0, (o) oI +ffi 7-or n@) @,)00

.Qs.34)

The integral in (29.3a) is convergent and can even be done by elementary rnethods. The result is

'+V'+T

t-

#l+-#-?-#)IF*T^*

n
l'-y;ry|

E q u a t i o n ( 2 9 . 3 5 ) a s s u m e st h a t

1++)

r

ol "t

l | I

fLhr or s

rr se rqurrrl l t

],,,

0 . O t h e r w i s et h e l o g -

arithm must be replaced by an arctan function. la2 l! | *,

,,,

For small values

ri cD

: _ !:, !_ +t ... . ft
(ze. 36) \!'

In this waythe theory gives well-defined expressions for the experimentally observable effects although the "charge renormalization" fr(o)19; is infinite. Fjrst, the renormalization of the charge is necessary in principle if the expressions of the theory are to be

C . f i i i t 6 n , Q u a n t u mE l e c t r o d y n a m i c s

138

Sec. 29

compared with experimental results. the infinite In addition, term in the expectation value of the current is therebv glnilnated. This is the great practical significance of the renormalization and this is the reason that a satisfactory formulation of quantum electrodynamics was not possible until a clear formulation of the renormalization principle had been given. D e s p-u],t',orlnn ite this, tfre-resutt (ZS .3 6) *a= J--"ri"uO fo"g ago bv- Ue-hlin9,l the method was not completely satisfactory from a theoretical standpoint.^ The first "modern" deduction of. (29.35) was given by S chwinger . z According to this calculation, the vacuum behaves like a polarizable medium with a "dielectric constant"

e(f,) : 1 Tho

ownraqqion

()Q

*gto) (p\ + n@ e) - i, e(p)Itot lpzl.

?7)

hac

an

imaainar\/

n^r+

Tn

tho

r*r-q r *r * a -]

Qe .37) r.r/ d Y ,

this must signify absorption of energy from the external field. According to (18.22) and (I8.24), the total energy given up by the external field is

aE--J aAtl')'1p'1 : op tprl ii" ,h" I

l''"'"" |

Comparison with (24.34) shows that this energy agrees with the energy of all pairs produced by an external field. of the equations derived here A direct experimental verification has not yet been made. In a later section we shall see that the term (29.36) contributes to the level shift in the hydrogen atom an amount which is about a hundred times larqer than the experimental uncertainty. In this way we can regard (29.36) as having been indirectiy confirmed. for In addition there are several effects,r example, the energy levels in a so-called ",a-mesic atom" (an atom where the electron has been replaced by a p-meson) and in where a small improvement in the the proton-proton scattering, present experimental accuracy should show the effect of the vac-

1. E. A. Uehling, Phys. Rev. 48, 55 (1935). See also W. H e i s e n b e r g ,Z . P h y s i k 9 0 , 2 0 9 ( 1 9 3 a ) ;W . H e i s e n b e r g a n d H . E u l e r , ; . F . W e i s s k o p f, D a n . M a t . F y s . M e d d . Z . P h y s i k 9 8, 7 I 4 ( 1 9 3 6 )V 14, No. 6 (1936). 2 . J . S c h w i n g e r ,P h y s . R e v . 7 5 , 6 5 1 ( i 9 4 9 ) . 3. L. L. Foldy and E. Eriksen,Phys. Rev.95,1048 (1954), ; . X r i k s e n ,L . L . f o l d y a n d W . R a r i t a , P h y s . R e v . 98, 775(1955)E 103, 7Bt 0956).

Sec. 30

Regularization and the Self-Energy of the Photon

139

uum polarization. In certain measurementsl on the x-rays from p,mesic atoms, it has even proved necessary to make a small correction for the vacuum polarization. This is done in order to obtain agreement between the mass of the pl-meson determined from these measurements and from other independent measurements. Regl{ler:!4a'[!p4 e4!l the Self -Energy of the Photon T h e t r s e vorf ycLr er rnuer rdar l ccoonnssf oi edred rl lao tnisO n s oorf _s ,\ /-m _ .m - . .e_ t r v s r r c h a * _s c v auge invariance and causality [in nq. (29.27)] *a. essential to our calculation of the effects of vacuum polarization in Sec. 29. In the present section we are going to show that these considerations saved us some work in the calculation as well as giving a precise meaning to the otherwise ambiguous, divergent integrals in (29.8). In order to show more clearly the ambiguity of the expressions (29. B), we attempt a direct evaluation of these integrals. We are here concerned only with the real part, since, bythe previous calculation, the imaginary part is unique and convergent. We have 30.

: - fio ap' neKr,(f)- R,,(p) I " - p,) -

- p f' I m,)fx + p"(pL p,) 6u,(p',

x lp;(p:

6 (-( p - p ' l z L n 2 ) \ ,^, -p l 6 ( p ' , t * , ) , T p'z 1 nz\ t P : P ' l z: , m z l'

1,,,,

To compute the last terms in brackets, we need the integral representations *o

d(a)

:

1f -' 212

I

da

cN@a

,l

(30.2)

t

- @

^1 I P-:-h" r l * l . ;

+a f .rat | ,Jr;'-!-"iwb ,'*1".,1" t

(30.3)

-€

according to which

(30.4) o{Y * +} : fi [f a*to*,Iffi * ffi),iw'a]iw,b. With

the change of variables ut:

u'&

(30.5a)

'

1s2:7e (l -

(30.5b)

d')/

we can transform (30.4) as follows:

p{jyL+ I

D

d(b)}:=! e

)

f fara"t*,t,',12+

+xtt.JJ

l ? r ll l c r l

+a :

,*-

a* [*

[

,1

- a-)riovaa+(r-o)al: I

l1-crll

1 t r n r i w t d a r ( r - a ).b )

I

l(30.6) j

l. S. Koslov, V. Fitch and J. Rainwater, Phys. Rev. 95,291 0954). See also the articie by S. Fh-iggein Vol. XLII of this handbook (HenQlush der Physik, edited by S. Fliigge, Springer-Verlag, Herdelberqr.

I4O

Sec. 30

G . K d t l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

Using (30.6) and making the change of variable (30 .I) , we obtain ioz -@ R u "( -P)

+@1 f

J _6

f

f

q:f'-af

-

ud u o" dq l 2q rq,-('t- 2n)( f,t"- t tr il- l J J o

in t

l(30.7) - 2 u ( i - d ) p u f , - 6 r , @ ,- f q ( r : 2 d . ) | ) - f, n ({ . + mr)f ei@lq'+l'u(L-dl+n2l ") On the basis of symmetry, we can drop the term linear in q inside the square brackets and replace gog,by tdu,q,:

-.p26p") : -# a*| a*[ arfz*rt- e)(p,p, R,,(p) + i; ],ro.r, ' * ur"(+ * P'n(r- o) + *)leiutq'+P'd(r-a)+n't i It is usefut to split (30.8) into two parts: R,,(p) : (pup"- 6r,pr) Fr(fr) -l 6u"Fr(fr),

( 3 0 .e )

1+6 i"2

f

f

f

-

F , ( b z \ : - + +1t- I a " I u i l u q . ( r - o ) l a q e ' t t r p ' d ( t - d ) + n '(t3, 0 . 1 0 ) J 'l J 0-@

1+€

r ' o z \ : - ty6 r-f J[- .a- -*J [ r a * J[ a "q, l q , l 2 f , a ( r- a ). *

-z\rt

o_el

l

.^

(30.11)

t

rivlq'1-f2a(r-e)lml) '

+ 2m'l

l

According to the considerations of Sec. 29, $ should vanish identically and F, should contain the result (29.35). First we evaluateF,. Sincewe can write[c.f . fq. (16.39)]

t

f d.q,"0' : have

we therefore

u l!', ul

r

( 30 . 1 2 )

,

+@

F , ( f z:\+ 0"" f a . g- a . ) d " | + + r i o t p ' a ( t - a \ ! m 2( 3 t .0 . 1 3 ) l*l _r* ! This corresponds The zz,-integrationin (30.13) diverges for w:0. to the divergence of the integral (29.32) tor a: e . Evaluating this according to the charge renormalization program, we find oz

'

1+@

Ebzl-E (0): -=I a(, , . ,_ 412"J

a) d.a [ J

O-@

!y/ur"e$-") lul-

11+@

:

:

,"^ f q.o - a\ dq. f as f lubzah

4n'J oo-o

-

I ozf,h2|

J

J

" . I q.(t -a), d.q.loel+ l !-a({ "l ,n'

2xt" J 0

_ t] ,iuin2 -

d@-d.\ ei@tt)'pd(r-q\+m") llul

-a)l.

(30.14)

Regularization and the Self-Energy of the photon

Sec. 30

141

The a-integral in (30.14) can be done by elementary methods and the result agrees with (29.35) . As expected, we have

-ntot 4(P\ - Fr(o):U(o) 1Pz1 @). For calculating

(30.ls)

,4 , we shall need the integral

I o- z 6- r r i * e ' - -

J

?'

a"lwl

which can be proved from (30.12). aJter performing the t i o n , w e c a n w r i t e ( 3 0 . 1 1 )a s

( 30 . 1 6 ) q-integra-

1+@ 'E 2 \ (ts2\t-

-a]'*m2

e' g',z

w

"!0"_[o*lhlry

* 2frn*w"tr-e)tm't-

u-@

1+@

:

^ ' i ' JI a " Jf a * ,l u' ,l o^a a l t , ; ' w o t t" -) + -) ' ] l : lu

6n"

I

:

i e2

f

( 3 0 . 1)7

-m

0

,

-q\'m"l

I ei@ l,'q(t

t ad.l + 1"t _' . J \ 0

u

I

lu-O

The function 4(P,) does not vanish; indeed, it is infinite! This is in flat contradiction to our considerations of Sec. 29, which led to (29.14). lt looks as if our theory were not gauge invariant, aIthough this would seem to be "impossible', on physical grounds. In Eqs. (29,17) and (29.18) we even "proved" explicitly the gauge invariance of the theory. As was observed after (29.18), what was essential to the proof was that certain integrals which actually are strongly divergent, were defined to be zero by symmetry. Also, it is immediately evident for physical reasons that the gauge invariant result of Sec. 29 must be the correct one. The symmerry properties of the integral (29.18) have obviously been lost in our explicit calculation above, where we interchanged the orders of integration several times, displaced the origin of the coordinate system, etc. In order to obtain trustworthy results from our calculations, we should cut off the integrals (29.18) in a sl,rnmetric and invariant fashion. Then we should carry through the whole calculation with a finite cutoff, and take the limit of the cutoff going to infinity only in the final answer. Such a calculation has been done by Pauli and Villars.r In their method, the integral (29.I9) alone is not considered. Rather, several hypothetical particles with various masses ,ni are introduced. We multiply the charges of these particles with "weight factors" C; so that in place of ( 29 . 1 8 ) , w e h a v e

I : I d-p' r, a(p',+ m?) pi. l If the quantities

mi and C; satisfy

(30.r8)

the relations

l. W. Pauli and F. Villars, Revs. Mod. Phys. 2I , 434 (1949).

r42

G. Kiill5n, Quantum Electrodynamics

4'

Sec.30

:o '

( 30 . 1 e )

l C a m !: o .

(30.20)

then (30.18) is convergent and actually vanishes. The new masses serve as cutoffs, and in the final result all masses, except the original one, must tend to infinity. If mr:1rv is the original mass, in this limit. By this process-then we must also have Ct:l convergent integral will clearly which is called regularization--a not have its value changed. A divergent integral will be evaluatedr exactly according to the prescription (29.18). In order that (30.19) and (30.20) be satisfied, some of the C; have to be negative. If we try to regard the auxiliary particles as real particles of spin I/2 , then we have to give them imaginary which is hard to ignore and which will probcharges--something It can be shown2 that relaably have unphysical consequences. t i o n s s i m i l a r t o ( 3 0 . 1 9 )a n d ( 3 0 . 2 0 ) c a n b e s a t i s f i e d b y r e a l c h a r g e s and masses if some of the particles have spin zero. However, in this procedure we obtain a relation between the masses of the elementary particles which does not seem to exist in fact. We shall not pursue the matter further, and regard the regularization only as a mathematical aid for cutting off our integrals in an invariant manner. rather than (30.17), we have Ajter regularization,

FFc(bz\:

n."" 4n"J

t I an(\ w =t' , c, siotp'e(t-a)qn])\

lu:o

0 1

l

1,"^ "'' I'"""

.s^r,, ll : : ' 4i enz," JIf da ,r *m ll*o( .) 1,_o " l -s)"^ c;* i l ,c,l pzu 1-u) Lu?'

,

0

U s i n g o n t y ( 3 0 . 1 9 )g i v e s

: - +7,C,*?. nR""(f,\ 4t'' 7

(30.22)

From (30.20) we conclude that

4*"t(P'): o.

(30.23)

Thus the results of this section aqree completely with our previous results. The use of only (30.19) for evaluating the vacuum polarization seems a trifle artificial and must obviouslv lead to a false result. 1. Other methods which give the same result are due to J. ; . C . P e a s l e e ,P h y s . R e v . S c h w i n g e r ,P h y s . R e v , 8 2 , 6 6 4 ( 1 9 5 1 )D 8 I , 1 0 7 ( i 9 5 1 ) c; . K a 1 1 6 nA, r k . F y s i k 5 , 1 3 0 ( 1 9 5 2 )a; n d S . N . G u p t a , Proc. Phys. Soc. Lond. A66, I29 0953). 2 . R . ] o s t a n d J . R a y s k i , H e I v . P h y s . A c t a 2 2 , 4 5 7 ( 1 9 4 9 ) ;s e e also J. Rayski, Acta Phys. Polon. 9, L29(1948).

Sec. 30 Regularization and the Self-Energy of the photon

I43

If one is not particularly careful, it is quite possible to carry out the calculation without regularization, so as to obtainl,2 the res uIt

F,(p\:- #*, .

( 3 0. 2 4 )

For the moment, if we were to treat3 a light quantum as we have treated the external field, we would conclude that the photon had a self-energy -p*z

1f

.

6E:11a2--;

-@N

ez

,Ft

m2

(30.2s)

This is also the result of Wentzel. According to the preceding arguments, such results are to be regarded as more or less accidental and one should expect different results from the various methods of calculation. In the old calculations, which were not formally covariant, the self-energy of the photon was often given4 as infinite. By the requirement that all the results of the theory be gauge invariant, the self-energy of the photon has been unequivocally determined to be zero. In conclusion, we can say that the vacuum polarization gives an infinite charge renormalization ("self-charge") which, in principle, is not observable. The physical consequences of the theory are all gauge invariant and finite if the calculations are done with sufficient care. The preceding calculation verified this for the lowest order term in a theory with spin I/2 particles, for a serles expansion in both the radiation field and the external fie1d. The charge renormalization has been calculated to the next higher order in the radiation field by Jost and Luttjnger5 and the observable terms have been calculated by Baranger, Dyson and Salpeter6 and by Kiill5n and Sabry./ As was already mentioned, the calculation has been done by Kroll and Wichmann8 without expanding

1 . G . W e n t z e l , P h y s . R e v . 7 4 , 1 0 7 0 ( 1 9 4 8. ) 2. One obtains this result from (30.I7), f.or example, if the inte qral - a ) - pn ,')' l/) [0.#*(!,o'l*@,o(r

J lul oa\@ is set equal to zero on the basis of symmetry. If the external field is a photon (p,:O), t h e n ( S O . Z + 1f o l l o w s f r o m ( 3 0 . 1 7 ). 3. See Eq. (29.4), where the external fietd and the radiation field enter symmetrically. 4. See, for examplerW. Heitler, Quantum Theory of Radiation, S e c o n d E d . , O x f o r d, 1 9 4 4 , p . 1 9 4 . 5. R. Jost and J. M. Luttinger, Helv. Phys. Acta23, 20I (1950). 6. M. Baranger, F. I. Dyson and E. E. Salpeter, phys. Rev.

8 8 , 6 8 00 9 s 2 ) . 7 . G . K i i 1 1 6ann dA . S a b r y ,D a n . M a t . F y s . M e d d . 2 9 , N o . 1 7

(less).

8 . E . H . W i c h m a n n a n d N . M . K r o l l, p h y s . R e v . 9 6 , 2 3 2 ( 1 9 s a ) 1; 0 1 ,8 4 3 ( 1 9 5 6 ) .

144

G. KdI16n, Quantum Electrodynamics

Sec. 31

in powers of the external field for the case that this fie-ld is a The higher powers of the external fieldr in a Coulomb field. gene^ralseries expansion in this field, as well as the regularizationz in the order ea have been treated. The results of all these calculations are essentialty the same as those of the lowest order. For particles of spin different from l/2, the vacuum polarization has been evaluated by Umezawa and Kawabe and by Feldman.3 Here it has been shown that for particles of spin L I the observable terms contain divergent integrals. For these problems, therefore, the charge renormalization is not able to give usable expressions for the physically important quantities. 31. The Lowest Order Radiative Corrections to the Current Operator Now we shall consider a problem which is formally more complicated: the next higher approximation to the current operator. For simplicity, we omit the external field and so we have just the operator equations which were studied in Sec. 17 and 18. We reproduce here the expression (18.12) for the current operator to order e3:

(*"),y,rptot (x")f\]tD*(x'-r")-l ,y*S*(r-x')y,lp'o'(*'),lq'o, ift@)=if lax'dx'nllq,(o'(r) (*')\-l (r")f {A',1' (*),TrSe(x-x')y,,5^(x']-x")y,"rp'0, @'),e',".' i! [nr' ar" Lrp,o, -][ (r")}+l(3I.1) y,,Sa(x'-x),yuS ^(x-x")T,"rp'o'(*")f {A',i'@),r4i:) !a*' a"" lrl,'o'(*')

+ |[ [h*' a*" 1{lrl,'o'(x"),y"rl,'o'(x")f ,rp'o'(x')\y,sn(x't-r),yurptn'(x\lD*(xa-x")-i!

[a*'

(x)f{A',\'(*'), n',"'1*"y .l yurl:to' y,,sa(x'-x), y,,L7(x'Lx') a*" l1to'(*")

The operator (3I.1) has non-vanishing matrix elements from, the vacuum to the following states: states with an electron-positron pair and no photons, states with one pair and two photons, and states with two pairs . We are not concerned with these two last classes of matrix elements here, because all the p-integrations can be done just by using the conservation of energy and momentum. Therefore there will not be any "inner" 1-integrations left over, i.e.. there are no summations over an infinite number of intermediate states. Things are quite different if there is only one pajr present in the final state. Then there is one 1-integration remaining and so we can study this lowest, non-vanishlng correction to the matrix element (C,C'li"@)10) or (Oli,@)lq,q') . I. G. Kdll5n, Helv. Phys. Acta 22,637 0949). 2. E. Karlson, Ark. Fysik 7, 221 (1954). 3. H. Umezawa and R. Kawabe, Prog. Theor. Phys. 4, 443 (1949), and earlier articles. D. Feldman, Phys. Rev. 76, 1369 (1949). See also J. McConneII, Phys. Rev. 81,275 (1951).

Sec. 3l

Radiative Corrections to the Current Operator

145

Similar considerations show that the quantities (q li"@)l q') are the only matrix elements which contain inner p -integrations if neither of the states is the vacuum. In this last expression, both states lq) and Iq') ar. either one-electron states or both are onepositron states. Using the methodsof Sec. 18, we have

(*),yu S* (* - *') @(r')ll q') * e"(q lil]'(*)lq') : + d'x' (q lllptot I +

Ia-

j s l 1 6 @ ' )S n ( * ' -x ) , y u g ' o ' ( x ) l l q ' ) - l

?l ?\ " (qllrt'o'("'), Kr(x' x,'x *"irp'o'1r"111q') d.x" d,x' I * + II -*') Do(r'- x")(qllrt'.o'(x"),y,rtro'1x"))lQ'> + + II d,x'dx"Kr,(*

and

(r),TpSp(x-x') A @')ll q,q'>+ es(o I ift (x)lI, Q') : :: ax'
(31.2a)

ar' ax" K u,(x- x') D^(x'- x" ) (o llrp,o, (x"),y,rt,r, (*" )fl q,q') . + ! [| In (31.2)and (3I. 2a), Kr(x' - x, x - x") is a c-number in the Hilbert space of the,particles but a matrjx in the "spin space" of the 7matrice s:

K,,(x'-x,x-x")- - f,r^fs(l)(r' - x) Toso(, - x") D*(*" - *') * | + S n ( x ' -* ) y u S ' t ' ( * -* " ) D o ( x ' - x " ) ! iGt.3) I

St(x' - x)yrSo(, - x") Dtrt(f - x")f yt .

I

Here @(x) is the operator ez f r^/r\, @(*) : - + I y^lSo (x-x') D1(x'-x) | S^(x-x') D@ (*'n))y^rp'o)(x') dtc'.(31.4)

fn" frn.tio n Kr,(x-z') is just the function which has been defined in Eq. (29.6) and studiedindetail inSec. 29 and 30. Because this quantity,

(*")), + I o*" D*(*' x") lt/o)(*"),yr',!,6\

h'r c\

is just the potential produced by the current TLrror(x"),yuv$lQl',)f we can interpret the last terms in (31.2) and (31.2a) as the polarization of the vacuum by the electron current. We shall now work

146

G. Kii116n,Quantum Electrodynamics

Sec.3I

out (31.4). It is convenient to go over to 1-space: @(*) : f dx' F(x -

(x') , x')tP{ot

(Jr. o/

F(* - x') : - lr^lt',1* - x')D,t@'-*)+ S^(x-x')Dot(x'-x))y^) :

a , Y [ a q e n o r ' - o t r 1' q 7

F(s):- ## IIdp d.k 6(q-p+k)ytiyf -m\ytx o(nt] * [r(p'+ m\[e$ - i n e(n) + { i n e(P)6(P'+ *")]].

+ d(ft'z) [, e**

f31'?)

1.,

(31.7) is again a retarded function and The function F(x-x')in consequently there should be a relation between real and imaginary parts of F(4) similar to (29.29). As before, we shall first calculate the imaginary part. From considerations of invariance, we can write

ImF(q) : F,.(s)* (iy q * m) F,(d ,

I?r q\

with 4i qrFr(q): Sp [2, Im F(q)] ,

(31.I0)

a ( F r ( q )* m F , ( q ) ) : S p [ I m F ( q ) ] .

(31.11)

F r o m ( 3 1 . 8 )w e h a v e

: 4i quF,(q) ;F :

-ipz

;;

apdk 6(q- P + h)Sply,y^(iy p - /r1) /rl x I x 6(p,+ rnz)6(k2)l' (A) e(p)l: f trr.rrt f I dk(q,+k,)6((q+k)'+*') 6(k'z)lt,(k) e(q*h)),J J JJ

(q+ h)].(31.i3) +(4 @)! mF,(q)\= Yu" I and ((s+ k),+ *r) d(ft,)[e(A)-e .\r\:'la\r'llfJ'',

If we multiply (31.12)by -liq, and use the properties of the delta function, after a calculation similar to (29.24) and (29.25), we find -e2

F,(q): ffi

r

l. - m2\ dk 6(kz)6(qz-r m2* zqk)le(ti-e(q+ A)l:'1 . \o ; ) J | ( 3 1r.4 )

: n e (q)2[o) (qr) ,

: #"[, (q,) zJo) ffYl@? q, m,),

)

(31.1s)

Sec. 3l

Radiative Corrections to the Current Operator

I47

P,(q) : n e(q) 2to\(q2\ ,

1?r rA\

q'- m'). (31.17) xlo)(q'): fr:; (t * *l )(t * #)o ? As before, we introduce the Hilbert transforms of these functions, @

(- a) p f _Elot ;*F ?E!o) 7 (or\ Oo \ y t _: _ ul Thus we finally

obtain for the function

(31.18)

F(g) ,

F(q): ZP @'){ in e(q)Zld(q')-f (iy q -f *)lELo)(q')| in e(q)Z[o)(q,)] . (sr.rg) Fnr

larna

rzairrac

nf

-Az r ,

iLhL a L v

frrnaf innS

r u r r v L f v r r r

t(0) /i2)

a ,

tencl

to

f inite

limits, just as the function nQ)Q2) did. The integrals in (31.18) thereforediverge at their upper limits, as did (29.32\. In Sec. 32 we shall discuss further the interpretation of these divergences. The evaluation of (gt.g) in full generality is possible and can be done by methods similar to those already used. In order not to complicate the formal operations excessively, we shall work out only the special case which is of interest: that Ku@'- x, x - x") appears with the operator g(0)(z)as in (31.2)and (31.2a).We write Ku(x'- x, x - x"):

(, 2. :n ,\ ," . J[ [. J

-o' ao "oan '-r i q ( l - z ) t i q ' l x - t ^"K^Jp \(YoJ.Ya/ ', \ \8 ' 1.20)

and consequently we are concerned with the function K,(q, q') only when g2:q'z:-mz and where a factor iyq+* on the left or a factor iyq'l * on the right can be set equal to zero. From (31.3) we obtain quite generally

-+;fr Klq,s'): !an\el(q-

6 (h')

h)' a m'zllk'- h),+ nx,f

* t effffn _u,*TiI r * o !,,[ln-_oo',',IT,?t - k)z-m')lt - e(q-k)e(q'.-k)l€(q'- q)+ + in P $ 6((t- k)'+m'\6((q' * i n P A _ # + a a g z ) 6 ( ( q ' - k ) 2/+' ) l r - e ( k )e ( k * q ' )u) ( q '-) (31.21) - i n P _# a 6 ((q- k),+ *,) 11- e(k)e(k- q)le(q)+ * a @,) U, * n 2 6 ( k z ) d ( ( q ' -k ) ' l * ' ) d ( ( q- k ) ' * * ' ) l e ( k ) e ( q- A )f e ( A x) x e(q'- k) * e(q- k) u(q'- n:;l\y^ (; y (s- h) - *) x xyo (i y (q' - k) - *) y^. Now because of qzapz:O,

t h e e x p r e s s i o n sA 2 a n d ( q - k ) r + m ,

Sec. 3I

G. Kii115n,Quantum Electrodynamics

I48

cannot vanish simultaneously.I 1y" can therefore drop many terms in (31.21) and write

(3r.22)

x u @ , r ' ): N f \ ( t , q '+) i n e ( q ' - q 7K p ( t , t ' ),

-')fi Jau{,6=a;ffima *ofi(ffffi* q')= xtt(q, r, },r,. - k )- * l v ( s- k )- * ) ozrfl

-

. ,O 6 ( h ' - h ) 2 4 m 2 \ \

- n , + '* 1 1 v^ l iv h zL @

|

^)

vr l i r @ '

- h),+ m\ n xl xf) (t,s'): -') tr',fJ ana(@- n),+ *,) 6((q' $" | (3r.241 Dzif

x lr - e (q- k) u (q'- k)) y ^ li y (q- k) - ml yu U y @'- k) - *l y^. )

As before, we start by evaluating the term with the function e(q'-q), ( 3 1. 2 4 ) . T h e f a c t o r w i t h t h e 7 - m a t r i c e s c a n b e s i m p l i f i e d i.e., considerably by means of iy qyuiy q' :

( i y S * m ) y r ( i r q ' I m ) - % T r , ( i yq ' * * ) -(;yq*m)yum*mryu-mzTu,

iyq' yriyq - -2im(q,*

\ (3r.2s) )'

- q ) 'r 7 m ' ) , s L )- v r ( @ '

(31.26)

etc. In a few terms the A-integration can be done directly, by elementary methods. By methods used several times in previous sections, we obtain

p [+ JR"

t

d.k a ((q- h)' + *') d ((q'- h)' + m2)lr - e(q-k) e(s'-k\]

_ ',u,,e)- (r;o"* quq'" ={+(,+S)(ra. * q*ql, +* u,,q')lX }h \ Ti I

n @(.......:: -A'

I

Q z- 4 m ' )

tl

4mz

t

)

v'n g,

( 3 1 . 2 7a )

Q:Q'-q,

P f b#6k-k), JR"

'7\

t m , ) 6 ( ( q ' _k ) 2| m z ) x

x [{ - e(q- k)u(q'- k)] : (qp+ il + The angular integration in

9J4.!., a}

V'* E

I l(31.28)

l

p [ + 6k - h),* m,)6((q, - h),]- m,)U - u(q- k)e(q'-ft)l (3r.2s) \\r h" J diverqes.

This is because the denominator vanishes

at the limits

1. Here we are ignoring an "infrared term" in which a1l compohave to deal with nents of ft, vanish. As we shall see below,we infrared terms by introducing a small photon mass p. We have omitted it here.

Sec. 31

Radiative Corrections to the Current Operator

149

of integration and so the principal value does not exist. We shall make this integral finite by formally introducing a small photon mass p. Equation (31.29)is therefore replaced by

Of course the cutoff p must drop out of the final result. W i t h t h e a i d o f ( 3 I . 2 7 ) , ( 3 I . 2 8 ) , ( 3 1 . 3 0 ), a n d t r a n s f o r m a t i o n s l i k e ( 3 1 . 2 5 ) a n d ( 3 1 . 2 6 ) ,w e c a n e v a l u a t e ( 3 I . 2 4 ) c o m p l e t e l y . T h e result is

Kllt (q,q) :yuRto)(qz11;

q'lJL

s(o)(?') ,

: -#{-( *#-)bsb- E#*'J**[,*#]] R(0)(02)

(31.31)

/2r 2r\

@( - Q 2 - + m z 1

"Ra, S(o)(0r):-

e2 rnz @ ( - Q 2 - 4 m , \

4"' A'

1I

4vn2

(3r.33)

V'*_F

T h e q u a n t i t y ( 3 I . 2 3 ) m u s t h a v e t h e same general form:

Ktt @,o'1: y,,R$)(Q\ -y ; q,'l-11" s(o)(0r).

(3r.34)

From (3I.3), we see that the function Ku(*,-x, x-x") vanishes if. r!o> xo or if x'o',> xo . Since the functions Kl? @,q,) depend essentially only on p , it is a natural assumption that these "causal." properties arise because of the following relations between R(0), S(o) and R(o), S(o).

pto)19r1 :p J 0

no?=!)

f

eLA'

@

:p S(o)(0r)

I .l

da

t

t,",\ ^!_ao . a *e2

/?l

?(\

(31.36)

0

This guess can be verified from (31.21)through (3L.24) by an ex-

150

G. Kdll6n, Quantum Electrodynamics

Sec. 32

*'=,t'^ 4nz _e2 so that the.integral (31.36) is convergent. for large values of -Q', f)n r'ha nthar hanrl the integral (31.35) iS divergent because the

plicit

The functions S(o)(02) go - to zero like

calculation.l

f u n c t i o n R ( 0 ) ( 0 r )s o e s l i k e

#"t#

for larse values of -Q'.

32. Mass Renormalization In Sec. 3I we have seen that the matrix elements (3I .2) and (31.2a) contain several divergent integrals. Our next task is to study these terms and, if possible, to interpret them physically and to removethem. We note first that the operator @(x)in Ql ,a) was also present in Eq. (18.2I) and therefore that it can be taken as the first radiative correction to the matrix element (Olrp(r)lq). In fact,

(ol v@)Iq) : (o l,!to\ @)|q>* / so@- "') (ol @(x')|q) dr' + ... (32. 1) or

I a , m\). ( o lt p ( x ) l q:) Q2.2) Qla@)V). ly# + Accordingto (31.15)and (31.I7),the quantities lld(-/nz) vanish. W i t h t h e s ea n d ( 3 1 . 6 ) ,( 3 1 . 7 ) ,a n d ( 3 I . 1 9 ) ,w e o b i a i nt r o m( 3 2 . 2 ) / a \,, ( 0lvb)(") |a) =],rr.,, 1-mz|1 + m) \ut !) \x) s ) - lE tu(- m') * (iv qI m1E lot \T 6*

:-rf')(- /n') .

To the present order of approximation,

)

we can write

(- *'))
(32.4)

rather than (32.3). From this, it follows that the infinite quantity can be regarded as the mass difference between the tlo)em\ f.

F r o m ( 3 1 . 2 1 )t h r o u s h ( 3 1 , 2 4 )w e s e e t h a t

xr(t,t'):x'i'k,q')+i,n"(q'-q)R'i'@'q'\:..li,1orll'{e,qo-ie:q"q6+i'u'), By a transformationsimilar to (30.6), we can write xft(q,q')as 111

- ,t\-.v' (pl\)at ^' 1

" I r d,adBdy6(r-o-F-y)x t6nzJJ.tI .

000

x Il rn r tlt r k(1- cr)- q'0\ - *f vpltv (q'(r - 0\- qa)- *lv e- 21,pL a - , ' S * a , ^ r , ."', l l *, T J V,T"'- ll , ltos l--)) A : q 2 c e *y q ' , f l y + Q 2 a F+ m r ( r - y ) : Q ,a F + m 2 ( r- y ) 2 . follows from this representation that for q2:q'z: -*z , Kt\(q, q') It function of. Q2 which is regular for all complex is an analytic values of B2 except the positive, real axis. From this, (31.35) and (31.36) follow immediately.

Sec. 32

Mass Renormalization

151

free particl€S dt le:-oo and the "physical" particles which are coupled to the electromagnetic fietd. In fact, the free particles at -oo are only a mathematical fiction (however, one which is essential for the whole interpretation of the formalism) and their mass is not an observable quantity. Indeed,we can choose these masses arbitrarily, as long as the mass of the physical particles, i.e. , the quantity rrx+t1o)(-m2), has the correct experimental value. It is now expedient to take the mass of the free particles anrrri in rha ovna'i-ental mass of the electron. We can do this :i1::ji :Y ::i:

by transforming the differential equation for the operator rp(t) as follows: la\ (32.s) l y , l m . " " l l p ( x l : t e y A ( x \ t p ( x *) 6 m r p ( x ) . YOz

That is, we add a term 6mtp(x) to both sides so that just the experimental mass appears on the left. From now on, we shall denote this (observed) mass simply by tn. It is always this mass which is to be used in the S-functions and other quantities that appear in the theory. The quantity 6m is then to be defined as such a function of e that the mass of the state lq) remains unchanged during the adiabatic switching. Clearty 6m-->0 for e->0 and the previous calculatj.on shows that the first term in a series development of 6rn in powers of e is given by

6 m : l 1 o t 1 _ n f+) . . .

(32.6)

Actually Eq. (32.5) means no change in the original differential equation, since the same term has been added to both sides, but it does change the adiabatic switching. The change is that only the self-mass on the right side is to be switched off for xo->- r:r., while the experimental mass on the left is formally independent of e . This procedure is knownl as "mass renormalization". lf now m is to be the experimental mass, then we can conclude from the differential equation (32.4) or from la\

: \y i; + m)(oly(x)lq) o that

(ol rp(x)Iq) : N(o I y,b\ @)|q>.

leo

a\

(32.8)

I. Even in the classical electron theory of Lorentz, a similar procedure was used in order to treat the effect of the self-field of an electron on its motion. The self-energy of a relativistic elect r o n w a s f i r s t s t u d i e d i n q u a n t u me l e c t r o d y n a m i c sb y V . F . W e i s s k o p f . , Z . P h y s i k 8 9 , 2 7 ( 1 9 3 4 ) , 9 _ g _ , 8 1 (21 9 3 4.) T h e p r i n c i p l e o f mass renormalization was first enunciated by, among others, A. Kramers, Report of the Solvay Conference, 1948. It was further d e v e l o p e db y H . A . B e t h e , P h y s . R e v . 7 2 , 3 3 9 Q 9 a 7 ) ; Z . K o b a a n d S . T o m o n a g a , P r o g r . T h e o r . P h y s . 3 , 2 9 0 ( 1 9 a 8 ) ;T . T a t i a n d S . T o m o n a g a , P r o g r .T h e o r .P h y s . 3 , S 9 1 ( 1 9 4 8 ) ;a n d J . S c h w i n g e r , P h y s . R e v . 7 5 , 6 5 1( 1 9 4 9 ) .

G . K i i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

152

Sec. 32

In principle t h e c o n s t a n t N c a n b e c a l c u l a t e d f r o m ( 3 2 . I ) , b u t some care is n e c e s s a r y , s i n c e a n e x p a n s i o n s u c h a s

(32.e)

I so(* x)Etot1-*,1(y-f,,+ *)
for example, is actually indeterminate. Because of the differential equation for ,/oJ(x), one might suspect that the expression vanHowever, by a partial integration the derivatives may be ishes. so that the integralthen moved over to the singular functions, appears as proportional to gr(0)(r) . This alternative form differs only q. consequently, we expectthat by a "surface term" for ro:6 this ambiguity is removed if the boundary condition for *o)1 (i.e. , the adiabatic switching) is treated with sufficient "uru. ( 3 1 . 7 ) ( 3 1 . 6 ) w r i t e p l a c e w e t h e r e f o r e of In , ,

( o l @ ( r , al)q ) :

L )l

,, 10,

I d X ' e d t r o + x i )y ^ X

-€ p r i l ( * ' - * ) - 5 t t ) ( z - x ' )D ( x ' - * ) f y ^ ( 0 l y ( o ) ( " ) l q ) fS("- x')

We have the factor eq('o+li')here because one factor e is associated with the point t and the other with *' according to (31.1) U s i n g t h e d e f i n i t i o n s ( 3 1 . 1 5 )a n d ( 3 1 . 1 7 ) , w e c a n through (3I.4). write (32.10) as

(ol @(x,c) | q) :

A*

i pIr-'' ) x x' eqzo+'i'r+ Op dL I

I

x lzlr,(p,) + (t;;

, (f) u (q)eiq'' : * m) z[ot 1pz1)

(32.11)

*e

:

i szax"l;q' [ ''

,l?o

x S,1,,gJ

e(D)

x "a.i1p._q.)

."^i1 * (i yoqu- T& ol rn)Z;o)(q2- Pill, @).

W e n o t e t h a t t h e r e i s a f a c t o r e 2 G ' 0p r e s e n t i n integral can be transformed as follows:

(32.I1).

The po-

- rB) =[ ap,zt)ta,+s)f^fudrfu]l (s2 , ;0) ffiffi -€@o*ltzz'tz) +@6

2podpo2lo'(s,-p3l_ f daz"Dt(- a\ _ f __H=tr;+iat__J _J ;r@=\q;l_rnr, 00 +@@

I

J I

G.

podfoe(il -2 z t o\ \: r( q 2 r_ob r ? \ :t("qu^ '* i"a .')o [J

Po-qo-i cr

Saa

LUOerS,

fr nv rr

a Jv \asmr n L r rl ya t v ,

F t

.

Z. Naturforsch.

T I dt

do,zPJ-"t, - ( q o i _ i a ),2" ./

al42

|

I (?2

\ r 4 ' r u1/3 )

Dyson, Phys. Rev. 83, 608 (1951);

2 0 60 e s 2 ) .

Sec. 32

153

Mass Renormalization @

II

( o l i \ ( x , u ) l q) -) o : r ' * "" J[ - , = u0 3 . ,= - , x a+e=@;@ x [I{0r1-a;-iaynE[ot1-a)] (01,p$)(x)lq). For very small values of or, (32.I4) becomes

(32.r4)

I

(ol@(x)lq):Xlo)(m - ')
(32.Is) according

(ol@(x,u)lq)- t'"'" Z!or(-m2)(01,tot@)lq): f

(

.:^-

) I

-z

-,,"^ do{rt,r-,)6=fiffi6 -i yno affi#w} " ft:z.ror | I

x ( o l ? ( o( *) ) l D .)

The overall factor e2o"o enters because we have to introduce the self-mass by means of a time-dependent charge. Instead of (32.1) we now nave

(ol tp(x) x I lq):(ol vft\@)lq>+;* f d.f i dx'eto{'-"t I l2n)" r

Qz'r?)

x o ( p , tm , ) e ( f ) ( i y- m f ) e 2 d i, 6i t . . ) u ( q ) e i t .r I o

The integrations over x and p in (32.17) can be done immediately and give

( o l v @ ) l q ) : ( o l v 6 ) '@ ) l q ) + ^ 1 - r ' " ' " 1 | =' 'q ; 1 "

@l .l I

2Qo

I

L2u'"q-m)-2@_

\ i|fx|

iy' x \ r t a g n t y + q o - 'm 2 -d( -q o + i d- - ) 2 ) +| } r s z r n t ) J)t l d .' ta- {' z o te -a- )t (. a - m '-\ :l a2+qqo l\u4'rul

0l llot (- a\

,

- urr a a qz---1oo,, i oy},o' ,,0,(x) lq) .

I

Here we have succeeded in making the calculation without encountering undetermined expressions like (32.9) . In the limit a-+0, we obtain from (3 2 .18) , @

(ol,t(x)lq): (ol?(o)(r) |q>- + Uy q - m1.[ a"x

" {W

-

( lp<,,1,y 1u>: 4'j" ;g} 1o1

/?,

I q\

: [,- +ft"L++- 2*#!#i] . r,.,r,r -0

Inthis approximation, the constant N in (32.8)is therefore given by N : I _ + (tj', (_ m2)*2 *2!0,' (_ *r)) ,

(32.20)

Sec. 33

G . K a 1 1 6 n ,Q u a n t u m t l e c t r o d y n a m i c s

154 tvrvrrhr eal rsa

2\ 2tli ( 0 ) ' /\ 4 r I

c li D

tLhr ra€

i rqrLal vi isr r o udeol rI V

o v rf

LFl t o l /\ ;yz \ )

rvnv rr Li fr hr

rr eeasyn ue v ct

fo

h2

As is evident from (31.15) and (3I.17), the integral for thjs derivative converges for a-->@ , although the integral for fjo (-*') | .rivr er or vr voro. s u

jn

Ilnrrerzer.

iha

fnrmar

infanrai

anothef

because the function tlt' (-a) only vanishes This leads to the divergence of the integral

afiseS

diffiCUlty

linearly

for a:m2.

@ €

f 2!o) ( - a\ da r(a-nt212 J o

f J

tk2

Elo't- ut @-nt2)2

(32.2r)

This divergence is of the same character as at the lower limit. the divergence discussed in (31.29) and (3I.30). we can circumvent it most readily by introducing formally a small but finite photon mass as a cutoff. That is, we replace the function d(42) in (32.12) W i t h t h i s t h e f u n c t i o n l l o t 1 p z 1b e c o m e s and (32.13) by 6(k'*p').

: #b- ry)v ?- -={)' (p,) 2t,

the The function (32.22) already vanishes for -p2<(*ltt)'and integral (32.2I) is also convergent at the lower limit. We now return to the matrjx element of the current operator (31.2). If we renormalize the mass in the equations of motion acc o r d i n g t o ( 3 2 . 5 ) , w e d o n o t o b t a i n 6 1 x 1 i n ( 3 1 . 2 ). R a t h e r , f o r the first two terms, we obtain forms like (32.16) which have been By a calculation similar to folded with a function S*(x-x'). (32.16) through (32.I9) and after carrying out the same charge renormalization as for an external fie1d, we finally get

(q I i,i,@)|q') : (q I i,f,(*)| a') l-il
(x)ytil(x):lq')[30)192)line(Qs(')(?')],f fi {t,+tl)(ql,,p@ Q : q' - q'

(32'24)

For the matrix element(31.2a) we have a similar expression, except at the appropriate places. that q is replaced by -I 33. The Maqnetic Moment of the Electron Considered as functions of q and q' ,the two termsinEq. (32.23) One term is proportional to i'i'@) show quite different behavior. and therefore has the same character as the "dielectric constant" In fact, part of (32.23) is the same as ofthe vacuum in (29.37). by the elec(29.37) and can be viewed as a vacuum polarization (3I.5). The with tron current, as was already noted in connection other parts of (32.23) have to be understood as the difference of as the interactions of an external current and the radiation field, opposed to the interaction of an electron and the radiation field. For example, an electron emitting a virtual photon picks up recoil momentum which is included in (32 .23) . Of course, this effect

The Magnetic Moment of the Electron

ICJ

would be absent for an external field. Moreover, (32.23) contains a term of completely different character; the term proportional to

-

(*) vb)@)|| q'>. rnL(qu* sL)Q | :rp6)

(33.r)

Using the Djrac equation for the free electron, we can write (33.I) as follows:1

(33.3)

or,: t (y,yu- yry,). We see that the electron gets an anomalous magnetic cause of its coupling with the electromagnetic field:

moment be-

(33. 4) ,*
,n-

value of (33.4) is

state the expectation

(ol:vkt (x)or,r!,(d (x):lq)s-(')(0).

(33.5)

Since the zero-th order current operatorcorresponds to a magnetic m^m6n'F

n i r z wa nu r Y f

l

h .' v /

(q l*,i',@)l q>: + (q l:'p@) (r)o,,rtbt(*):lq), in this approximation be come s

the total

magnetic

(33. 6)

moment of the electron

l t + S t o r l o(yql l * , ; , , ( n ) l q ) . As has beeh noted above, from (31.33) we have

the integrut

: 5ror(o)

(33.2)

5tol(0) is convergent,

e2

d,

8lr2

2n

and

(JJ.

t

O,

where2 ez1 4tt

1 3 7. 0 3 6 4

( 3 3 .B a )

Thus the electron has a g -factor which is not exactly 2, as it would be in the Dirac theory. In this approximation its value is3

c:2(r ++):2.t.oo1t6t4 1 . W . G o r d o n , Z . P h y s i k 5 0 , 6 3 0 ( 1 9 2 8 .) 2. C. f. Sec. 36 and 37. 3. The anomalous magnetic moment of the electron was calculated by J. Schwinger, Phys. Rev. 73, 416 (1948).

r?? o\

first

ls6

G. Kii116n,Quantum Electrodynamics

Sec.33

It is remarkable that the finite magnetic moment of the electron can be singled out even in (3i.31) and (31.34) by considerations of invariance alone. It is not absolutely necessary to carry out mass and charge renormalizations if only the magnetic moment is to be calculated.l Experimental investigations of the magnetic moment of the electron were performed in 1947 with sufficient precisionso that there were doubts about the validity of the Dirac theory.z This was due parlly to the measurements of the hyperfine structure of the hydrogen atom and of the deuterium atom and partly to exact measurements of the Zeeman effect in a few complex atoms (Na, Ga, In). G. Breit had shown rather early3 that the equality of the present deviations of the hyperfine structure in hydrogen and deuterium could be interpreted in a natural way as an anomalous magnetic moment of the electron. The later theoretical va-lue (33.9) deduced bySchwinger agreed very well with the original measurements. Recent more exact measurements4 of the magnetic moment of the _electrondo not differ very much from (33.9). Franken and Liebeso find the value

. E : 2' (r.oor167+ 0.000005)

( 3 3. r 0 )

Equation (33.10) agrees completely with (33.9) to within the limits oferror. Since Eq. (33.9) is onlythe firstterm in a series in or, one may ask how large the next term in this series is and whether it will disturb the agreement between theory and experiment. Thg term proportio_nal to a2 has been calculated by Karplds and Kroll,o Sommerfield,T and Petermann.5 gy quite involved calculations which we shall not go into here, they obtain the result

: c: tln+ *+#Gr *4 ++e())- *n'l"c2)] -

*a

*21

I + 0 . 5: - - O 3 Z g : = l : 2 . 1 . 0 0 1 1 5,9 6 J .

" v

)

,,, Io,

I. See also J. Luttinger, Phys. Rev. 74, 893 (1948). 2. I. E. Nafe, E. B. Ne1son and I. I. Rabi, Phys. Rev. 71,9Ia(L9a7); D. E. Nagel, R. S. Julian and J. R. Zacharias, Phys. Rev. 72, 97I 09a7); P. Kusch and H. M. Foley, Phys. Rev. 72,1256 (1947). 3. G. Breit, Phys. Rev. 72,984 (1947). 4. S. H. Koenig,A. G. Prodelland P. Kusch, Phys. Rev. 88, i 9 1 ( 1 9 5 2 ) ;R . B e r i n g e r a n d M . A . H e a l d , P h y s . R e v . ! ! , I 4 7 a $ 9 5 4 ) ; P. Frankenand S. Liebes, Phys. Rev. I04, LI97 0957). Areview of the older measurements is given by F. B1och, Physica, Haag 19, 821 0953). More complete references are to be found here. See a l s o E . R . C o h e n a n d J . W . M . D u M o n d i n V o l . } C C ( Vo f t h i s handbook Eandbuch der Physik, edited by S. Fliiqge, SpringerVerlag, Heidelberg). 5 . A . P e t e r m a n n ,H e l v . P h y s . A c t a 3 0 , 4 0 7 ( 1 9 5 7 ) . 6. R. Karplus and N. M. Itoll , Phys. Rev. 77, 536 (1950). See also A. Petermann,Nuclear Phys. !, 689 (1957). 7 . C . M . S o m m e r f i e l d ,P h y s . R e v . L 0 7 , 3 2 8 0 9 5 7 ) .

Sec. 34 with

Charge Renormalization a

c0 : Z *

of the Electron State

: 1 . 2 0 2.0 6

I57

( 33 . 1 1 a )

I

The numerical agreement of (33.10) and (Sg.tt) is excellent. The experimental limits of error are so large that the term proportional to a2 in (33.11) can scarcely be viewed as confirmed. On the other hand, there is no discrepancy between theory and experiment: the quantum electrodynamic description of the magneticmoment of the electron is certainly correct in its essential points. This is a great success of the theory and gives one justification for the mathematically faulty formalism in which infinite expressions are treated as if they were finite. The Charge Renormalization of the Electron State We now return to the complete current operator (32.23) write it as 34.

and

(Q')* n o,(o)-ine(QII(o)(0,) Qli'i'@)lq'): (qli'fl@)la'){-fi
space,

Q I Q l q ' ) =4 [ d s x ( q l , j n @ ) l q ' I) =d-' *i ( s l i f , @*)i L ' ,( r )+ . . . 1 q ' ) = l : e|l+ R-(0) (0)- 5ior(0) lts+.21 - t;r,(- m2)- 2mtlo,'(- lrxr)+ ...f . l In order to obtain (34.2) we have used the same charge renormalization for the electron as for the external field. However, from (34.2) we see that the charge of an eiectron state does not seem to be equal to e, but that an additional charge renormalization

(o) - tio,(- m2)- 2m 2rov(- mz) ;ror (0) - 5
(34 .3)

enters. This expression is not present for an external field. It contains two infinite integrals, namely R(.)(0) and i;or (-m2), and these are present with opposite signs. All other expressions in (34.I) are finite. We have two possible choices at this point. The first is to say that an additional charge renormalization for the electron, if it exists, must be unobservable and therefore that the corresponding terms in (34.1) can be dropped. This choice was made by the earlier workers in the subject. A second choice is possible, however. We can cut off the two infinite terms in (34.3) by a suitable llmiting process and then consider a possible cancellation of both. The assertion that an expression like (34.3) (which actually contains infinite quantities) has a finite value or vanishes is clearly nonsense; ong can give it a well defined

Sec. 34

G. KAI15n, Quantum Electrodynamics

158

meaning only by means of a limiting process. To do this, we cannot simply introduce an upper limit in the integrals (3I.35) and This (3I.IB) and then set these two cutoffs equal to each other. is not possible because the two variables of integration 4 have nothing to do with each other and cannot be directly compared with each other. lnstead of this, we must return to the original integrals (3I.8) and (31.21), cut off both of these integrals inthe same way and then work out the expression (34.3) once more. We shaII take the regularization method developed in Sec. 30 as the We modify it here by thinking of the electrons cutoff procedure. as being coupled to several kinds of "photons " of different masses. Zto' , ft(0) ,and In principle, we should recalculate the functions S(0) for various photon masses in order to form such linear combifunctions (31.18), nations that the integrals for the "barred " (31.35), (31.36) converge. Such a calculation can probably be done; however, it is a little more efficient to use (31.8) and (31.19) as follows:

., ,,l

: # ouliv(q+ k)| zmfx * (iys + rn)Eio'(q')1."* f4g,(q') I

.n. \ - r [ a ( t qr H 2| m 2 '),r d ( A ' / : ) I *-pi 1o-t1z1nz)' +"nlIn the final result we shall take Cr->\,

IC1:O

,

/h+0,

Fr.+ a

(i'+D

,

are the other "photon masses". where h U*I) Using (30.6)and a few simple transformations, we obtain from (34.4), ',nz)frec: [t]t (- ilxz) + 2vt, f;$'(I

: -

r

+F

:6'n- " [ a " ] U - " ) J

*2n1r T

l'

+) ] [ w a w JI a n47 , C ; e i ' & ' t n 1 d 2 + 4 i t r - q

J

-a

a(t;a'z\ ."r+*r(r*")J \-

^

: - #2c, orL--

' 2:n-

^1

{

(34.5)

(

2a(l-u2\ r ,,? r\, -a)log(a, + ------,?+ f, U - 6,)) I aal( 1

In the first term of (34.5) we integrate and make use of

1

t,

, ",ail(t-,I) by parts with respect to c(

***

*"'*' ; : c, I | -i!#@Zr, I a 2 l ! : ir:,,, i J a2l!4(r-u\ i J (r-a\ ) an P? (l -c()/(c()

(34.6)

to obtain

[IJo'(- n2) | zrnlioi',- *\1'"r : t -za*Ia| _Lrl _ : _ _ , , y c . f t : !_ - Tl 4 Fi . o o J , l _"1 l;;4(, _")

(34.7)

Sec. 34

Charge Renormalization

159

of the Electron State

The integral (34.7) can be done by elementary methodsi however, we shall leave it in its present form until later. In a si.milar way, we find for the constants ptot(0) and S(0)(0): rer- - ((o)161r vr t t Llp(0r161 " \",/ \"i.1 "

(q- k1- n) y,(i y tqt A\ y t \r :;l O r r J I a n y,g

hl- m) y ^xl I

x t c -', J| (q , - h)2 , , .+, tn2f d (l(q' o -- 1,h12 ,rfa,f lntz) ), = '+

, '

+ d((q hf-nf1 , (h, pit -tq'- k)2r mz) '

I d(rq'-h121*21 ) | - h ) 2- l * r l J q _ q , ' t p z' p , , i ) ' \ q J

The expression (34.8) is to be evaluated underthe made after (31.20) . By means ofl

!b9' r- + '

ltro.ul

restrictions

1d

A(6) jg: a ' c +' o ' b

" d , ,\ -(-ar p' . _ b t a -r ' t + c , - , , ) ) ( 3 4 . e ) [ 0 "- J I- -dr p J

we can combine the terms inside the curly brackets of (34.8): ld

{ . . . } n : t : I d aI d p 6 " ( a ( 1- kq ) z+ l n ) + ( k ,+ p ? ) ( r- . ) ) : I 00

1

I (34'10) I

. : I u d . u d , ' ( (-A q d - ) 2 + m z a 2 * p i U -d,)). 0')

S u b s t i t u t i o n o f ( 3 4 . 1 0 ) i n t o E q . ( 3 4 . 8 ) a n d u s e o f t h e s y m m e t r yr n gives fi':k-qu

--5
I

r, 6"1k'z1 rnzu2I tt?( I -o.))'i-A,, Tu* 4mzyu(r.- - - *ll 4

The A'-integration result

is now convergent,

-itorlsl]rec : -;*4t,1 [R(o)(o) {

,

,,?

(34' 11)

I

can be done, and gives the

I

adqX ,

,('-"-+)l

x jrog(",+Su_,.))*+ il+l

uz+4u-a\)

t

I }(sq.rz)

I

l

H e r e w e c a n t r a n s f o r m ( 3 4 . 1 2 ) i n a m a n n e rs i m i l a r t o ( 3 4 . 5 ) [ b y applyins (34.6)] and we obtain

- 5-ro, *r: ;+z r, ++l' ':;.I "" (o) (0)] [R-(0) I +LI( : + . :+ + \ t _ d.* ' t o ld-

: tr-rr)(-mz)| 2mr1o),(_ *r)f,"t.

_l f

I

tt)

Equation (34.3) now vanishes by the use of the identity (34.i3) and, in this approximation, we have the same charge renormalization for an electron as for an external field. In a later section, .i. I-S"h*i"S"r,

Phys. Rev. 76 , 7g0 0949) .

I60

G . K A l 1 6 n ,Q u a n t u m E l e c t r o d y n a m i c s

Sec. 34

we shall see that this result is true not only in this order but can be proved quite generally. The relation between i-nfinite constants expressed in (34.13) was first used by Schwingerr and later shown by Ward2 to be true in general . The equation to which (34.13) is the first approximation is often referred to in the literature as the The second approximation form

to the current operator now takes the

_, (ql if\ @)lq,): (qlilil@)lq,) * Rr"1nz, [_z(d(0,)+ rr]0)(0) - R t o r ( 0-)s ( 0 ) ( ? r )+ s ( 0 ) ( 0-)i n e"(,Q ) o), Q r . -t r o\ rt p/z l

_ Ro)(0') * sot1n,,,,_t i Q,(q lmf)(x)lt,) x . x fStorlgz, *ine(Q s(.)(0,)]

AII expressions occurring in (34.I4) are convergent. (24.35), (31.32) and (3I.33) we have pz

2m2\t r 7 $ ) ( Q 2 )1.:2=1 fl" r\ - ; ) 1 1 r

l-----

t*

^.

From Eqs.

(34.ls)

*II@(Q\ 1fftor1o1: ez 10,,, s , I :r;dltr"-T+(1

R(o)(02) :_

I

--l

''lt'*#l l,rn.,u, +$*,^^ -l + m 2l l I 1l

2mz\

l^o--l

Q'I

11r-

L'

4m'

"l

I

l ' _ Vr * _ T | )

f ,t'n'' .2 l-, nz e2++nz1 l__j_1." ^ _ 1 ,

6fl'l

l(34.14)

I )

Q ,- 4 m , ) ,

+ #vo ( -

1,.

uz

|

I j1 tf2 l

I

Q'

d*l

1 V +i

-

Ot- Q2-4n2),(34.17)

R-c)(0r) _ R(o)(0) :

T.#),-

, ,^_,ll'*#lt+f | -+#bc; -;rogz,--r-=-----:-, l J''t

r*"FY'$ll, L-2*'

t-

t,-m tl

"l

( 3 4r. 8 )

\re v;g

- ' t o- sP,l! l , _ ' -=J-tosl - 1 , _ L lyl , -_on*,, l,

: -#6 S(o)(?r)

'+/i;,1

\,

@(- Qz- +*z\ | [-

'(34.19)

4tt'z

V'+_F

I . J . S c h w i n g e r ,P h y s . R e v . 7 6 , 7 9 0 ( 1 9 a 9 )s; e e e q s . ( 1. 9 5 ) , ( 1 .9 7 ) a n d ( 1 . 1 0 0.) 2 , I , C . W a r d , P h y s . R e v . 7 8 , I 8 2 ( 1 9 5 0. )

Charge Renormalization of the Electron State

16t

nf

S(.)(0')

(34.20)

In (34.18), @(r) stands for the function f

)t

< D ( * ) :l # t o s l l - ' r l .

(34.2r)

1

It satisfies many functional lowing examples herer

equations

of which we give the fol-

e, < D' -(,x*'\ @ ( 1\ - - a l o ' - " o rt -""l, - 4 @ ? 2 \xJ 2 " \

x).

.-t t

(\ r3+ a ,2\ .LLl

r D ( r+) A e 1 - x ): - * + t o g l x l t olgr + x l , 8 4 . 2 3 ) (-x \Q ( - x \ : @ I' - "xI )' . \(v3" 4 -q\D - ( - - 3 - l - O^l 3 : - l *' t 4- n ' 'o" (--\ e ". ? . 4 \ "/ t-.rJ t+xl \

\

I n ( 3 4 . 1 6 ) , ( 3 4 . 1 8 ) a n d ( 3 4 . 2 0 ) , i t i s a s s u m e dt h a t

,' rt+ a T = ' i " Q2

positive. If this is not the case, the logarithms in these equations and in (34.21) must be replaced by arctan functions. For smal1 values of Qzlmzwe find

R ( 0 ) (-0R,-)( 0 ): (-0 )_ 3 "# l " r # -

:#-#"#* s(o)(o,)

+]*,

G4.2s) t J z + .z o /

1 . S e e , f o r e x a m p l e ,B . K . M i t c h e l l , P h i l . M a g . 4 0 , 3 5 1 ( 1 9 4 9 ) and W. Grbbner and N. Hofreiter, Integraltafeln, Vienna and Innsb r u c k , 1 9 5 0 . E q u a t i o n( 3 4 . 2 4 ) ,w h i c h i s n o t g i v e n b y t h e s e a u t h o r s , can be shown in the following way: tt

< D (-xe\ ( - u ): [ * t" r t r ! / l : [ 4 z" r l j"-1l1+- zl l J"

- @ ( -r ) .

I

The substitution

1 +y:'

a @ \- o ( - , ) : I+x

:-{ r, -1

*

-

L 1+

in the integral

gives

+€

y -@(- 1)- @(- 1- o o, +)Log |,1 | I l+ #l -6

1 L e

: - o ( - 3 2 \ + f 4 r o g l r - r l - n z @ ( -1- x ): \t+x/ J t

- (,(-3:-)++ - n2@(-,-,), : * (#) since @(-r) is equal to -!

accordinqto (34.22),

logIr-Frl=

162

G. Kiillen, Quantum Electrodynamics

Fromthis and (29.36) follows

: #l^r; - #]$ tz+'zzt + #zret*lf)@)tq') +.... I The expressions obtained so far contain only integrals which are convergent for large values of the integration variable a.and which can be calculated. Equation (34.18) and hence (34.27) alsodepend upon the quantity 1t', The iimit pr+g cannot be taken here without encountering new diverqences. Therefore our results cannot be compared directly with experiment. What we must do isto discuss the experimental conditions more carefully in order to decide how ,a is to be interpreted in each special case. 35.

Radiative Corrections to the Scattering in an External Field. The Infrared Catastrophe As an application of the results of Sec. 34, we now examine the radiative corrections to the scattering cross section for an electron in an external field. The lowest order approximation to this cross section has beenworked out in Sec. 24. With the assumption that the external field can be treatedl in Born approximation, we can write the S-matrjx as follows:

s - i / dxiu@)A"*"*(r) .

(JC.r/

The quantity ir@) in (35.1) is to be understood as the complete We obtain the result of Sec. 24 if we consider current operator. only the lowest order approximation to the current. The radiative corrections can be obtained by using higher order approximations for the current. Using the quantities defined in Sec. 34, the next higher approximation to the S-matrix becomes

( s I s l q ' ) : i ( q l i , l q ' >A i i ' "( Q )z n6( Q s ) ,

( es. z1

where

- R(0)(0) (q li,,lq') : (q li(f)lq') j - fr rot l 11(0)(0) + R(0)(Q'z) + 1gz1 I,r, - s(0)(0,)j; Ar1 n;12(q\t-V),) 1 Sioi(o)l Q:q'-q. In Eq. (35.3)we have used the fact that the vector Q is Iike, so that the " unbarred " functions R , S, and 11 Using the methods explained in previous sections, we cross section for unpolarized electron scattering in an field:

.,, ( 3 s. 4 )

spacevanish. find the external

1. The second Born approximation in the external field has been calculated by R. C. Newton, Phys. Rev. 97, 1162(1955). See also R. G. Newton, Phys. Rev.9B,1514 (i955); M. Chr5tien, Phys. Rev. 9 8 , I 5 I 5 ( 1 9 5 5 ) ;H . S u u r a , P h y s . R e v . 9 9 , 1 0 2 0 ( 1 9 5 5 .)

Sec.35

#

:

The Infrared Catastrophe

163

-a tu\ ror @r -n
in (35.5), terms of order ao have been dropped, since (35.3) should be improved if this order is to be retained. AIso, it has been assumed in (35.5) that the gauge of the externat field satisfies the Lorentz condition QrAii*(9):0. In particular, if the external field iq a (lnrrlnmh ficlrt then

4,(Q):-i6u4--=o.

(3s.6)

4P2stn2,

Using the lowest order cross section ofSec.24, do@) I ,r, na : t;;;ql \ -

\, l-"

," "@\ \''- p"sln't

/? q

7\

2/

we can rewrite (35.5) as follows:

, ,^'[ -'.or1ovn'',?,:!,')+s.'',n', :L: #lt -z@*t@ I,,,.r, -S'(o)(0)) - *'^u 'U 14_r"cos2|

zS.r(?r)1, I

Q' : qh'sin'3

I

,|

( 3 s. 9 )

i n ( 3 5 . 8 ) t h e f u n c t i o n p t o t1 g z 1 - p t o )( 0 ) i s s t i t l p r e s e n t ; b v ( 3 a . 1 8 ) this expression contains the small photon mass lz. In order to interpret this quantity, we note that in any actual experiment it is never possible to measure only the ejastic scattering (35.8). One can never be sure that the electron has not radiated a photon of very low energy; the detector always has a finite resolving power and electrons of energy only slightly less than the incident energywill be counted as having been elastically scattered. This effect is one form of bremsstrahlung, and so we can take the cross section of the process from Sec. 26. If the energy of the light q u a n t u mi n E q s . ( 2 6 . 3 ) , ( 2 6 . 1 0 ) , a n d ( 2 6 . 1 3 )i s t a k e n a s v e r y l o w with respect to the energy of the electrons, we obtain dlostrahr dQ

da\o) e2 S orol'*"*4p25;nzo(2n13 J z. I hq . hq' Itrl< lE

dQ

(h ql'z

#1^3s'10)

Here we have integrated over al"l emitted photons of energy smaller than the resolution AE of the measuring apparatus. The integral (35.10) over the emitted photons diverges for small energies and again we shall cut it off by giving the photon a small mass. Thus,

Sec. 35

G. KaII6n, Quantum Electrodynamics

164

instead of (35.10), we have to examine 4n'"'-'o

d o s 6 a-6d1o-ao ,,,1'*'* , , , , , ^ -r -vo: to) ez f u dfilhl<.tlE n6(kz lf)o (k)l-#L

l

-,#-

#]'

as'ul

Since we now have cut off (35.1I) and (35.8) in the same way, we can add both cross sections and see if the limit pl->0 can be taken in the total cross section. In order to compute (35.11), we have to work out the integral

I : J analn'+p')o(k)r;rV: t l.oiu

: ' f ,4 4 - f !

/8

r

-

oau

,

yna+r,, J (pk- EVFar9 @,n- nll@Jp\

f

zJ oto

:-

(3s.12)

1

kzdlkl

f ,

dQ,

f

^r-.

J'*J

IF;7

itp+9") k-E\?tr1p2_z

For the last transformation in (35.12) we have used the identityl I

lfdu

(3s.13)

t-

a.b

J

lau*b(l

0

11-1sA-integration and gives

in

(35.12) can be done by elementary methods

I

r -c* r -'ru

f du !'t^o2/E J ,4 t^"" tt u

E tno" L '. '>"t @ - J ' " o

I

.-

tVL'-A

I

|,

( 3 s. 1 4 ) (3s.r5)

A:m2lQza(l-a).

B y t h e u s e o f ( 3 5 . 1 4 )a n d ( 3 5 . 1 5 ), w e c a n w r i t e ( 3 5 . 1 1a) s f o l l o w s : do(o) e2 f, 2AE -TTZFltoE r

ddstrabl --7d-:

tr:

-

I

t,

rI

J

l-, -

-l rzl '

( 3 5. l 6 )

"t - t

l

(3s.17)

),

02

mzaaz

r,2=.la.d lA1-6a, *, 0-

i.d e f *++ gr^, =,

-

E tllEj-frii77"EF *\. F lr

I ^^-

-

E_ r

IF_gu-g*"1 T.

I.-A

2P'"o E+p

- 1,.. ,o'

ll'""""'

T h e a - i n t e g r a t i o n i n ( 3 5 . I 7 ) c a n b e done easily, although the integ r a t i o n i n ( 3 5 . 1 8 ) i s q u i t e t e d i o u s . 1 A / oa i r r e n n l v t h e r e s u l t s : Y v v

l.

Y r v v

R. P. Feynman, Phys. Rev. 76, 785 (1949).

The Infrared Catastrophe

Sec.35

2,rtz t, _rtr14\_

r\z lt, ",_,/,^,VTry_l - f

o,=--Zl '(-F V, y r'*@/-t-u\'"-re-') r

. E-

E+b

*IoE B_p

I' ' L 2 m z

lll@. ub Bn - .V r u'g' 1 - ; l L r c *rE

vt*-F

vt

-

e"

' '

l

+, L+(Et ,t\r- ! ,, \ t

e, I

at_l

' : ;kgt._? (#)')-'(- ffi))* - bg++ r"rs##1,

+tq# bgYig!{

@\ ^l P I t + s t n -- '1> ' t ' f;: -r t @\ ^f F{t A stn-1 y:--------=:11 rTU

(3s.20)

'

t

,,, |,,, (3s.22a) r?q ?rh\

d:

(3s,22c)

P:*.,.

("E,

ttA\

T h e f u n c t i o n @ ( x )u s e d i n ( 3 5 . 2 0 ) a n d ( 3 5 . 2 1 ) h a s b e e n d e f i n e d i n ( 3 4. 2 I ) . A d d i n g ( 3 5 . 8 ) a n d ( 3 5 . 1 6 )i n o r d e r t o g e t t h e t o t a l c r o s s s e c t i o n , t h e m a s s p cd r o p s o u t o f t h e r e s u l t b y t h e u s e o f ( 3 4 . 1 8 )a n d ( 3 S . 1 9 ) . '9It i. therefore independent of the phoThe observable quantitv dQ ton mass and the role of the cutoff has-now been assumed by the resolution of the measuring apparatus.r For very small values of 1JE , the probability of emission of many photons has to be- considered also, as Bloch and Nordsieck and Tauch and Rohrlichrhave 1. The first discussion of the infrared problem was given by F. Blochand A. Nordsieck, Phys. Rev. 52, 54 0937). See also W. Pauli and M. Fierz, Nuovo Cim. 15,167 (1938). The treatment used here is due to ]. Schwinger, Phys. Rev. 76,790 (1949). A similar discussion of the higher orders of the problem has been given by J. M. lauch and F. Rohrlich, Helv. Phys. Acta 27, 613 (1954). See also p. 258 of the book by these authors.

166

Sec. 35 d?It re-

G . K i i l 1 5 n , Q u a n t u mE l e c t r o d y n a m i c s By these considerations, it can be shown that

shown.

dQ

mains finite even in the limit AE-->0. In this problem we can regard the motion of the electron as given, so that this is really a problem of the interaction of photonswith a classical current. Just this problem was already discussed in Sec. lland we shall not repeat the details here. At least for the accuracy of measurement presentl_y attainable, such a refinement of the theory is not neces sary. * The complete expression for the total cross section can certainly be obtained from the above equations, but it is rather comnlinrtarl \A/a fharafnge restrict ourselves tO the extreme relativistiC limit. Under these conditions , the cross section simplifies to yrrveLev.

dotot

dt)

doro, -_ --aT ,\t^ -

with (1, f,s d" : ; 'etzl t " g {/ 2; lEsl i " :@ l Jl ' - ; lr[l o

E

rE

-

A\ u)t

(3s.23)

131 17

1

; l + ; - ; s i n 2 ; I"lO , l

(, ^3 s . 2 4 )

and -r -

r

| / n\ -2 / l o ) l - s i n 2 : ) + + - l o g { s i-n- o 2 \i l- l^o' -SO l cto, s z |: - " O ,lll . / : ) l - I' " - ' 2 ) ' t 2 2J--or"-- 2Jl' 2sinzi' 2

3 !S . 2 5 ) \( u

or nf2, the function I For special angles, for example, @:n can be evaluated exactly by means of the formulas given in(34.22) through (34.24). For arbitrary angles there are detailed tablesz of the function @. That a finite cross section results after the renormalization of charge and masswas noted by several authorsJ in 1948. The first careful calculation of the cross sectionwas carried out by J. Schwinger4 and we have used what is essentially his method. A more exact experimental verification of Eqs. (35.23) through (35.25) is still lacking. Certainly many measurements have been made of the scattering of high-energy electrons by nuclei.5 First, the accuracy of these measurements is not high enough so that the results can be said to confirm the theory. Furthermore. composite and for these the spatial extension nuclei are usually involved, lndeed, of the nuclear charge distribution has sizable effects. the deviation from Eq. (35.7) caused bythe nuclear size is much

l . S e e ,h o w e v e r , E . L o m o n ,N u c l e a r P h y s . 1 , 1 0 I( 1 9 5 5 ) ;D . R . Y e n n i e a n d H . S u u r a , P h y s . R e v . 1 0 5 , l 3 7 B ( 1 9 5 7.) 2 . S e e , f o r e x a m p l e ,B . K . M i t c h e l l , P h i I. M a q . 4 0 , 3 5 1 ( 1 9 4 9 ) . , r o g r .T h e o r . P h y s . 3 , 2 9 0 ( 1 9 a 8 ) ; 3 . Z . K o b a a n d S . T o m o n a q aP H . W . L e w i s , P h y s . R e v . 7 3 , I 7 3 ( 1 9 a 8 ) J; . S c h w i n g e r , P h y s . R e v . 7 3 , 4 1 6( 1 9 4 8 ) . 4 . J . S c h w i n g e r ,P h y s . R e v . 7 6 , 7 9 0 0 9 4 9 ) . S e e a l s o L . R . B . E l t o n a n d H . H . R o b e r t s o nP, r o c .P h y s .S o c . L o n d . , A 6 5 ,1 4 5( I 9 5 2 ) . (

Soa

fl vnr r

a vamnla uzrqrrryre,

JL -r rhs o

nnmnrahanqirra vvrlryrv

R e v . M o d . P h y s. 2 8 , 2 I 4 0 9 5 6 ) .

rarriarrz

hrr

v1

P

r\.

T{ofqiedfar

rrvleLsglv'

Sec. 36

The Hyperfine

Structure of the Hydrogen Atom

167

larger than the radiative corrections of (35.23). These measurements are therefore more a determination of the nuclear charge distribution than a verification t\4nrc roconrlrz lhsss -;; of the theorv";;";;";;;"

havebeen

r,trT;t

;;i;;

;il

;;";i:;r""'

by protons "*p"ri." and here the charge distribution is not such a big effect. The experiments have been done so as to determine first the structure of the proton from the angular distribution of the scattered electrons. Then the results can be used for a comparisonZ of the theoretical and experimental values for the total cross section. With an electron energy of about 140 MeV, Tautfest and Panofsky obtained

o""p/o,n"o.:0,988+O,021. Since the theoretically

important

is nf nrrler nf maonitud" ]tog!-, n-'m

/?R

part of the radiative i.e.,

,A\

corrections

about 4% at I40 MeV, the

experimental result (SS.Z0) cannot be viewed as an exact verification of the theory. At least these results are not in contradict i o n w i t h E q s . ( 3 5. 2 3 ) t h r o u s h ( 3 5 . 2 5 ) . 36.

The Hvperfine Structure ofthe HvdrogenAto* As the next application of the results derived in Sec. 35, and as preparation for the discussion of the Lamb shift of Sec. 37, we shall now study the hyperfine structure in the I s state of the hydrogen atom. We are therefore considering a system in which the electromagnetic field is the sum of the external Coulomb fieid of the proton, the field of the magnetic dipole of the proton, and the radiation field. we take the electron in the electrostatic field as the unperturbed system with energy levels given by the Dirac theory of the hydrogen atom. We are concerned with the splitting ofthe Is level underthe influence of the magnetic dipole of the proton, and corrections arising from the coupling of the electron to the electromagnetic radiation field will have to be considered. First we shall consider only the magnetic dipole of the proton as the perturbation and, from the elementary principles of quantum mechanics, we obtain the following expression for the splitting of fha

anarntr

Y7

rl au -v' va fl ., .

6E : - [ d.3x(nl ir@) ln) At r (n) ,

(36.I)

: +# Af;ae(r) Here pr is the magnetic

moment of the

(36.2) proton and

( " 1 i , @ )l n ) i s

1 . R . W . McAllister and R. Hofstadter, Phys. Rev. 102, 851 (1956)E ; . E . Chambers and R. Hofstadter, Phys. Rev. 103, 1454

(1es 6).

2 . G . W . Tautfest and W. K. H. Panofsky, IJOc' (IYb //.

Phys.

Rev. 105 ,

G . K i i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s

168

Sec. 36

the expectation value of the current operator in the unperturbed For the "unperturbed state" we must state under consideration. Since an exact exconsider the electron and its radiation field. as an pression for this expectation value is clearly not possible, we shall develop it in powers of the additional approximation, radiation field. We cannot use the results of Sec.34 directly befield is strong enough so that it must be cause the electrostatic In principle, we treated exactly in every step of the calculation. can do this by transforming the differential equations for the field

operators, la\

Vz; + m)v@)i ey (Ast'"rt(x) f :

,{coulomu (r))',p(*),

(36. 3) r?A 1\

J Alt'^ht (x) into inte gral equations :

(36' 5) p(x')dx' , v@): rp"(x)- ie J so@,*') y trst'at'r(x') D o ? - x ' ) l r i ? ' ) , v , v ( x ' ) l d x ' .( 3 6 . 6 ) ) ': , 4 f " " \ r (:x )A ' f ' ( r + I In (36.5), yf(x) is the solution of the equation * *)v'@):

Q*

i g v t r c o t t o m o ( r ) r t," @ )

(36.7)

and S^ (x, x') is given by t a \r )* ( x , x ' ) : m ) 5 " 1 x , x ' )- i e y X c o u t o n b (S l, * + So(x, x'):g

,

for

- d(, - *'), (36.8)

xol xt

/? A

o\

This function is a singular function of the same kind as was studied an explicit calculation of this function in Sec. 16. In principle, is possible, since the eigenvalues and eigenfunctions of (36.7) are known. Up to now, this has not been done. Next, if we consider the radiation field in (36.5) as a small quantity, we can iterate these equations in the usual way and thus obtain the first This correction is non-zero correction to the current operator. exactly what results from (31.2) throuqh (31.4) if the singular funcSa(*-*'),and S ( 1()x - x ' ) a r e tions of the free electron, So(r-x'), replaced everywhere by the corresponding functions for the bound S n @ , r ' ) , a n d 5 t t ' ( x , . r ' ). I n p r i n c i p l e , t h e S*(x, x') , electron, expression obtained in this way takes account of the Coulomb field exactly. This procedure leads to

dE:

d E r o*r d E , t r ,

( 36 . 1 0 )

: - + f #x(nllrt"(r),y,rtc(*)lln)Af,s(n), (36.l1) dE(o) 2.1 d r ( l ): -

[

a ' r 3 1 6 i * @ ) l nA) f s ( n ),

/?A ir\

Sec. 36

The Hyperfine Structure of the Hydrogen Atom

169

(x),yuSo@, x')@"(r')l'r ::[ax'76c(x')Sn(x' , *),yr.t"@)l airt l : * Idx' lrt'c --'i- a- a*" lrtc(x'),y^(st1r1z', x)yrSn(x,x") Do(x"- x')I [ /2A

r1\

+Sn(x', x)y,,S(tt(x, x") D^(x'-x")+ So@', x)yoSo@,v") Dtrt(x'-x"))y^tp9(*") (r, r')y,S(r)(*', r)] iltc" (Sp +u:: [[ d.tc' L'P \ T [z,So 4 JJ + Sp [2, 5ttt(x, x')y,St(x', *))) D^("' - ,")lrtt" @"),y,rt" @")f .

@c(x): -t

x')Dn@'-x)*So(r,x')Dtt(x'-x)fy^rp'(*')d'x'.(36.14) Ir^[S(rr(r, We beginby evaluating(36.11). From(36.2)we find d E ( 0 ) :- # l

Fxu,(r)rnun(r)VP.

(36.15)

of the time-independent Here wn(u) is the eigenfunction equation corresponding to the state Iz) :

u,(n) s-i E"xo.

hydrogen atom with

(3 6 .16)

nuclear

u,(r):#V,0,;; i'3, A,i'-,'' :;TI :#tt,t,o * ;n' f;T - o+:;d tI -t o r s , : u^(n) tLv.

l?):

(2nzdr)Q l l @r Q a + 1 ) r ll

Djrac

-z

charge Ze we

-rr l -;- /

e - , f t u ,,

/

( 36 . 1 7 a ) (JO.r/O,

( 36 . 1 8 )

e:1ft:Zza

( 36 . 1 e )

From these, we find after a simple calculation iu*(u)ynu,(r):

tn x Efh

- t9::n

7!Q

r

,

(36.20)

where s is the vector 5 :

(36.20a)

(0, 0, s,).

Substitution into (36.15)gives

d E (') :- : - -

, ", 2 [ dsx1@, +((g s)zr- krc).( sr ):)

: - ; 1 +,a : ; " , ' *( f, iosr)t JV-,J 4 r l ' \ / ) : - i

I 4 e , ,mtzrcr (lrs) 2* z AEa;|

I l(36.21)

,| We can expand t:3.fgl in powers of Zafor small values of this quantity. Accordingly, (36.21)simplifies to

t€ll : - +* tp") dE(o)

+ ) z,",+ ...1.

(36.22)

Sec. 36

G. Kiill6n, Quantum Electrodynamics

I7O

Moreover, since the magnetic moment of the nucleus is proportional to its spin 1, we get

( r , s:) *2,1(1t"(- I + 1' )- 1 ( r + 1 )' - +4)l

(36.23)

from the well known rules for the composition of angular momentum. In (36.23),' I stands for the total angular momentum of the we have whole atom. For J:11'{, I"A

ktd:*, a n df o r I : I - 4 ,

Hence ttle ls ferencel

: -*?+|) (r,s) state

into two

split

is

, A\

tJo.zc,

an energy dif-

with

states

zo:ta,ur4;!t+-lr+j2,",+...], e

(36.26) (Jb.z0a,

lt"-- ,*

The energy difference (36.22) is of the order msp"prZsas-E* ,wLth If we were not interested a correction term of the order Zza2'E-. in this last term, we could have derived the first approximation to (36.22) in a simpler way. By the use of the Gordon decomposition of the current operator [c.f . Eq. (33.2)], we have

-,

i u ( n ) y o u (:n+)( r " "

h)t

r'*

- f , , @ o o 1 u ) .P 6 . 2 7 )

Now by the use of (36.17)it follows that i

-

lAu

2m\*0"-u 1

n*

?u\ rro):

-

ucnru --

lt(v\ azZz [r> slb I z m n \ 1+ a Y - - t ' :

l2(r)

t^

2 m x t ( r+ p l [ "

-

u2Z2 (st)r'nl

T- p

*

)* ,

(36'28)

(/ ?r r^ " ?' -ql ' l

If all terms with a factor and cyclic permutations of A , I , tu. Zzaz can be dropped, then we can write approximately i

IAA

-0u\

"*\i"-'*u):o

1 Zi

uoatu:

a 0 \ : J : (' z * z

llz

,

oz(v\ , , ;ilm \s)n ,

"\a

r-Zmar.

(36.30) (36.31) (36.32)

The function E?) in(36.32) is the radial wave function of the nonrelativistic Schroedinger equation for the hydrogen atom. In this approximation the current of the electron is just the current of the I . E . F e r m i , Z . P h y s i k 6 0 , 3 2 0 ( 1 9 3 0 ) ;G . B r e i t , P h v s . R e v . 3 5 , 1 4 4 7( 1 9 3 0 ) .

Sec. 36

The Hyperfine

Structure of the Hydrogen Atom l7l l spin. Integration by parts in

magnetic dipole of the electron (Jb..r5,Inen qlves

is "3 : - -+(s.ot /.ot4])= I [ F x q2(r\ /" rot ) [ # x o2( ' \ r\ I , tn I l6nziuJ r'l I

6 Eto): - -:-

16n'mJ

:-r-:,-

r

f -L| ,a, l d . t x q z l r l s t s r a d t s ryat cI _r , t r A\ tftaJ t t-

[,"^

16n.m,t

:

e(su\

:

24n.m.l

f ,-

-,,,/t\

-1u"G &r) r ' - l lr @ r \ - (/ r0 ) 1 2 = 3reYr-/

I d r x c p z (A4l\ -y)t= - ; - ( s s ) 6nnr-

4n

|

I )

Because of (36.32), there is complete agreement between (36.33) and the fjrst approximation to (36.22) . From (36.12) we obtain the corrections to this result which are of the order of E^a. That is, the corrections to (36.22) evidently are not to be dropped even if Z is of the order of I (for ercample, in hydrogen), since further terms of the same order or even larger order result from (36. 12). We shall now evaluate the term of order E*a. Here it is

c l e a r l vr r

Jq qt r1f lfriec ri uc rnrt

to

rrsc fhe

simnlifiod

i-hoorrz nf

fho

c rn-i.n. _

rJ l v e n

above. Moreover, we can replace the singnrlar functions for the Coul o m b f i e l d b y t h o s e f o r a f r e e p a r t i c l e t h r o u g h o u t ( 3 6 . 1 3 )a n d ( 3 6 . 1 4 ) because the terms neglected by this aII contain a further factor of Zu . The quantity di" in (36.13)then becomes the same as those expressions which were treated inSec.3l to 34, except that we have to use solutions of the non-relativistic Schroedinger equation for initial and final states, rather than plane waves. Certainly the Dirac equation for free particles was used for initial and final states at several places inSec. 31. This is not strictly correct for the present calculation; however, the error involved is again of order E*Zaz, and thus not disturbing. In this approximation we obtain

da(il: - oLI a"*V:*

II W

eiq(r-rlf;,n,1*,1yuu,,(n,) x

x (-t(0) (n\ +fr,o,(0)+R(0)(g,)-n,0, (o)-$
p*@'\fo oo * p,stot @')fr (n,@')

* f. Ia',fi(v#b) II q#r-eiq(n-el 4? an"

(36.34)

x [S(0)(o) +Rtol(n')-O.,(o) -fitot(A') *fr.t1o)J:

v;a) : - +h osz (srot I IIY*r x

".,

(\Jo.JJ/

l' ^* t,n' t"t'2,

,t

e

eiq(a-rl v,?')x

- t ( 0 ) ( q , )+ f i ( d ( o ) ] . [] +n*,(q2)-R-(0)(0)

1. This is not true inthe exact theory, since the spin and orbital angular momentum are not (separately) good quantum numbers. In the ls state with total angular momentum I,/2, the electron is principallyin the state with /=0 and s'=I/2; however, there is a small probability of order Z2azthat the electron can be found in the state with /=1, 7n=1,s"= -1/2. This isthe origin of the expression (36.28) and the last term in (36.29). There is an accidental cancellation among the contributions ro (36.27), so that the actual difference between (36.20) and the result of l' the approximations made here is given by replacing bV 4 . 2 t+8

G. K51i5n, Quantum Electrodynamics

I72

Sec. 36

Now we have

=M$LV##W, (r,) Ez rc#,-tan'

(Jb.JJ,,

and

:+" [tr," l - g+tq]. I d "xe i u,(-.'" t,!e :'t )

( 36.36)

Therefore, from (36.34) we find

4 . ro,(o)1, l+..l: d E ( l ) :- 3 p " Q r E T J f_zzz:g_qtq)_f ,r lz, , I 36.37)

r r-ffi#i#f** : dE(.)

u't(n')-o''(o)-ttor(q') +il-(')(o)]'

0

The square brackets in (36.37) vary slowlywith q2becausethe dimensionless variable in this expression is qzfmz . The first factor is almost zero for q2fm2)422a2 ' Its effect is therefore almost the same as a delta function and, up to terms of order E-22*s , we obtain dE(1): 5ptd !-

,

: )- \ p ,@"4) ? U dE (o) + 6 E t1

(36.38)

* i") .

( 36.3e)

In this fashion, the anomalous magnetic moment of the electron enters directly into the expression for the hyperfine structure splitting, as would be expected naively. From the above discussion, it follows that (36 .39) actually should contain a further correction of the ord.er E*Z *2 . This term can be calculated from the preceding expressions if the Coulomb field of the nucleus is included in the next order. we shall not go_further into these rather complicated calculations. The result is' dE:-

4 , , m 3 Z 3 u s l . , u o . 3 2 8 -uzza - r , ' " b - / l "-r(!-1qgZ)lL [ f'+, 1zZ- r a-.'r)]' ., $ O . S O ) nz 3,r/"(gs) Itt ,n-

tn (36.40) the fourth order magnetic moment has been included since, for small Z , this contribution is of the same order as the other new terms in (36.40). Exact measurements of the hyperfine structure splitting in hydrogen have been carried out by Prodell and Kusch2 and by Wittke and Dicke.2 Because the ratio of the magnetic moments of the 1 . R . K a r p l u s a n d A . K l e i n . P h y s . R e v . ! 9 , 9 7 2 ( 1 9 s 2 ) ;N . M . Itoll and F. Pollock, Phys. Rev. 86, 876 (1952). ; . 2. A. G. Prodelland P. Kusch, Phys. Rev. 88, l8a (1952)P D i c k e, ( 1 9 5 5 ) ; H . a n d R . P . W i t t k e 1 0 0 , I l 8 8 R e v . P h y s . Kusch, J. Phys. Rev. 103, 620 (1956).

Sec. 36

The Hyperfine

Structure of the Hydrogen Atom

173

proton and electron and the quantity $mu2 (the Rydberg constant) are known to greater accuracy,'Eq. (36.40) is suitable for an exact determination of the fine structure constantz a. It must be remembered that the mass m in(36,40) comes from the non-relativistic wave function of the hydrogen atom, and therefore should be taken a S t h e r e d g 6 e d m a s S i . h r r r l r n n a n r n + l - r i gw a y t h e r e s u l t

*:tlr-olr4+o.oool.

( 36 . 4 1 )

is obtained. The limits of error given here come from the paper of Kroil and Pollock, which was mentioned above, and contain only the uncertainty in the measured values. In addition, there are the terms neglected in (36.40). These are, first, the contributions of order ocs, which are probably very small . Second, the motion of the nucleus must be considered. In (36.40) this motion has been described only by means of the reduced mass, but it can give rlse *' to terms of the order o . Third, it is possible that the asmp sumption made here to regard the proton as a pojnt particle is not sufficient.r The corrections arising from this cannot be evaluated without special assumptions; however, it is quite probable that they affect the last two figures in the above result. Finally, it should be mentioned that the hyperfine structure splitting of the 2 s state in hydrogen has also been measured.4 Theoretically, it would be expected that this energy difference would also be given by (36.40), if only the last factor were modifiedS to (!{22u2) and a factor B introduced into the denominator, so that ( 4' I dE(zst I .r.oooorll. :

R: ;;i;,;

The experimental

result

(36.42)

; l' +;2, n,]: u

of Reich, Heberle,

and Kusch is

R : 3-[1.ooo 0)46+o.oooooo]l

(36.43)

R : + [1.ooo 0142+0.0000006]

( 3 6. 4 4 )

for hydrogen4 and

f o r d e u t e r i u m . b T h e e q u a l i t y o f t h e t w o r e s u l t s ( 3 6 . 4 3 )a n d ( 3 6 . 4 4 ) indicates that the deviation between them and (ZA.+21 cannot be I. S. Koenig, A. G. Prodelland P. Kusch, phys. Rev. 88, IgI ( 1 9 5 2 ) ;I . W . M . D u M o n d a n d E . R . C o h e n , p h y s . R e v . B t , S 5 5

(lesr).

2. H. A. Bethe and C. Longmire, phys. Rev. 75, 306 0949), and the references cited in footnote l, p.I72. 3. E. E. Salpeter and W. A. Newcomb, phys. Rev. 87, IS0 ( 1 9 5 2 ) ;R . A r n o w i t t , P h y s . R e v . 9 2 , 1 0 0 2 ( 1 9 5 3 ) ;A . C . Z e m a c h , Phys. Rev. 104,I77I (1956). 4. H. Reich, J. Heberle and P. Kusch, phys. Rev. 98, ll94 ( 1 9 5 5 ) , I 0 I , 6 1 2( 1 9 s 6 ) . 5 . S e eG . B r e i t , P h y s . R e v . 3 5 , 1 4 4 7 ( 1 9 3 0 ) . 6 . H . R e i c h , J . H e b e r l e a n d P . K u s c h ,p h y s . R e v . 1 0 4 , t 5 g 5 0 g S 6 ) .

Sec. 37

G. K5ti6n, Quantum Electrodynamics

I74

explained by nuilear effects such as a finite extension of the A theoretical evaluation of the radiative nucleus or its recoil. gives the result by Mittleman' order of as corrections

R:+tt

+ E Z ' e 2+ 5 . 2 8 n :' l* '

(36'4s)

1.0000J54.

in place of the relation (36.42). The difference and experiment has not yet been explained.

between theory

Level Shifts in the Hydrogen Atom: The Lamb Shift After these preliminaries, we are ready to discuss the level shift in the hydrogen atom which was.discovered in I947 by Lamb and RetherfordZ and which was a major stimulus in the development of modern quantum electrodynamics . We are going to use the methods of Sec. 36; i.e. , we attempt to treat the Coulomb field of the nucleus as a small perturbation except in the wave funcIn order to be consistent, tions of the initial and final states. of the electron spin given in theory we must use the approximate sec. 36. From this we obtain immediately the first order shift in 37.

the state lz):

b(n)dsx dE : - | (nl6i,@)1n;Afio"to ,

( 37 . 1 )

x n' ) ( n l 6 i r ( x ) \ ">:;;FJ' d 'q,'nr*-''ra*@' ) vr u,( l ' 2) -nto)(o) $ror1nz1 -11(0) + (0) lsro)(0)l xf 1nz; 1fitot +F(0) 1oz1 l(37 )(g(r')o ,) ,,u ,( n' ) ) i + t # | d "qrun t*-')s(0*tu we have completely neglected3 the effect of the For simplicity, magnetic dipole of the nucleus in (37.1) and (37.2). We again m ake the approximation used in (36.37) and expand the functions -nrct(qr) (34 .27) we , R(o)(qt) , and S(o)(gr) in powers of q,. From find

qnt6i,@)tD * #l"r+-

y,u,(r)) * #] $ lr,{*) ,c(A

* * Now, using

l ctosu r o m\b (' r :)

t",;i(r"(tr) o,u,(r)).

t,' ,, )

Ze _ i 6.."1_, *-

( 3 7. 4 )

41,

we find

dE:

: - #l^rt dr(l)

(37.5)

d E ( 1*) 6 E r z t ,

- ?J# o"*r-1*1 a(]) : y^"-14 [

: +#l^r# -#)tq'(o)t' ,

,,, u, |

Phys. Rev. 1o7,lr7o (1957). @, 2 . W . E . L a m b a n d R . C . R e t h e r f o r d ,P h v s . R e v . 7 2 , 2 4 I ( 1 9 4 7 ) ' 3. In the final comparison with experiment, clearly both effects have to be included together.

Sec. 37

The Lamb Shift 6E@:-Y

Ze

ti F'i

f

J

-^

d "* ;

I

By the use of the Dirac equation, A

,-,

?

"

A

* u ( i l " ( r )y n y o u " ( n .) )

G 7. 7 )

we have

t

&,

;(r"(r)ynyu",@)): ^@f : r l-^

175

( u\ ) y u u , ( ":) )

l (, 3 7 . 8 ) [

I zut_ , . u , ( r ) - 2 m u f ,( r ) u , ( r ) . lE"l ;lr"(r) J

For our purposes it is sufficiently accurate to express the small components of the function uo(n) in terms of the large ones by

dsmalt@) : i;o

(ez. g)

Bradrzlurn.(r) .

From (37.8) we then find 2 ^l^

l E) )" - r n ! 7zl ct vl ,, @ ) l ' a , h \ u t r @ ) v o ux*2( n

I

- 4.1 . 7 tgraoqtr@) gradp,(o) + 2i s fgradq|(n) x gradq,tr)] ]=l (37.10) -1 I -; 2 i s . : j lSradrf (r) x gradq,(r)f , ;Alq*Gr)|, where g"(r) is again the non-relatj.vistic Schroedinger wave function. In (37.10) we have consistently dropped all terms with a factor Zzuz and have used the wave equation for the function g,(c) in obtaining the last result. Integrating (37.10) by parts, it follows that

: #tr,(o)t'+ I ++QI@)vuu,(:r)) '

i(i+ 1)-l(/+l)-+

fdsxlq-17112 J*

l

(37 .,,)

where the last term is defined to be zero tor l:0. Using the familiar expressions for the non-relativistic wave functions, we obtain from (37.6),(37.7), and (37.11),

: +t#["rt dE(1) -6EcD:

1 mZaa, Ct,i 27t n3 ;i-:7

." , , r_l# -l (-Z

- #]uL,,

(37.r2)

'

( 37 . 1 3 )

ror i:t++,) ,

.

lor

t:t-Z')

,, I

(37.r4)

For states of non-zero orbital angular momentum,therefore, (37.12) vanishes, and only the term (37.13) coming from the anomalous magnetic moment of the electron gives a shift to the state. For

r76

G. Klit15n, Quantum Electrodynamics

aan

'2.7

s-states (37.12) does not vanish; however, it contains the small photon mass 1z. ]ust as in Sec.35, the occurrence of this quantity in our result means that we have neglected something essential in In this case, the mistake is that the formal deour calculation. velopment in powers of Za is actually not possible for the bound In a virtual state, if a photon enters with an energy electron. which is small compared to the binding enerqy of the electron, then the electron can no longer be regarded as almost free and the s i n g u l a r S - f u n c t i o n s i n E q s . ( 3 6 . 1 3 )a n d ( 3 6 . 1 4 ) c a n n o t b e r e p l a c e d In order to take this into account, by those of the free electron. we first note that the binding energy of the electron is of the order much smaller than the rest mass of of magnitude mZza2, i.e., the electron. It is therefore possible to break our calculation into First we consider only virtual states in which the photwo parts. ton has an energy which is much smaller than some limiting energy K. For K we choose an energy which is much larger than the binding energy but much smaller than the rest mags of the electron. By this device, it is possible to treat the electrons in the theory. This means virtual states by means of the non-relativistic FinaIIy, in the formal calculations. a considerable simplification for the other states, for which the energy of the photon is larger we can neglect the effect of the binding and use the prethanK, In this part we can set p equal to zero;howceding calculation. Rather ever, we are not to integrate over the whole of A-space. than the small mass p, the energyK enters as the cutoff quantity The desired connection between p and K is obtained in (37 .I2). most simply from the calculation of Sec. 35. If we introduce the quantity K rather than pr into Eq . (3 5 .12) , we get /lE

r

I

J

K
I

, . _ : , .o ( h \: l _ I 'n a n[ d " . [ , . . , o 2 o = , , , , ^ : z.l'""" J^"J Lrp+Q")rr-Elt{ll'z h q . h q '" , '

)b^tbz\

K

o

,

(37.Is)

: z n t o s :-f- ; -l : f l , a n ds o w e h a v e

We are interestedonly in the case # l * doStrahl

d!)

-

dolo) e2 ,__lE -"d K dQ 2n2

I 3

Q2 tn2'

( 37 . 1 6 )

The two functions -{ and F, in Eq. (35.16) have the power series tOz

(37.17)

4 -: _ - x3 m - *" . . . , rE1- 2 : - e * ,5+ .Q. 2. .

,

(37.18)

a s i s r e a d i l y v e r i f i e d . B y c o m p a r i s o no f ( 3 7 . 1 6 ) a n d ( 3 5 . 1 6 ) , i t f o l l o w s b y t h e u s e o f ( 3 7 . I 7 ) a n d ( 3 7 . 1 8 )t h a t

Sec.37

]ne tamp Snrlt

177

,2Kq I o-gp_ - - : : 0 . o

l.?7

lo'l

From this we havel

: +t#[^r dE(1) rafhar

+L h ^h r r s r r

f? 7

'l

\ v t . t 4 t .

# + fi,]a,s,

( 3 7. 2 0 )

,)

For discussion of the non-relativistic intermediate states, we begin with the Schroedinger equation -

1 Za ,1-1Jt / s t r a h\ l. l. rt \A q - j V : e 2mur , ,n,--, axt

V.

(37.2I)

The term with the radiation field is taken as a small perturbation and in the lowest non-vanishing order it gives2

/\ ","4t nF- \ - v

lH|,',)-+*l'

L)-E-=-F-==

\J / . ZZI

,

m,k,l-11

H l iL,*: - el l t(ml ai l n >l +;, ( n l u , l n ') :

- j - [ a " r o ! . @ 1 3 ! t @, ; n * . '*"' mJ Axt

(17

C"\

(37.24)

As before, in (37. 22) through (37.24) q, (c) stands for the eigenfunction of the state ln) of the unperturbed problem. The corresponding energy is En . The limit V-+a and the summation over the directions of polarization ,1 of the photon give t^r\ \AI1 ).::-

2 3

K a. f \r I todro ) 4 xr.' o*

( n l u n l m () m l a n l n ) E,r-

E*-

to

( 3 7. 2 s )

We have set the exponential factor equal to I in (37.24), as is allowed by the restrictions made on K. The expression (37.25) contains the total energy change of the state I n) ,i.e., the change in the non-relativistic kinetic energy which is due tothe self-mass. Ifthe external field is zero, then (37.25) contains only this last contribution. In this case the inte-

l . J u s t i f i c a t i o n f o r t h e u s e o f t h e r e s u l t ( 3 7 .1 9 ) i n ( 3 7 . I 2 ) c o m e s about as follows: As can be seen from (31.23), the infrared term L n ( 3 7 . 1 2 )a r i s e s f r o m a n i n t e g r a l o f t h e t y p e ( 3 5 . 1 2 ) . t f r u s i n b o t h cases it is a question of one cutoff in the same ft-space. 2. In this treatment of the non-relativistic problem, we regard the longitudinal and scalar degrees of freedom of the radiation field as having been replaced by the electrostatic interaction between the particles (c.f. Sec. 38a). Since there is only one electron present here, and since the electrostatic self-energy is the same for a free particle or a bound state and can therefore be neglected, only the transverse electromagnetic field enters (37.2I). T h e s u m m a t i o no v e r , t i n ( 3 7 . 2 2 ) i s h e n c e o n l y o v e r I a n d 2 .

Sec. 37

G. Klill6n, Quantum Electrodynamics

I78

gral (37.24) vanishes for

n*/n

, and we have

n ) : f, a*?ft,' l n) , ( 37.26) @n)1,'": " - i * o
^ +u-. dm: -1;r'.

( 3 7. 2 7 )

The actual leve1 shift is therefore

\

dE(3):(AE),- (AE)iy"":

r l wd.,: w-l(3 7.28) :- " n | (nta,t *^;^a"t 1,.. \ " ' , ' f t 1 n) " / Jf f:__+__ +I r) 3;+\"tLht"'/ lE"-E*-@

:'," 3

I l ( z l t l r n \ 1I 2m ( E * - E - llLon-g --E4 , :-. i l ', , )

n z!-lr"r-t'/t

I

' I

l

The sum ,nf{r.28) has to be worked out numerically for each state In order to get a feeling for the order of magnitude, we reln) place the energy in the logarithm by an average value (E) , and find

5 6 ra):4 *t" r*

\-/

E^- E*) . zm l @l ulm )(1,

( 37.2s)

Equation (37.29) can be regarded as a definition of the mean value <E> . This quantity will probably be of the order of the binding energy. The sum in (37.29) can be worked out exactly:

r In) : (nli lt1tot ,rlln) : zl@lo lrn>r@*- E,): (nlLo,/1(0)l Z a l r t, ,l , , IZa l,.r : Z A \ \ r t L p ' ; l p t n ) + \ n t p l ; , P y t t ',1sI :

Za

f

;A .l

,,/1\ d3x sradl-)gradI p" (r)12

2nZa,

74-4

lq, (o)l':2-+

56tt):*t#^s$a,,,.

) L

t(37.30) |

m 6t,o,l

(37.3r)

F r o m ( 3 7 . 3 1 ) , ( 9 2 . 2 0 ) , a n d ( 3 7 . 1 3 ) ,t h e t o t a l l e v e l s h i f t b e c o m e s [6-

4 mZaus { r c n - - \ - + g },

3n

6E: .1

nt

l.,o 2 (Es)

*zln'{to* jL

31T n3

for t:0,

Q7.32a)

30J

r, 3 c r , ; } f o r I + 0 . 8 zt + r J t^," (gr)

( 3 7. 3 z b )

The first term in (37 .32b) has been added in order to take account of the fact that the sum (37.28) does not exactly vanish f.or l# 0, as it should do from (37.30). In this case, it is independent of K. The non-relativistic part of the above calculation was first p e r f o r m e db y B e t h e . l T h . l a s t t e r m s i n ( 3 7 . 3 2 ) w e r e g i v e n 2 i n d e The pendently by Kroll and Lamb, and by French and Weisskopf. 1. H. A. Bethe, Phys. Rev. 72, 339 (1947). 2. N. M. Kroll and W. E. Lamb,Phys. Rev. 75,388 (1949); B J . . F r e n c ha n d V . F . W e i s s k o p f , P h y s . R e v . 7 5 , 1 2 4 0 ( 1 9 4 9 ) .

Sec.37

The Lamb Shift

179

method used here follows a calculation of Feynman.I Independently, Tomonaga and coworker=2 have derived the result (g7.32) by similar methods. Finally, it must be mentioned that l(ramers gaveo a semi-classical discussion of the level shift from the electromagnetic interaction prior to the authors mentioned above. For comparison with experiment, the mean value <E> must first be calculated. This has been done l"rv Refhe Rrnrarn and

Stehn,4 as wen as by Harriman.5 w" *rjtJir.,;;;"tr (8,):

1 6 , 6 4 0R y

for the 2s state,

(37.33a)

(Ep):

0.9704Ry

for the 2 p state .

/?z

For the two states 2p, and 2f ahard\z

-'

diffaran-o

;"'

^f

t

"

?e1-\

in hydrogen, (37.32b) gives an

6E: *-!-m. 32xt

G7.34)

.1.*

From the Dirac theory, there is also an energy difference

) ' E : m 3l f2t[ + l o , + .' . . 1 -'8-'

hz

eR\

)t

so that the total energy difference

is given by

6 E+ 6E ' : / E : * l * * , i ; o u+ * ( t - y P * , 1.]r s z . s o r In this equation we have replaced

L

from (ZZ .Z) [and hence also

from (37.34)] by the fourth order magnetic moment (c.f . Sec. 33), so as to take account of the next term in the series expansion in the radiation field. using the value (36.41) for the fine structure constant, we obtain for deuterium

AE:'16971.6 Mc/sec. This agrees very well with the measurement wasser, and Lamb, who obtained6 ;;;;;1"""

( 3 7. 3 7 ) of T)arzhnff L L V L L 'T1iebtt

A E : ( 1 0 9 7 r .+60 . 2 ) M c / s e c .

(37.38)

1. R. P. Feynman,PhysR . e v . 2 4 , 1 4 3 0( l 9 4 g ) ;7 6 , 7 6 9 0 9 4 9 ) . 2. H. Fukuda, Y. Miyamoto and S. Tomonaga, progr. Theor. P h y f . 4 l 4 7 , I 2 I ( 1 9 4 9.) S e e a l s o y . . N a m b u ,p r o g r .T h e o r .p h y s . 4 , 8 2 ( 1 9 4 9 ) ,a s w e l l a s O . H a r a a n d T . T o k a n o , p r o g r . T h e o r .p h y s . ! - , 103(]e4e). 3 . H . A . K r a m e r s , R e p o r t S o l v a y C o n f e r e n c e1 9 4 g , B r u s s e l s , 1950. 4 . H . A . B e t h e , L . M . B r o w n a n d J . R . S t e h r y p h y s .R e v . 7 7 , 370 (les0) 5. I. M. Harriman,phys. Rev. l0l, S94 0956). 6 . E . S . D a y h o f f, S . T r i e b w a s s e ra n d W . E . L a m b , p h y s . R e v . 8 9 , 1 0 6( 1 9 s 3 ) .

I80

G. fa116n, Quantum Electrodynamics

Sec.37

Alternatively, one can take these measurements as an independent determination of the fine structure constant, and this is justified in view of the uncertainty in the _interpretation of the hyperfine structure. In this way one obtainsl

L:1\7.0181+ 0,0012. a'

(37.3s)

The words "Lamb shift" refer not to the energy difference (37.36), but to the splitting of the 2s, and 2p, levels of the hydrogen atom. According to the Dirac theory these two levels should have exactly the same energy, but according to (37.32) the 2s, level is a trifle higher in energy than the other state. We find

-L : 6 "E - \ - '/i2 t s,)

6 E( z p ,:) , = f r o u , ? ( l r ) , - r - j3' ( 1) l .-. l e \ v7 ' .40) bn " zny(Ej-L

W i t h t h e n u m e r i c a l r e s u l t s ( 3 7 . 3 3 )a n d t h e v a l u e ( 3 7 . 3 9 ) l o r ( 3 6 . 4 1 ) ] for a one finds L : lo52.l Mc/sec.

la.7

A't\

The experimental result is2 L:

la'7 Ac\

( 1 o 5 7 .f8 0 . 1 )M c / s e c .

The difference of (37.42) and (37.4I) is far outside the experimental limits of error. As several authors3 have shown, this discrepancy can be explained to a large extent by the next term in the series in Za. The corresponding contribution is of the order of 7 Mc/sec. If additional small effects, such as terms of order a2 from the functions fr(pr) , F(pr) , anO Slpz; and the finite extensionand recoil of the nucleus are considered -4 the theoretical value is changed to

L : (1057.9+ 0.2)Mc/sec.

( 3 7. 4 3 )

To within experimental error, (37.43) and (37.42) agree completely. Similar measurements of the level shifts have been done in deuterium2 (for n:2), in ordinary hydrogenS (for_n:3) , inionized helium6 (He+ for n:2), and in neutral heliumT (He). The results

1. See footnote6, p. 179 2 . E . S . D a y h o f f , S . T r i e b w a s s e ra n d W . E . L a m b , P h y s . R e v . 89, 98 (19s3). 3. M. Baranger, H. A. Bethe and R. P. Feynman, Phys. Rev. . 9 2 , 4 8 2 ( 1 9 5 3 ) ;R . K a r p l u s , A . K l e i n a n d J . S c h w i n g e r , p h y s . R e v . 86, 288 (19s2). 4 . E . E . S a l p e t e r ,P h y s . R e v . 8 9 , 9 2 ( 1 9 5 3 ) ;C . M . S o m m e r f i e l d , P h y s . R e v . 1 0 7 , 3 2 8 ( 1 9 5 7 ) ;A . P e t e r m a n n , H e l v . P h y s . A c t a 3 0 ,

407 (r9s7). 5 . W . E . L a m ba n d T . M . S a u n d e r sP, h y s .R e v .1 0 3 , ? l ? 6 -

R - \ T ov rv zr iv ,c\ k,

1 r s 3( r 9 s s ) .

!l l.

T l r .yi rnyrvarz n r f h

and

rP .

rF.

Vr earrvnr irnr ,

l rryr oc . rD h

/lqcA'l

R e v . 1 0 0,

7. I. Wieder and W. E . L a m b , P h y s . R e v . 1 0 7 , I 2 5 0 9 5 7 ) .

Sec. 37

The Lamb Shift

181

are 0 0 , 1 ) M c / s e c i n D e u t e r i u m , ( 3 7. 4 4 ) d E ( z s } )- 6 E ( z p s ) : ( 1 0 5 9 , + dE(ls;) - 6A\ps):

(jt5 * to) Mc/sec in Hydrosen, 6E(2sa) 6E(zpa): U4o4, !1)) Mc/sec in He+, 6E(2pr\ - 6E(2pr) : (229r,7* 0.4)rvrc,/secin Herium,

(37.4s)

d E ( ] p J - 6 E ( l O o ;: ( 8 1 1 1 .+8 O . 2 ) M c / s e ci n H e l i u m , 6 E ( J p t ) - 6 E ( l p J : ( 6 5 8 , 6* : 0 . 2 ) t n c / s e c i n H e l i u m ,

(37.48)

(37.46) (37.47)

\J / . +J,'

The corresponding theoretical values are d E ( z s 1 -) 6 E ( z p a ) : ( 1 0 5 9 . 1t 0 . 2 ) M c / s e c i n D e u t e r i u m , l ( 3 7 . 5 0 ) d E ( 1 s ; )- 6 E ( ) p , ) : 0 1 4 . 9 5+ 0 . 0 5 M ) c/sec in Hydrogen,2 (37.s1) dE(2st) 6E(zp6):(404) +)) Mc/sec in He+,3 $z.sz) Within the limits of error, all these values agree very well with the measurements. For the two-electron problem in helium, the mathematical difficulties are so great that up to now there has been no sufficiently accurate theory with which to compare (37.47) t h r o u g h( 3 7 . 4 9 ) . B y o p t i c a l m e t h o d s , G . H e r z b e r g h a s m e a s u r e d 4 +ha

a n a r n . 'y r

vA fi rf fr ^vrl ^vhr ^r 6 v

ho+rrraan

f4htds

o A L.pil

- nr d q ru

II Js .,

s fLaof e _ Ls D

vo rf

d cg iul fLecf i U m u

with sufficient accuracy so that a value for the shift of the i s level has been obtained. The accuracy of the measured shift obtained is much lower than for the previously cited measurements. The result is (0.26 + 0.04) cry1-l and agrees with the theoretical value of 0.2726 cm-r. Finally, it should also be noted that optical measurements of the x-ray spectra of a few heavy elements have shown the influence of the radiative shifts5 of the energy levels. it must be said that the good agreement between In conclusion, theory and experiment indicated here can be regarded as a confirmation of the validity of the general ideas of quantum electrodynamics and the renormalization program. As was said in Sec. 29, the function fitot1pz1-n'(0)(0) in (37.2)gives a contribution (of about 27 Mc/sec) which is much larger than the experimental uncertainty, so that the good agreement of calculated and measured values for the Lamb shift may also be considered as proof of the existence of the vacuum polarization. 1. See footnote 4, p. IB0. 2. I. M. Harriman, Phys. Rev. 101, 594 (1956). In going to (37.51) we have changed the theoretical value of the level shift given in this paper so as to include the corrected (theoretical) value for the fourth order anomalous moment of the electron. 3. See footnote 6, p. IB0. 4, G. Herzberg, Proc. Roy. Soc. Lond., A234,526 (1956). 5. R. L. Shacklett and l. W. M. DuMond, phys. Rev. 106,

s 0 10 9 s 7 ) .

r82

G . K t i 1 1 5 n ,Q u a n t u mE l e c t r o d y n a m i c s

Dec.

Jd

3 8 . Positroniuml F n r c o m n l c f p n p s sw e s h a l l n o w d i s c u s s t h e b o u n d s t a t e s o f a n r

vr

vvrrryrv

L

electron and a positron, caIIed "positronium" . ln the lowest nonapproximation the binding energy is determined by relativistic interaction of the two particles, the instantaneous electrostatic just as in the hydrogen atom. The sole difference is that here the reduced mass is half the electron mass, so that the energy eigenvalues are given by m a2

(o) -Ft u

4

(38.I)

nz

The spectrum of positronium is therefore strongly degenerate in as is the corresponding spectrum for the nonthis approximation, hydrogen atom. A more exact treatment gives a fine relativistic in which the level splitting is of the structure to this spectrum order of mua . Just as in hydrogen, this splitting comes partly terms in the kinetic energy, partly from spin effrom relativistic fcefs. and oartlv from corrections to the electrostatic interaction due to the finite speed of propagation of electromagnetic effects. Our discussion of positronium will be done so that we first isolate the electrostatic interaction of the particles and treat this exactly, and afterwards consider the other terms by means of a perturbation calculation. a. The Electrostatic Interaction The complete Hamiltonian for a system of electrons and the electromagnetic field can be written in the following form:

H : Hl,o) + Hl-'i+ Hlo,+ H:2'+ HY,+ Hy + Hg.

(38.2)

of the free electrons, and Here the first term is the Hamiltonian tha sccond - third - and fourth terms are the Hamiltonian of free and scalar polarizations, photons with transverse, longitudinal, respectively (c.f . Sec. 6) . Finally, the last three terms give the Acinteraction of the electrons with the three kinds of photons. in ,-space we shall split up the electromagnetic field cordingly, into three parts as follows:

(38.3) ( 3B . 4 ) -A - (\ "x/ \

:

- :4n

f J

d3x' 7A,*(t'):1-t

DA - n ( ' t

d*'h

Arn

lxt

,

(38 ' 5)

^n-.o

A n @ :) i V ( x ) .

tJ6.

O,l

1. A more detailed discussion of positronium will be found ln the arricle hv T,^ Simons in Vol. nO(ry of this handbook (Handbuch der Physik, edited by S. Fiiigge, Springer-Verlag, Heidelberg).

Raa

Positronium

?Q

183

We define

x@):"#:aA(x)try' ./..\

A2Au(x)

AzV\y) ,

nAAel

,r0,4t.vt

::-;-,--J-----;-:Jt X , \ x ) - - o; + xoox! ox6 dxo

,

;"'

dr,

(38.7)

i l - ( x, ) ,l1 |r oc (\x' ' /\ . (\ 3 8 . 8 )

The last form in (38.8) follows from the equations of motion for V(*) ; g(r) is the charge density. Using these equations and integrating by parts a few times, we can write the sum of the terms

Hlo) , H:P , and f1,[ as

+HtP +Hs: t I a",tffiffi | ##ffi 4(o) -

? v ( x \ a v ( x ) - a v Q \ 0 v 1 x I1

t-;

_

f ," v(x): J d'xp(x)

tt)+

(38.e)

) y (x\ (v (r\ /-, (r\- a4o) : H. i ! [ a'*l(Ll (r\- ?Y^Q \ * \ + o\ / A x o ]\ t +" \r,t] q "rl " 2J " ?xo l* t' [\

with

H,:# III#pe)pe)

(38.9a)

'o:tir

lgnoring the terms which contain the subsidiary condition (3g.7) and its time derivative, the sum of the three terms in (3g.9) is just the instantaneous electrostatic interaction of the electrons. We now make the gauge transformation tp(x) -tp(x)

H;u' + Htw -->H;o' ,

ei'tr(') ,

(3B.10)

and obtain

H: Hio'+ H f )t H c * n n + J ; d , * l ( . . ) x @ * )( . ) i @ ) 1 , ( 3 8 . 1 1 ) I

Hr;:+I'""les#!e#*ry\Wl r

f

| ^J

t-,\

Ht': + I a'xlv (rr* + *\,p@)), @), Hil:

-

f

;"

: | d|x.ilu@)io@) +

,

(38.11a) ( 3B. 1 1 b )

r

(38.11c) | dtx.{o@)lrp(r),yrve)1.

instead oi trr. 2). Forpf,y.l"uify interesting states, for which the subsidiary condition must be satisfied, the last two terms in (3g.ll) have no significance and can be dropped. In this way the effect of the longitudinal and scalar photons has been replacedl by the Coulomb energy (38.9a).By this procedure,the explicit relativistic l.See Sec.B where it was shown that these degrees of freedom of the field can have no effect for free photons. The method grven there for treating the state vector by means of a limit must actually be used here also in order to eliminate the last two terms in a consistent manner.

G . K t i 1 1 5 n ,Q u a n t u m E l e c t r o d y n a m i c s

I84

covariance has been lost, troublesome.

but for our applications this

S e c . 38 is not

b. The Unperturbed Problem For the unperturbed state of positronium we write the state vector in the form

@ @ ,q ' ) l q ,q ' ) , l z ) : qX ,q

( 3 I. r 2 )

and drop the perturbation Hfr and the last integral in (38.11). The symbol lg, q') stands for an eigenstate of H(0' containlng one elecThe Schroedinger equation tron-positron pair.r

qi)=l(ae.rs) o(q, s')lq,q')+ Zlq, q')(q,q'lHclq,,Q,)@(e,, ^2,@,+ Eq') I

q,q'

:EI

follows

from this.

In r-space

@@,q')lq,q'))

the non-relativistic

approximation

1S

l_

1

A'

L 2m a4

_

r _22

r* fu

-

,2

f...,- _\ : * n2),$B'r4) ;i]'t'@'' n') (E 2m)tp(n''

with ttt(ar, rz) :

I

etttr'+ta'*, @.(q, q') .

q,q'

The binding energy E-2m

takes the values given in (38.1).

c. Treatment of the Transverse Photons by Perturbation Theory For a more complete treatment of the problem, we must also consider state.s with transverse photons in (3 8 .12) . Thus , in the next approximation we write

l z ) : I @ @ ,q ' ) l q ,q ' ) * l a " l n , k ) . q,q'

(38.r6)

n

The second term in (38.16) contains only states which can be produced by the operator H{, from a state with one pair. Thus, except Forfor the photon, the states contain zero, one, or two pairs. mally we write the vector lz) in (38.16)as

lz): edslv) .

( 38 . 1 6a )

Here the vector lq)is aqain of the form (38.12) and the transformation matrix S is determined from HY, in the following way:

H'lv):E19>, H ' - e - i s H e i sx H + i l H , S l - + [ [ H , S ] , S ] a r

s], sl , I x Hs* HY"+ t,tsr,sl + i lHy,,sl- + [tr10, Ho: H$v I H)o',* H"

(JU.r/'

tse'tzut ( 3I . 1 7 b )

l . S t a t e s o f t h i s k i n d m u s t b e clearly differentiated from states with a given number of incoming p a r t i c l e s . C . f . S e c . I l , e s p e c i a i i y E q s . ( 1 1 . 4 7t)h r o u s h ( 1 1 . 5 3 ) .

Sec. 38 Rrr

maan

Pos itr onium c

n{

+ha

185

-n h. .n -i - c e

rtH.,sl:-H{,

( 38 . 1 8 )

all terms which are linear in the creation or annihilation operators of the photons are eliminated in this order. Since aII diagonal elements of the operator l1ff vanish, Eq. (38.18) can be satisfjed by a non-singular matrix S . The commutator on the left side of (38.18) is formally the time derivative of the operator S in an interl^v ' r+' Eq. action picture which is associated with lha nnararnr rrrr 0 L (2.e)1. Equation (3B.IB) is therefore.J.;;;;;v!

S:

- @

J H{ (x'o)d.x".

(38.I8a)

Here HY,@o)is the time-dependent operator represented in this p i c t u r e . w e s u b s t i t u t e ( 3 B . l 8 a ) i n t o ( 3 8 . 1 7 a )a n d t h e n e w H a m iltonian becomes ro

H' : H!,0,+ Hf) + H"- * [ luTt*,), Hfl(xL)ld.x[. -_;

(38.le)

Since the last term in (38.19) contains the commutator of the photon operators (i.e., is a c-number) the photon variables of the problem have been formally eliminated. Using the equation

uf l M u U ) , d , ( x ' ) )-: -(+2 t 1 "t [ "d k- -r , o , ' - "n\ d " t (" \A' "rt )Le " h( tA , €)2f af^, ,l -. \(u3u8 . 2 0 ) l, we then obtain

n{@L)l: LHil@o),

I

: [[ a'*d,x'i,(x)i,(x'\ f(38'2I) ' ' Q n--!1 " t ' . "[- dk rtuu'-od(ft2) " \ ' - l "e(ft) \ ' ' l LIa.," R ' i+Ll k, JJ

lJ

For the transition to a Schroedinger picture, all operators present in fl' have to be taken at the same time. We therefore write the current operator j,(x')in the form

i,@')=-t r(:

at*" dBx",Ly;(x,,,)ynS(x,,,-x,),y15(x,_x,,)yEyt(x,,)J.9g.22) 1.1 ri!:ao

In (38.22)the singular functions of the complete, unperturbed probIem ought to be used. That is. they should include the influence of the term -F1. in the unperturbed Hamiltonian. As in the previous sections, we shall approximate these functions by the singular functions for free particles. Then the time integrai (38.19) can be done and, upon qoing over to the Schroedinger picture, we have

xd(q+ q',- q,- a)far,xlv@'),yrq@'')l

186

G . K l i 1 1 5 n ,Q u a n t u mE l e i t r o d y n a m i c s

Sec. 38

withl V@) -

(38.23a)

[ dsxe-i,t'tr(r) 1,":0.

d. The Fine Structure of Positronium The remainder of the calculation is simple in principle, but somewhat tedious. In the barycentric system the total perturbation enerqv is

d H,+ 6HcI6Hr,+

dll:

dHexchange,

Q , - s l 6 H , t q- ,r,) : - # -r) : (q,- ql6H"lq,, #n:*

(38.24)

d ( q- q , ) ,

(38.2aa)

I (38.24b)

"

l

x lu*Gt (q)uG)(qr)u*G)(q)u( ) (qr)- t7

(q,- ql6H,)q,,-8): -#G:

I

x rF,G)(q)y,ue)(q,)

- (q qr)i(q : qJr u( ) (q.\ x uGtb\ \ 1 L t f 5^, \ r ' t,, t t _ (q

l-a'

q,,1,

''n''

, )" u

-

, , e x c h a n g e< q ,- q t "o n

e2 I r l h , - 8 t )- \ r 4 r -+ t / F ; 6 x f (,r. rn6l x n(-)(q)y* u( ) (q)u( ) @r)yru' n (qr). J

In (38.24), 6H, contains the relativistic corrections to the kinetic energy; 6H",the corresponding terms for the electrostatic energy; rnrl

A Ir

,

fLh1 !av

anmjna

fLaL rr m c

1 '-^rr vrrr

/ 2 ov \v

.

t4 ? \. ul

TO

this

Ofder

We

mUSf

also include exchange effects for the electrons, and this is the origin of the term (38 .24d). In order to obtain this expression, it is simplest to startl with the sum 11, * 6Hr,. Using the approximations

nt*g.ra) :ir-#l''r,

(3B.2sa)

'!.arr (t)- lll'> and the exact relation

(c.f.

Sec. 14)

uc)(-q) : - c uG)(q), l.

(3B.2sb)

'C:TzTr,

G8.26)

If (38.23) and (38.9a) are added, we obtain the result ?2

1Ht ^ u t 3I H e- .z .- -t .r f

I

=X

S f

s3-

x I '" I #

(22)13

.t3.,

d ( s + q , - % - e ' ) L E @ ty) ,

l E @ ' )y, p q ( q ' , ) ) .

v@)l J J \q-qt)' This is just the interaction operator of Sec. 27 f.or the scattering of two electrons from each other. There, however, the whole expresslon was treated as a small perturbation; this is not allowed here.

Sec.38

Positronium

IB7

where I s) is the non-relativistic spin function for the unperturbed problem, we can develop the expression (38.24) in powers of. q2fm2 and (g- q')rlmr. AJter some rather lengthy computations, we ger 6 H : 6 H , + 6 H ,+ 6 H t , + d 4 + d l l e x c h a n s e ,

: -,*l#@(* -ogt1 - #1, (6H,)

G8.27)

Q: e - e,, G8-2za)

(d1/,,):-&+g#y,

(38.27b)

:&#l,L-s2* w], (d11,)

(38.27c)

(d rl e xch a n se ): ##

v6.t/a)

Here S is the total spin of the positronium and the terms (38.27) have been grouped according to their dependence on the spin. The expectation values of the expression (38.27) can then be cat* culated for the solutions of Eq. (38.14). In this way we obtain the perturbed energy levels of positronium: S:0: ^ rE _m t -177-7u o : -a-arl-t o

+

j

(38.28a)

rill ,

s:{:

6": #t+ - rh]

+* # 6,,, +ffilP

A\j(3B.zlb)

with

I 3t+4 ior l(t+1)(2t+31

a, 1 ' , 1i: 1 -

l\ From this,

ror

1w I

:,-, I(2t -

we find the splitting

1)

; _I !-r-T

|

^ I

(38.28c)

i:t

ror of the states

{3S, and

A E : 1, m,qL: 2,044 . ,tOE Mc/sec.

1156 ,

( 38 . 2 s )

Formulas for the fine structure of positronium have been qiven bv Pirenne,l Landau and Berestetski,2 and by FerreIl.3 Expeiimentai investigations of the energy difference (38.29) have been carried 1 . J . P i r e n n e ,A r c h . S c i . P h y s . N a t . 2 8 , 2 3 3 ( 1 9 4 6 ) ; 2 9 , I 2 I, 2 0 7, 2 6 s ( 1 9 4 7 ) . 2 . L . D . L a n d a ua n d V . B . B e r e s t e t s k i , I . E x p . T h e o r . P h y s . ; . B . B e r e s t e t s k i ,J . E x p . T h e o r . P h y s . U S S R U S S R1 9 , 6 7 3 ( 1 9 4 9 ) V 19, lr3o (1949). 3 . R . F e r r e l l , P h y s . R e v . t 4 - , B 5 B( 1 9 5 1 )t;h e s i s , P r i n c e t o n , 1 9 5 1 .

Sec. 38

G . K d U 5 n , Q u a n t u mE l e c t r o d y n a m i c s

188

out by Deutsch and coworkers.I

They have obtained the value

/E : (2.o))8 + 0.0004)105Mc/sec.

(38.30)

The difference between (38.29) and (38.30) is ^fully explained by Karplus and Kleinz have calculated the next higher term in o( . fLhr roe

cncrorz

el r ur YJ

differcnr:c

account terms of order ma'. ,r

zJtt:

maL 17

a

r- L:

4 3S.

hefuroan

P

:{': n\9

2'

and

4r r qJ o

sf ^fps uLeLvu,

.

takino

into

They find the result 'l

+ 2 l o"g , 2 ) l: 2 . 0 ) } 7 . 1 0 5 M c / s e c . ll

(38.31)

Similar results for the 25 and 2P states have been given by FuIton and Martin.3 e. The Lifetime of Positronium Whether they are bound in positronium or are free particles, an electron and a positron can annihilate each other and radiate two or more photons. This means that positronium is not a stable form, We shall now compute this lifebut that it has a finite lifetime. time. The element of the S-matrjx for a transition from.a state lq, q') with one pair to a state with two photons is givena by the rules

of Sec. 2I and 22 as ( q , q ' I S l k , , k , ) : 6 ( q+ Q '- k t - k , ) t ' ; ) # = u t ' t( q )x , 2V@t@2 ,.liyt\(i /rl._ L

y1q-hr)-*)iyrQ) , iyt(2)(iylq Ar) n\iye{t)l-,,.r, -,\ T\-1). -2ghz _ 1- .ab' t l^ I

(38.32)

The vectors ,(D and e(2)ut" the polarization vectors of the two photons At and ftr. By means of the methods used repeatedly in Chap. V, we obtain the transition probability per unit time zs' for the annihilation of a state le):

l e ) : I @ @ )l q , Q ' ) l q + q , : 0 ,

(38.33)

q

1 . M . D e u t s c h a n d S . B r o w n , P h y s . R e v . 8 5 , 1 0 4 7 ( 1 9 5 2 ); R . Weinstein, M. Deutsch and S. Brown, Phys..Rev. 98, 223 (i955). See also V.W. Hughes, S. Marder and C. S. Wu, Phys. Rev.106,

934 (r9s7). 2 . R . K a r p l u sa n d A . K l e i n , P h y s . R e v . 8 7 , 8 4 8 0 9 5 2 ) . 3 . T . F u l t o n a n d P . C . M a r t i n , P h y s .R e v . 9 5 , 8 I l ( 1 9 5 4 ) . 4. As emphasized before, the states with incoming particles in (38.32) are quite different from the states with "particles at time zero" in (38.12)and (38.16). Only if there is no interaction are these two kinds of states the same. Despite this, we can identify them with each other here because we are concerned with the first non-vanishing approximation in a perturbation calculation. In higher orders this identification is not allowed.

189

( 38 . 3 4 )

U( q , k ) : J L 1 l . t @ ) x X

-h1) -m)i l t y e { t ) l iy ( q -tq", t

y e\z)

' l u ' - ' \ q. r) l-*,-,*,:*

iye@fiy(q*hr)-ml;yeQl,.

- 2 qh z

L"o..t

, p ( r ): * 7

(38.36) o@)',o, VVT tn (38.34) we can go to the non-relativistic limit by the use of ( 3 8 . 2 5 ) a n d ( 3 8 . 2 6 ) ( i . e . , w h e r e l q l < m a n d a rx m ) . If the two photons are polarized parallel to each other, we obtain

U(q,It): s

for e0)le@.

( 38 . 3 7 a )

If the two photons are polarized perpendicular to each other, we have instead

u(s,ts):

i#(,

-

+) ror ei)Leo\.

(38. 37b)

In (38.37b), S again stands for the total spin of the positronrum. Consequently, for triplet states (38.37b) vanishes. Hence these states cannot decay by two photons.l For singlet states,2 ho*e v e r , w e c a n u s e ( 3 8 . 3 4 ) a n d ( 3 8 . 3 7 b )t o o b t a i n y#lg(0)lr: ? ? F i n s-l r e i

ffa,,Z/triplet -

0.

080-1010 d 7 , 6 s e c - 1 ,( 3 8 . 3 g )

(38.3e)

The result (38.39) does not mean that the triplet state is stable, but only that it must decay by at least three photons. The transition probability of this decay must be of the order of ma6 and a detailed calculation using the previous methods 9ives3 the result l. This has been shown only for the order being consi-dered. By the use of conservation of angular momentum, the statement can be shown to be an exact selection rule for a 35-state: Because of momentum conservation, the two photons must be emitted in opposite directions. Either they have total angular momentumzero (mutually perpendicular polarization directions), or they have total angular momentum 2 (mutually parallet polarization directions). Neither case can originate from a 3S-state. See also L. Michel, Nuovo Cim. I0, 3I9 0953). 2. J. A. Wheeler, Ann. N. Y. Acad. Sci. 46, 22I (1946). 3 . A . O r e a n d J . L . P o w e l l, P h y s . R e v . 7 5 , 1 6 9 6 ( 1 9 4 9 ) ;R . Ferrell, thesis, Princeton, 1951.

G. Kiil16n, Quantum Hlectrodynamics

I90

? e r r i p -r c t *

Sec.39

o.7z_'uro, u , , os e c - 1

@,_ g)+6,,r-

(38.40)

The difference between (38.38) and (38.40) has been used experA direct imentally in verifying the existence of positronium.I l:0 has also been made by meas.urement of u)trlpretfor n: I, The resuk, Deutsch.I

,l'ifit", : (0.68+ 0.07). 10?sec{,

( 38 . 4 r )

is in good agreement with (38.40). 39.

A Survev of Radiative Corrections

in Other Processes

a. Compton Scattering Radiative corrections for the Klein-Nishina formula (25.23J have been worked out by several authors. Jost and Corinaldesiz have rroafod thc cnrrcsnnnding problem for particles of spin 0. In the non-relativistic limit, their result isJ L r r v

ao:

v v . r v v y v

t^ / 3 U+ c o s 2 @ d)Q u )1 , t - 9 4

1

lt-i*n

"

[,

-3(i-lcoso*c9s1@)rcos3o l' "obe L LI ( \l -r *^ \ f

c o s O ) l o g = { = . *1 " zlal

a,

l+cos2@

.l,i("'t' 11

where e2

r^:-:-'

4nnt

"

d,

/?q

1n

and

t\

(3e.3)

@<m.

is the resHere a,' isthe frequency of the incident photon and/E olution of the measuring apparatus (as in Sec. 35). As before, the term with AE is present because of the infrared divergence which appears in an integration over virtual photons. Just as in is divergence this fie1d, in an external the scattering of electrons to be understood in terms of properties of the measuring apparatus. For particles with spinl/2 one also obtainsa (39.I) in the nonthe cross section has a rather comrelativistic limit. For ulm, In this case plicated form; we shail forego reproducing it here.a also there is an infrared divergence which is treated in a similar \V^v Ozy^. r ' i.e.,

{\ J?J .9L -l l ) h T form L La - -s t h e r v vn llm g v Lc c f i o n f a c f o r ir rn l l lh s a c the radiative

corrections

vanish

for

at(,/n.

l- + O t uI ' 1 ,.t \2 , |

),

This is not only

l . M . D e u t s c h , P h y s . R e v . 8 2 , 4 5 5 ( 1 9 5 1 )8,3 , 8 6 6 0 9 5 1 ) . 2 . E . C o r i n a l d e s ia n d R . J o s t , H e I v . P h y s . A c t a 2 1 , 1 8 3 0 9 4 8 ) . 3. In this limit the cross section for Compton scattering goes over into the classical Thompsonformula both for particles of spin zero and for the particies of spin I/2 considered previously. At higher energriesthe formula for spin zero differs from the KleinNishina formula. 4 . L . M . B r o w na n d R . P . F e y n m a n ,P h y s . R e v . 8 5 , 2 3 1 0 9 5 2 ). An earlier discussion of the same problem was given by M. R. S c h a f r o t h ,H e l v . P h y s . A c t a . 2 2 , 5 U ( t 9 4 9 ) ; 2 3 , 5 4 2 0 9 5 0 ) .

Sec. 39

Radiative Corrections in Other Processes

tgl

true in lowest order, but can be proved in general.l b. Electron-Electron Scattering The corrections to Eq. (27.20) have been evaluated by Redheadi Here also the complete result is very complicated and we shall restrict ourselves to the non-relativistic limit. If one electron is at rest while the other is incident with velocity u and is scattered through the angle O , then we can write the formula with correc-

tions as follows:

d,o:+nftY!!9 + =cos= + (v

- rr1) d.sin @(t - sin@))* [#t t, - rzacos@ ( l * cos @)) U \

lr Ll:3-tr '' sin20coszO\^' 2

(3e.4)

sin@- cos@))],

(39.aa)

a KI.

In the general case when (39.4a) is not satisfied, the radiative correction terms to this cross section also contain infrared terms which must be treated as in Sec.35. These terms have been dropped from (39.4), since they are multiplied by u2 and this is neglected 1" ^-^

c. The Self-Stress of the Electron Ana+lnr ^-^h'i^* WhiCh Can be treated qimnlrr lrrz fha mctfu6jg J r l l l y l y v y L l l g l l ' g L l

darrolnnad

To do this,

hara

ic

f-ln-e-

SO-Cal.LeO

r r - 1 r - Sr + r ^ ^ ^ r r ^ { S€lf u re > > ur

+L lr -r c' ^ c^ rl c^u^u+r 'i O f l

.

we examine the energy-momentum tensor of our system:

Tr,(x): rjl,)(x)+ Tl?)(*) , r/lt (x): i (lv t*1,T,0,yt(x)l* lrt:@),y, ar y @))\ - lAfitp{x),y,tp@)1 - rctrp (x),yr,,p (x))), J T,!7)@):-+{F,7,1}**d,,4n1n,

a,:+-ieA,(x), -t ti , , _,

aA,@) )Au@) a;____d;.

(39.s) (39. 6) (38.7) /eo

a\

t r ?o

o\

Consider the expectation value of the tensor Tu, for a state with one electron. In the rest frame of this electron, the expectation value of all components of Tu" vanishes, except for the diagonal terms. This follows directly from symmetry considerations. We now define 1 . W . T h i r r i n g , P h i l . M a g . 4 l , 1 1 9 3( 1 9 5 0 ) ;F . E . L o w , P h y s . R e v . 9 6 , 1 4 2 8 ( 1 9 5 4 ) ;M . G e l l - M a n n a n d M . L . G o l d b e r g e r , P h y s . Rev. 96, 1433 0954). 2 . M . L . G . R e d h e a d ,P r o c . R o y . S o c . L o n d . , A 2 2 0 , 2 i 9 ( 1 9 5 3 ) . limit is given byAkAnother treatment of the extreme relativistic hiezer and Polovin, Akad. Nauk USSR90, 55 (1953).

G. K5115n, QuantumElectrodynamics

192

Sec.39

E ( 0 ): * I Q l r n n l q ) l n : o d T x , s(0): I klrul Q)lq:oct"x. Clearly call

E(O) is the total energy of an electron

.S/O) the

been written

Sinco

"splf-stroqq"

for the unrenormalized E(0) :

f\ v?v q. v , ( l

t! Y- 'n. c

theory,

%"*p: mol

narynafia

d^lf-^h^-^y.

rL hr r "t ^v g"Ynr h r

/aa

\ - - . , /

7 ) l' l a v e

we have

(3e.r2)

and dm is the electro-

FfOm

- t

: - t lv $),'p(r)1, T,l',,' rl'): o, --

it follows

(39.11)

at rest and we shall

6 n c,

where rno is the mass of the bare electron

( 3e . 1 0 )

,'rr-

wn

r-,

,

(3s.13a) (3e.13b)

that

3s ( o ) - E(o ): I a " *3 9 ,,1 4 1s)l qo' --im^ f a"* Qll't'(xl'tPtr)Jlt)la:o ,] 1

(3e.r4)

or

m o l 6 m* 2 o ' - ( q l l l t ) u )t ,p ( x ) ) l q ) l q : o3:S ( 0 ) . ( 3 e . r s ) I I f . a i s t h e H a m i l t o n i a n o f t h e u n r e n o r m a l i z e dt h e o r y l S q . 0 Z . t g ) t h r o u s h ( 1 7. 2 2 ) l , t h e n 1

f

-.

hH

I

a

z J d ' x ( q l l v t x l , , t @ l l,s=>Ql sl f t l c ) l no : f t { * , + 6 m ) . ( 3 e . 1 6 ) From (3 f . i5) and (3 f . i6) we concludel

- - tl*,# - u*l:- +hW) s(o)

\JJ.1/,1

Thus it is possible in principle to obtain the self-stress from Jhe self-mass by differentiation. In fact the self-mass is infinite lat least in lowest order perturbation theory; c.f. (32.6)], so that Eq. (39.17) actually has no meaning. We will now show from considerations of invariance that the self-stress (39.17) must vanish (39.17) can then be used to define a identically. In particular, cutoff ,4 in (31.18) as a function of the mass /n0. In order to show this, we consider the electron in a system where it is moving with velocity r; in the x-direction. We denote all quantities in the new coordinate system by a prime, so that the following transformations hold :

-,, (r,,)), (ri): 7)7 { l.

/?q ra\

A . P a i s a n d S . T . E p s t e i n ,R e v . M o d . P h y s . 2 I , 4 4 5 ( 1 9 4 9 ) .

Sec. 39

Radiative Corrections

in Other Processes

i(ri): ,+ ((r,,)-(4,)),

I

-t\ r- l/ t , \ : /t -Px \(,,\ "r -

-"

a

( 3e .r 9 ) (39.20)

dsx,_ll_uzdsx. For the energy E(u) and the momentum \(u) frame, we therefore obtain

I93

in the new coordinate

l l t n \ . '' z r S ( 0 ) )

t"t"'

/tr/')\ -rr S(0)) . \"\"/

,

l | ?q

,11

(3e.22)

J/i_32

Hence, in order that E(u) and {(z) have the transformation properties of a four-vector, it is necessary that S(0) vanish identically. Therefore we must have A l|ml otno \, rnn I

^

(3e.23)

Consequently, the dimensionless quantity 6mlmo cannot depend on ?r,r,o.In a convergent theory this would also be impossible for reasons of dimensionality, since the only dimen'sionless variable which could occur here would be e2. However, in a theory with a cutoff, the mass mo can occur in the combination Al*3 .-(Here a ic t1-.,a a'tnrf I .rur result shows that ,4 must protrruJ uE be L ultu_vlt chosenl portional to mzo. Finally, we remark that a cutoff using the regularization prqcedure of Sec. 30 also givesz the result zero for the self-stress. d.

Scatterinq of Liqht Quanta by an External Field (Delbriick Scatterinq) and Photon-Photon Scattering In Sec. 29 we have seen that an external field can produce virtual pairs and that the vacuum behaves like a polarizable medium. An indirect confirmation of this effect is obtained by the exact ineasurement of the Lamb shift. For a free photonwith energy-momentum vector satisfying h2:0, the effect vanishes, as demonstrated by the charge renormalization of Eq. (29.23). However, ifthe photon passes through an external field, then it can produce virtual pairs which can be scatteredJ by the o
l . S . B o r o w i t z a n d W . K o h n , P h y s . R e v . 8 6 , 9 8 5 ( 1 9 5 2 ) ;y . Takahashi and H. Umezawa, Progr. Theor. Phys. B, 193 (1952). 2 . F . R o h r l i c h , P h y s . R e v . 7 7 , 3 S Z ( 1 9 5 0 ) ;F . V i l l a r s , p h y s . Kev. /Y, rzz \ry5u). 3.-From the charge symmetry of the theory it can be shown that the vjrtual particles must be scattered at least twice by the external field if a non-zero result is to be obtained. 4. M. Delbriick, Z. Physik 84,144 0933).

194

(v .l l.: e r rKv r H r r Yl le 5 g rn r L e.r r rO ! r r v vr La r vnu t J rrm Electrndrznamics

Sec.

39

a scattering angle of zero, Rohrlich and Glucksterni have calculated the scattering amplitude and Bethe and Rohrlichz have evalFor uated the cross section for small angles and large energies. a Coulomb field with charge Ze the cross section is of the order of magnitude

6 - (Z a.)arf;.

(3e.24)

The total cross section is much smaller than the cross section for Compton scattering ( o.o*0,o.-rfr ) ; however, the angular distribuThe Deltion is quite different from that of Compton scattering. brijck scattering has a very sharp maximum for small angles and this may make possible its experimental detection.r If the external field of Delbriick scattering is produced by a second photon, then one has a process in which two photons scatter from ebch other by means of virtual pairs. A careful evaluation of the cross section for thisprocess hasbeen made by Karplus and Neumann.4 Here the cross section is.of the order of magnitude

o -x2rB ,

(3e.2s)

and the whole effect is scarcely observable with the present experimental techniques. In principle this effect represents a most interesting departure from the superposition princlple of classical electrodynamics .

1 . F . R o h r l i c h a n d R . L . G l u c k s t e r n , P h y s .R e v . 8 6 , 1 0 9 5 2 ) . 2. H. A. Betheand F. Rohrlich, Phys. Rev. 86, 10 (1952). 3 . S e eR . R . W i l s o n , P h y s . R e v . 9 0 , 7 2 0 0 9 5 3 ) . 4. R. Karplus and M. Neuman, Phys. Rev. 83, 776 (1950). Certain special cases were investigated previously by A. I. Akh i e z e r , P h y s . Z . S o w j t . 1 1, 2 6 3 ( 1 9 3 7 )a n d H . E u l e r , A n n . P h y s . 26, 398 0936).

CHAPTER

GENERAL

THEORY

OF

viI RENORMALIZATION*

40.

General Definition of Particle Numbers In the Jast chapter we worked out various examples of radiative corrections. We proved that the infinite quantities, appearing in the lowest approximation, may be interpreted physically by means of the renormalization of the charge and mass of the electron and so may be eliminated. The remaining finite terms can be compared with experimental measurements. The calculated and measured values show remarkably good agreement. In a few cases the results of calculations of terms of higher order were given also, but this was done without further discussion of the treatment of the infinite quantities which would occur. It should be noted, thouqh, that the idea of renormalization is not necessariiy tied to the occurrence of infinite quantities. Even if the self-mass and selfcharge were finite, it would be necessary first to isolate them, before a comparison with experiment could be made. In this chapter we shall attempt to give as general a discussion of the renormalization procedure as possible, and so we shall not rely on perturbation theory. Also, we will not be preoccupied lvith whether or not the intermediate results are finite. In this discussion we shall vorfLt ver n r

sr ynu eq a , \k

of

sfAi-cq

vn!f

a u

nirron eavulr

nrrmhar

nf

narfinlac

Tn :

i, l, -I e o r y

with interactions, the definition of the particle number is nor trivial . We therefore begin with a discussion of this concept. In a special example in Sec. 11,we saw that a theory with interactions allows various possibilities for the definition of states with a given number of particles. We recall the most important point: particle numbers were defined by the use of the Fourier de* (Translator's note) A survey of recent developments in the formal theoryof quantum fields has been given by G. Kiill6n in Frndamental Problems in Elementary Particle Physics (Proceedings of the 14th Physics Conference of the Solvay Institute), Wiley, New york, 1968. Other articlaq hrz rri116n will be found in T or f i r rh u e O a rh r ce u r o r _vv u v u uu r ra L unr rtr u m T fields (Proceedings of the 12th Physics Conference of the Solvay Institute), edited by R. Stoops, lnterscience-Wiley, New york, 1961; and Proceedings of the 1967 International Conference on particles and fieids, edited by C.R. Hagen, G. Guralnik and V. S. Mathur, Interscience-WiIey, New York, 1967. For an introduction to some other viewpoints, see P.C.T., Spin, Statistics and AII That, R.F. Streater and A. S. Wightman, Benjamin, New York, 1964; and Stephen Gasiorowicz, Elementary Particle Physics, Wiley, New York, 1966; and references cited there.

196

G . K b I 1 6 n , Q u a n t u mE l e c t r o d y n a m i c s

Jec. +u

composition of a free field in which the coefficients were taken operators. Here the canonical as the creation or annihilation commutation relations played a decisive role and determined the eigenvalues of the particle numbers. We have seen thisrepeatedly in Sec . 6, 12, and 13. From the fact that the free and coupled fields have the same canonical commutation relations for equal and times, it follows that for every time T, free fields A9@,I) , ! r ( o ) ( * , 7 ) l s e e ( I 1. 4 7 ) ] c a n b e i n t r o d u c e d w h i c h a r e d e f i n e d b y the equations

,p{o)(x, T) : - i I S(x -

x')ynty(x')d.Ix',

(40.r)

zi,:T

At ts P @ , r ) :- [ l u na :x il i ' )D ( * - x ' ) + A u 1,x 1 DO D (4L]Ia " r ' . ( s o . z ) r' xo Jl ti:T

These quantities obviously satisfy the equations for the free fields and are identical with the coupled fields lor x,o: I. From this we can now construct a system of state vectors in which every Vector is an eigenvector of the Hamiltonian

(* , T)). H(o)(r) : H@(Af) (, , T) , ,t@)

(40.3)

Thus the operatorlt'(o)is a sum of the operators (I7.20) and (17.21). B y m e a n s o f t h e e q u a t i o n s o f m o t i o n ( 4 0 . I )a n d ( 4 0 . 2 ) , i t c a n e a s i l y be shown that the operator (40.3) is independent of xo. It depends only on the "initial time " I . From this it does not follow that the operator (40.3) commutes with the complete Hamiltonian operator o f t h e s y s t e m H ( A * ( x ) , r p ( x ) ), b e c a u s e t h e o p e r a t o r s ( 4 0 . 1 ) a n d (40.2) and thus (40.3) contain the time coordinate explicitly in the singular functions. In fact, we see ,nu, B'tol(I) is identicai with p\o)(A,(*),rp(*)) for xo: I and, since the latter quantity does not c o m m r i t e * ' , n 1 1 t t r ( l, ( r ) , ' , 1 @ ) ) f o r r o : T , n e i t h e r d o e s ( 4 0 . 3 ) commute with the complete Hamiltonian of the system. We conclude that the state vectors introduced above for an arbitrary time 7' are not eigenstates of the (complete)Hamiltonian. The particle numbers introduced in this way therefore cannot describe the actual "physical" particles. Despite this, the corresponding states were often employed, especially in the older literature, and they The physical states were calledr "states of free particles". have to be formed fiom linear combinations of states of free particles. From the usual formalism of quantum theory (conservation of probability and completeness relation), it follows that the transformation from the "free" to the "physical" states is described by a unitary matrix. In this formulation the principal goal of the theory is to determine this matrjx. By the example of Sec. 1l and 1. Previously we have used states with free particles ofthis t y p e i n S e c . 3 8 , E q . ( 3 8 . 1 2 ), e t c . , i n t r e a t i n g p o s i t r o n i u m . I n particular, we put Zs 0 and went over from a Heisenberg picture to a Scliroedinger picture at this time. There the physical states were made up of linear combinations of states of free particles.

Sec. 40

General Definition

of Particle Numbers

197

also by the discussion of Sec. 17 and its subsequent applications, we have seen that the theory can be formulated in another way and that this alternative has proven very useful in the application of perturbation theory. With this in mind, we let the time ? in (40. l) and (40.2)tend toward - oo. By this, the corresponding free fields b e c o m e f o r m a l l y t h e " i n - f i e 1 6 r t t 7 { 0 )( x ) a n d r p ( o ' ( xw ; hich were used previously. As equations of motion for the operators in the Heisenberg picture, we obtain the Eqs. (I7.14) and (17.15), which we write here as

,p(x): rrot(x)- I So@- x,)I @,)d,x,,

( 4 0. 4 )

f ( * ) : i e y A ( x ) y ( x ),

( a 0 .a a )

A,(") : AI|)@)* I Do@- *')i,@')dr' ,

ioi) - j fvOl,yuv4)1.

( 4 0. 5 ) ( a 0 .s a )

By means of these free fields, we can again introduce a system of state vectors with definite numbers of particles. From now on we shall call these particles the "incominq" or "in-particles". The corresponding state vectors lEin) (lin)) are therefore eigenvalue s of the operator H@)(AY Q), ,p@) (r)) , i. e . ,

H@(AP @),y
(40.6)

The essential difference between these states and those which are eigenstates of (40.3) for finite I is the following: the particles defined by means of (40.I) and (40.2) would appear if the interaction of the system suddenly vanished at time Z. Because the interaction changes the number of these particles continuously, one cannot ascribe any actual physical significance to them. However, the particles of Eq. (40.6) are physical because they have existed in the system for a very long time. These particles have problem we are a physical significance because in a collision concerned with a certain number of incoming particles which scatter from each other and then move away from each other as different free particles. In the process, the number of final particles can certainly be larger or smaller than the original number. The mathematical advantage of the in-particles is that we have at our 'infinite length of time in which to "switch on" the disposal an of interaction the particles. to the description by In contrast me_ans of the particles defined by Eq. (40.3) , we can now switch onr the interaction (charge) adiabatically. By the adiabatic theorem of quantum mechanics, the state lEin) (lin)) is also an eigen-

l. As we have seen in perturbation theory, such a switching is necessary so that certain oscillating integrals will assume a well-defined value.

198

G. Ki[16n, Quantum Electrodynamics

Sec. 41

state of the complete Hamiltonian of the system. Up to the present, the most complete proof of the adiabatic theorem has been givenr by Born and Fock; however, as was noted in Sec. 11, there proof for a system with an infinite exists no really satisfactory number of degrees of freedom. We have indicated in Sec. lB how a proof can be givenz for each term in a power series in the charge. Without further discussion, we shal1 now assume that the adiabatic theorem is valid for quantum eleckodynamics, even if the solution is not represented by a perturbation series. AJternatively, we can (or any other property) of a solution which say that the existence satisfies the adiabatic theorem is meant if we speak of the existence (or of any other property) of a solution. We can make no mathematical statements about other, more singular solutions. Furthermore, we shall understand that we always refer to a state with in-particles if we speak of a state with a definite number of particles. Such a state is therefore an eigenstate of the operator (40.6) as well as of the complete Hamiltonian for the interacting fields.o Perhaps it should be explained that we have had to make no assumption about the completeness of our system of states. It is completely acceptable mathematically if there exist other eigenstates of the complete Hamiltonian, and therefore if the in-states form only a subspace in the Hilbert space. The other states would then correspond to bound states of two or more particles. Certainly in quantum electrodynamics there is no reason to suspect that such states exist;4 however, if the corresponding argument is used in a meson theory, one should be prepared for the emergence of such states (the deuteron, etc.). 41.

Mass Renormalization of the Electron We are now ready to discuss the mass renormalization of the electron (Sec. 32) in greater generality. We require that a state an in-state) with a physical electron (i.e., should have mass equal to m,tlne mass appearing on the left side of the Dirac equation. In order to do this, we add a term 6m to the right side and obtain in place of (40.4) and (40.4a),

la\

ll--tmlU ' l x' lJ: (f \l x' -l r Y Ax

1\"t t

( 4 1r. )

I. See footnote 2 , p. 54. 2. In the renormalized theory, a rather complete discussion for the perturbation-theoretic treatment of the problem has been given by F. I. Dyson, Phys. Rev. 82, 428 0951). 3. For an alternative formulation of the "adiabatic hypothesis", s e e H . L e h m a n n , K . S y m a n z i k a n d W . Z i m m e r m a n n ,N u o v o C i m . l ,

2 0 s( l s 5 s ) .

4. The positronium discussed in Sec. 38 is not stable and therefore is not an actual eigenstate of the complete Hamiltonian operator. In this connection, see W. Glaser and G. Kdll5n, Nucl . Phys. 2, 706 (1957).

Sec. 4I

Mass Renormalization l(r) :

199

of the Electron

i ey A(x)rp(x)-f 6mrp(x).

(4r.2)

The object of this section isto give a definition, even if it is only 6m. an implicit one, for theterm This definition should not be tied to any explicit perturbation calculation. We start by examining the matrjx elements of the operator rp(x) between the vacuum (i.e., the physical vacuum, which is the same as the vacuum for in-states) and a state I q) of an electron with momentum q . Thus we seek the quantity (Olp(r)l q). From Eq. (3.2I) the r-dependence is given by

( o l r p " ( x ) l q ): ( o l r l t o l q )r " o " After carrying

out the mass renormalization,

(4r.3) the time component

qo in (4I .3) is to be given Av l[q'+*" . This is the definition of mass renormalization. From this we conclude that the r-dependence of the matrix element of the in-field is just the same as the r -dependence of the matrix element (4I.3) , since in the former quantity the m of (41.1) is the mass by definition. Thus we can write

(ollpf)(x)lq): (o lq|qI q) r0,",

(A1 A\

where [c.f . Eq. (12.22)] the quantities (0lrpf)lq) are siven by t h e f u n c t i o n . , r 1 ;u t ) @ ) o t S e c . 1 2 . vv In order to make a connection between the x-independent factors in (41.3)and (41.4) and to obtain an equation for the self-mass 6m in (4I.2), we need to inkoduce the following device: From the equation of motion for rp(r),

,p(*) : ,{ot(x) - I I @") So(*"- x) dx" , ( 4 1 s. ) and from the integral equation , p ( r ) : - [ l @ " ) S ( x "* x ) d x " - i I , l @ " ) y n S ( * " - x ) d 3 x " , ( 4 I . 6 ) x'

it follows ,tot(x):

th.at

-i I f (*,,) S(.x,,- x)d,x',

I,l("")ynS(""-

x)dsx".

(4I.7)

In (41.6) and hence in (41.7) also, the time xi is arbitrary. Equation (41.7) allows us to express the in-field .1pto)(r) as a surface integral over the field rp(r) at an arbitrary time zi and a fourdimensional volume integral from - oo to x'o. In these equations only the limits for the time integrations have been shown explicitly; the spatial integrals always run over the whole volume V, If we set the arbitrary time in (41.7) equal to the time in some arbitrary operator F(r), then we obtain

(x'\} {F(x\. s (x,,- x,)d.x,, t " " ) ' Y u@\ \ " t ) : i gal,l@,,)} @ - i { F ( x ) , ' p ( x " ) } y + S( x " [ t":tn

L \

x ' )d t x " . I

(at.g)

Sec' 4l

G. fir116n, Quantum Electrodynamics

200

The last integral in (4I.8) contains only operators for which the time coordinates are equal and can therefore be calculated from the canonical commutation relations . We now examine the vacuum expectation value on the left of (aI.B): (o | {r(r), yro)(x')}| o; :

I

I's

lr) T Irn r

nrincinlc.

}JIrrIUryfs,

iLhr rev

rsul rtr-r

n v wr vr ra r

scLf var tv a v

c

ll -zl \

ir n "

(4I' e) | z) (zl F(x)i o) . i "x,) (4I.9) has a cOntribution

In particular, from each state in a complete set of state vectors. though, if we choose this set as the states of the physical parsupplemented, if necessary, by thebound ticles (the in-states), states), then it follows by definition that the states lz) between the operators must contain just one electron for the first term and just one positron for the second term. Therefore if we compute the vacuum expectation value of the quantity (4I.8), we obtain the of the operator F(x) between the vacuum qld the glgggllq MtrlI Moreover, we recall Eqs' o ne-positron states. o r one-electron (fS.f) tfrro"gtr (I5.3), according to which the function S(x" - x') onthe right side of Eq. (41.8),can be written as a sum overthe matrix elements (4I .4). Hence, by comparison of the coefficients ot (qlqtot (x') | 0) , we find r.l

( 0 l F ( ' ) l q ) : i / ( 0 1 1 r 1 r 1l,@ " ) j l 0 ) ( 6 l r p ( o r ( x " 1 1 q )+a x " \ - @

+ ( o l, :) t1p (9* )], " l/ o ) (\ -o' l ,t p ' o\ \" t(/ r ) l q )

/

(41.10) \ l

|

,

Equation (41.I0) allows us to express the matrix element between the vacuum and a one-electron state for an arbitrary operator F(r) in terms of the vacuum expectation value of the anticommutator of The the. same operator with the right side of the Dirac equation' under invariant is vacuum advantage of this procedure is that the (this is not true for the one-particle a Lorentz transformation states) and so we can use simple considerations of invariance. tp(x). We find As anexample, we take the operator F(x)tobe

(01tp"(x)lq) : I (41.11) , l _ : ; f (ol {rp,(r),7p@"\} (.) lD .l l0) (0 | y'(f\(x") lq) dx"'-l (o I p[o) -@

By for clarity' The spinor indices have been shown explicitly, o n t h e o n l v d e p e n d c a n (3.2I) the expression (01{p(r),1(r")\||0) x-x" , and, by using invariance condifference of coordinates it can be written as siderations,

-'") (0 I {v" (r), IeQ")} | 0) : / dp eiot' \6"eA (f) 1 i (y) " nP, B (p)}. (41.12) we now make an assumption which we shall frequently use later: vector of every physical We assume that the energy-momentum

Sec.41

Mass Renormalization

of the Electron

201

and that the vacuum is the state of lowest enstate is time-like ergy. The energy of the vacuum is taken as zero. Equation (4I.I2) then follows from the fact that from /u and po only the two combinations dou and OP),B have the correct transformation properties. The two invariant functions,4 and B d.epend only on the vector p, and, of course, in an invariant way.r Since the quantity (41.I2) vanishes for space-like x-x",

o (x- x")(ol{,t@),7@")} lo> is also an invariant.

i @(x- x")
We can therefore write

.,r, :' ^!'l r).|);.-.,,, IA,b\* i,,b B,6\t |,n, \zn)-J "''t )

where A'(p) and B'(p) are different functions. Then from (41.11) it follows that

( o l , p ( "|)q ): l r + A '( q I) i y q B ' ( s )(lo l 'y, ' ( 0 ) /r| .-r\r -- r

.\-/ \ : [ , 1+ A ' ( q )- m B ' ( d ) ( o l ? ( o ) ( r ) l q ) . J

(4i.14)

The equation of motion for tp\ot(x)has been used in the last srep in (4I.I4). For each one-particle state lg) we have q2:-mz and general the form 0. From of the functions A'(q) and B'(q) Io) we then conclude that the factor 1-A'(q)-mB'(q) is independent g. ofthe spatial partqof Accordingly, we can write (41.14) as follows:

(ol,p(x)lq) : N(o | ,pb)(x)lq) ,

(4r.1s)

w h e r e I y ' i s a " u n i v e r s a l " c o n s t a n t ( i . e . , i n d e p e n d e n to f r a n d a ) . From (41.15) it now follows that

(olt(")tr>: (,* * *) et,t(4tD: o.

(41.16)

The two equations (41.10) and (41.16) can now be used to obtain a n i m p l i c i t e q u a t i o n f o r t h e s e l f - m a s s 6 m i n ( 4 I. 2 ) . T o d o t h i s we take F(r) in (41.10)equal to l(r). This gives

o : i I @ @- x ' ) ( o l { l U ) , 1 ( 'l' o) })( 0 1? ( o@ ) ' ) l q ) d r+' \ (r) Is) a li ey,(olA,(r)lo) | 6m)(0I ?(o) )

Gr.rz)

or

6 m ( ovl n @ ) l q > : : - i I O ( x - x ' ) ( o l { l ( r () *, 7' ) r , 1(00)1, t o t @t q , )> a -) , 1 ( 4 I ' 1 8 ) since the vacuum expectation value of the potential vanishes in our gauge. To reduce the right side of (4I.18) we return toEq. (3.21). 1. The most general possible form fon I is there f ore Ar(P') I A r(P"l e (p), p . It can be shown where A;(pz\ is non-zero only for time-like (see below) that inthis case At(P\:0. The corresponding statements also hold for the function B(p) .

202

Sec. 41

G. Kiill5n, Quantum Electrodynamics

Using it,

we can write

( o l { L ( *) , [ ( ,Yto ')]):,;#

I

d 'pe i f('-r') x

Folo

(4r1 . e)

a'pr'ntt-'tx x [4., (f') -f (iy f ]- m)Z[" (f'))*p+ 1r,1,, f P u> 0

x [r{-) (p,)+ (- iy f + m)2}-t(p\f "B with l - F ( +\ r)t/ A_2| L\ t\ ";f. , rA ) |, m ) 2 l ' . ) ( p r ) f , e : - v

L-t

. p \Iz() 0 : p1 / e l

z) (zll"l0),(41.19a)

(hz\t (lL rP o>.(41.19b) \ r , / I r r r i,t h + /n)ZP (pr))_e: V Zell*l z) (z l-lpt -l p\z):p

The four functions )J*) *u defined by Eqs. ( I.19a) and (4I.I9b). The summations in these equations run over all physical states f o r w h i c h t h e t o t a l e n e r g y - m o m e n t u mv e c t o r i s p ( o r - 2 ) . A g a i n , Z is the volume and we have taken the limit V'->a in (41.19). Frnm fhc

qrzmmctrrr n vrf

cv r hr \ 4aYrv c c

Jl r r r r r r eLJ/

tL h, r cv

tL hr r cv vnr l r, r z

We

Can now

show

that

only two of the functions IJt) atu independent. We start with

If'plz)(z Il;l 0). (4r.zo) lrl,) (p,)+ (iy f t m)rl,t (t'\)"e: - ro,F_fo Now we apply the matrjx

C of Eq. (14.3) to obtain

-V (o Il,l z) (z IloIo) (C')e l lrtr (p,)* (i y f * m)2/,t (p,)7 " " e= p k2C", t:p : c", [x,l-)@,)+ (- iyf +m) z]-t(h,)7"uQ-'),p: : [:J-)(f,) r (iy f -t m)2? (f,))"e.

I

i(41.21) ) I

I n ( 4 1 . 2 1 )w e h a v e u s e d ( 1 4 . 4 ) t h r o u g h ( 1 4 . 6 ) . F r o m t h i s i t f o l l o w s that

,j,) (pr): zt-) (fr)- zo(fr) . Consequently we can write (41.19) as an integral -space: 2

( 41.22) over the

entire

(o|{1"(,),f,(*')}lo>: :

(4r.23) d- p eipv- 4 lz, (p2) + (iv p+m)z,(p\)"Be(p)' ] .rtr J -1

f

N o w w e s h a l l c o n s i d e r t h e e x p r e s s i o n @ ( * - x ' ) ( O l { f ( x ) ,l( L and use the Fourier We write @(x-x'):+(ae1x-x')) tation for e(x-x'),

;(x - x,'\ - l-p .t?v

,J

'S*0". r,,u, 'r,. T

)Jlv).

(4r.24)

Equation for the Constant Iy'

Sec. 42 'I

h l

C

203

^1 1 rA e

tf

anl"

e(x- '')

J

d'Pe;iu-z')t1ilZr(P\:

': : ;;i I aP"'v-"o I #+,2,(t] @oti'): _: -

);

:;4

f

f dbeit6-APf

\2r)"J

(4r.2s)

daz'( '-al alP'

J

and

*

tf

\2n)"

:*

e ( x - x '' ) I d b e i p ( r - a '()i y f * m ) e ( f ) r r @ \ : J

f d 6 r , r r , - ,('i,v b * * l p i ! : t zo t- -- f-'4 - -

t2nJ.J

-

);

f

(4r.26)

I

*o f

;",

tA

dp e l p v-r')Po" O fJffi J J

| -\

2 ,(p,- ( fo* r ) ,) .

The last integral in (41.26) vanishes by reason of symmetry we nave

- i @(x- x'): d,.St e;otr-,) * ;* I (4r.27) -l x lE,(f,){ i n t (p)Z,(p,) (iy h * n) (E,e\ + i t e(p)Z2ez))1, x , ( f \ : P I W- *T , ^- "X , ( - o ) . J Y

( 4 t . 2 7a )

0

Substitution,of(41.27)into (41.18)gives 6 m ( o l p ( o ) ( rD) l: l E r ( where the equation

l:';l'i"::',|i:l;n

m 2+ ) inI,(-

of motion

for

?;,'Hl':"

l n z ) l( o W ( o ) ( rs) >

t @r.zl)

rp(0)(z) has been used.

From

(4I're) ihat ' '-,nz) ' ir rorlows o

6nt

o-t , , ( - n ') : P I t ' ( O o. a-m'

(4r.2e)

J 0

This result is an implicit equation determining the self-mass. It is implicit because the function Zr(P\ has been defined by the use of the right side of the Dirac equation, and from (41.2) this already contains the self-mass. 42. Renormalization of the DiracField, Equation for !l]9 ConstantN W e s a w i n E q . ( 4 1 . 1 5 )t f t operator rp(r) between the vacuum and the one-electron state is proportional to the corresponding matrix element for the in-field.

204

Sec.42

G . K i i 1 1 5 n ,Q u a n t u m E l e c t r o d y n a m i c s

We will now rescale the Dirac field so that the constant of proportionality in (al.15) is equal to 1. This is done most simply by dividing each matrjx element (olrp(*) lb) bv this constant ly'. In other words, we introduce a new operator

,t'@):#,t@) . This new "renormalized" relation

field

satisfies

the following

(42.r) commutation

: #r, 6(u- n'). {'p'("),y'(x')}"":,,"

(42.2)

Since each term in the Dirac equation contains just one operator tp(x) , the factor ly' cancels out and the equation for tp'(x) appears formally the same as that for tp(x), ta\ ly-]- *mlu'(x\: Y

ax

t"

f ' \t*-1\,

) 6mrp'(x). l ' ( x ) : i e yA ( x ) v ' @+ With this new f'(x) we again define the function" ,;(p\ and obviously we have

,; (p\: # ,,(p'),

#: z;(-m')'

tA?..?\ \aL'a'

( 4 2. 4 ) bv (ai.23)

lAc

r\

(42.6)

Henceforth we shall be concerned only with the renormalized operator and the old operator g(r) will no longer be used. Accordingly, it is not necessary to denote the renormalized operator by a prime and we shall drop it, denoting the renormalized operator simply In a similar fashion, l(x) and Z,(pz) now bl ,p@) from now on. If all the quantities present denote the renormalized functions. of the field operators would be were finite , this renormalization A s _w e s h a l l s o o n s e e , t h e c o n without any great significance. stant N-1 is not a finite quantity. L See also Eq , (32.20). ) This quite useful because it allows us to is therefore renormalization isolate an infinite quantity in the theory and to study this quantity separately. An equation for the constant N similar to (42.6) can be obtained in the following way: We write the integral equation for the renormalized field asr l. The constant i/ is dependent upon the charge o and therefore becomes a time-dependent function N(.{0):N(e(ro)) during the adiabatic switching. Actually the integral in(42.7) should appear as follows: ,, N(z[) r. , JsRlx-x) tt1*It\*lox. we have not written the details out explicitly. For simplicity,

Equation for the Constant N

5ec.4z

,p(x): {v$) @)-,[r" @- ,,) l(x,)d.r,. The anticommutator (42.2) can then be written

(., rp(*')}: fi {v.,tr),v@) @')}+ {,t@), + ft (r),rt'@)-#l', ("')} + l 1'

t

)

|

- frv,o'{*),y,@t(,.)1+ +d,{ot"f (

I

|

.

.'.

t

fuz.at

d/" . I + JJ t.(*' - *"){f{*"'),I @")}sn(*"'- *)d)c" ff

-

|

If we take the vacuum expectation value, the second and third terms in (42.8) contain only the matrix elements (Olrp(x')lq) and ( O l r p ( x )l q ) [ c . f . t h e d i s c u s s i o n f o l l o w i n s E q . ( a 1 . 9 ]) . B y m e a n s of the definition

(r) | q) ,

(42.s)

and using the functions Zi(fz) we get

( o ) { ' , p ( x ) , , t @ )-} 1i s0()x: ' - d

[#*

''toi

" l-

I (42.10)

l -c+l o,,ip(,'-x) p+m)2,(p,)lffil ,@) ##lE,(f,)*(iy With

the identity iyP-*

iyp-m T F | / . . . .r , . " " \n 1 -tr;-;a p, -- mr-lzr+ \t Y P t m)LzJ

:

|,n'.", :(iyf-*)l##-ffi,1-#* ,J

it follows from (42.10) for xo:xL that

j,ru6(u-u'):

€

'

1 .' . : rE6(r-n') lt-$" +J'a.{\t::t-##r.'1, | * , N-r N2

-

r 2mE!(- *')) , | {Z,t- rn'z)

daEt( Ei (- mr\ ' t -- - t =1). J

*,

(42.13) (42.13a)

1n-*'1'

0

The integral equation (42.7) and the equation following from it, (42.I3), require a more detailed discussion, since an uncriticai use of these equations can very easily lead to contradictions, Thus, for example, one might readily conclude from(42.7) and (41.16) that the last term of (42.7) vanishes for transitions between the vacuum and a one-particle state, and therefore that

(ol,p(*)Is) : * (oI v0)(")I s>

(At

1A\

206

G. Klill6n, Quantum Electrodynamics

Sec.42

ought to ho1d. Comparison of (42.14) and (42.9) then shows N:1. In a similar way one could conclude from (42.9) and the fact that the functions Et(f') vanish for p2:*mz lsee Eq. (41.28) and the remark following it] that the first term on the right side of (42.10) ought to appear multiplied by a factor l, rather than by I ('r t '- 2- t -1 A / - 4- )t \l ' \'

N2

The answer to both these puzzles is that although does vanish, the convolution integral N-1,^,

"N'

r^r,

( 0 l r t o r ( x ) l q ' ,-

Iso(r-

x')(oll(x')lq)dr'

(Oll(x)lq'2

G2.r5)

is different from zero. In the proof that (Oll(x)lg) vanishes, of the constant ly'. This is we neglected the time derivativel allowed for all finite times because of the adiabatic character of fhc swifchinoT\Teolect of the time derivative is not allowed if integrals from -oo are involved, as in (42.15). Neglecting the time In derivative in such cases can lead to apparent contradictions. Sec. 32, Eq. (32.10) and following, we worked out in detail the approximation to this phenomenon. first perturbation-theoretic There we saw that after the subtraction of the term with tl0) (-*'), vanished because of an the matrix element of l(x) [eq. (:2.i4)] expre s s lon

(q'* m') 6(qz! nf) .

(42.16)

In the convolution (42.I5) there is a factor q2+rnz in the denominator, so that the right side actually has the indeterminate form

n i I*i 6 (q2 * m2 ) q't m'

(42.r7)

This hoids if the adiabatic switching is not taken into account. However, if the switching is carefully carried through, rather than (42,I0), we obtain aterm of the order of magnitude o( [Eq. (32.16)]. A similar term appears IEq. (32.I3)] in place of the denominator qztm2. The behavior of numerator and denominator is well defined in the limit a-+ 0 and gives the unambiguous, non-zero result 'l

I -

N

i n ( 4 2 . 1 5 ) . F o r u n d e r s t a n d i n gt h e e q u a t i o n s a b o v e , i t i s

important to remember that the operator l(r) contains these sin'l nrrlar 9uru

lL- so rr rmr rc D .

S ri lnl nu as u

fha

frrnnfinnqtrr

jfi .\ ff f i 2 \ , /

L-,,^ llqVg

^^Uqglr

'.]^fi-^d ucflll€u

l ay\ 7 u

means of l(x), they contain singular pieces also. This explains If the adiabatic switching is the formal contradiction in (42.10). done carefully, it is evident from the foregoing discussion that the two functions Xr(12) must contain terms of the form

q.'6(p'* m'). These terms give a non-zero contribution l.

See footnote 1. n.

204

(42.r8) in the integral (42.13a)

Sec.43

The Renormalization

of the Charqe

207

w h i c h j u s t e n a b l e s u s t o w r i t e ( 4 2. I 0 ) a s

(ol{,p(x), y(x'))lor:#l

orri!(r'-,) ,@1,6(p, +m")(iy?-m)+ \

l*,.,,

Lzfr(p,)t (iy f t m) zi"r(fz)l#;#]

+ #;#

J

The two new functions ZlrE(fz) are defined so that they are equal to the original functions I,(pz) everVwhere except at the point The singular expressions (42.18) at the point bz:-mz h2:-ruz. hrrzo

haon

ca rvynrlri ua ri tul ryl /

SUD b tIrf aa C C tt ee dd

tf fr O m

ZE, F c ( \br2 \t ,.

sv o _

t.h, .a_t .

.n. o_ f"

Only

does X'"s(- mz) vanish, but also (fra*r1-, Z;,r(pr) can be set equal t:o zero for .pz:-7r2. With these new functions, we obtain from (42.19) not (42.13), but rather

t + ti"s(- m2)+ 2mE'rec, (- m2).

S:

(42.20)

For the following it will usually be simplest to calculate formally with the originai functions J,(p2) and to make the explicit subtraction of the singular parts (42.18) only in the results. 43.

The Renormalization of the Charge We have formally renormalized the mass and the field operator of the electron in Sec. 4l and 42 without the use of perturbation theory. In this section we are going to give a similar treatment for the charge renormalization. Because of the gauge invariance of the theory, the photon has no self-mass and therefore it is not necessary to carry out a renormalization of the energy of the photon. Accordingly, for the matrix elements between the vacuum and a one-photon state lA), there is the same r-dependence for the complete potential Ar(x) as for the in-field A,f,(*) . In this case the most general relativistically covariant relation between the two matrix elements is

'

( o l A , ( ' ) l A )c: l a , , +M z : 3 ; - l ( 0, l , a i ' ) ( x ) l A )(.4 3 . r ) Axpdxyl'

The two constants C and M in(a3.L) are independent of k and. x. A formal proof for (43.1) can be given following the method of proof o f E q . ( 4 1 . 1 5 )i n S e c . 4 1 . U s i n g t h e e q u a t i o n f^, A ' , i ' Q )- . l o ( * - x " ) i p Q " )d x " -

-l

( 4 3. 2 )

D(*- x,,) s1o'nr.{', ) -*r n pAr1r"1ap(*u:,")larx,, \L )oxo l

--JL-6-u\^-^

I c . f . E q . ( 4 I . 7 )] a n d t h e c a n o n i c a l c o m m u t a t i o nr e l a t i o n s , w e f i n d for an arbitrary operator F(.r) , x

: I o Q'* x") F(x),i,@"))dx"-i D (x,,-x) lF(x),A'If'("')l ::+ -* w " t t \ ^ t +l *

. oD(x'- x)

'---

a;l-

oF{x)

" a--d-" Apvl

'sl [t+:

I )

C. fiitt5n, Quantum Electrodynamics

208

In the last two equations /u(.r) stanas equation of motion for A,,(r) :

for the right

Sec. 43 side of the

( 4 3. 4 )

nAu(x):-i,@).

The detailed structure of the current operator is not important for (43.2) and (43.3). Just as in Sec. 4I, the vacuum expectation value of (43.3) gives an expression for the matrix elements of F(,v) between the vacuum and a one-photon state. We have

I f rn,.rr

I

If we take F(r) as 4,,(x) , then (43.1) follows from (43.5) and the values use of invariance properties of the vacuum expectation which appear. The constant C in (43.1) corresponds closely to the constant Al in (41.15) but the constant M has its origin in the particular If the state lA) contains a vector character of the potential. drops out. The contransverse photon, this term automatically stant N of Sec. 42 was not eliminated from the theory by a counte-r term. [The onty counter term in (4I .2) contains the self-mass.J It was isolated by renormalization of the field operator 9(r) and The situation is somewhat it was then considered separately. different for the constant C of (49.1), since now the observable fietd strengths and are therequantities are the electromagnetic The constant C has a fore linear combinations of the potentials. in so far as it connects the units of the significance "physical" also the charge) of the complete fields field strengths (i.e., 4(orlY\ For simnljcitv, we 4 (v\ with fhose of thc in-ficlds shall use the same units for these two kinds of fields and thereThis can only be done by means of a fore we have to take C:1. we shall add an extra term to the curcounter term; accordingly, rent operator: ! ' l t

/ t u \ ^ t

i r@): tt#

\ . " 1

. f, (,),v,,t @l - L J A,"(x)

( 4 3. 6 )

One can substitute this expression with the charge renormalization constant I into (43.4), transfer the term in Z to the left side and divide by 1 - L . This factor can then be understood as a renormalization of the charge. The factor N2 enters the first term in (43.6) because y(x) is now the renormalized Dirac field. In what follows, it will be useful for us to define the charge renormalizaThis is most tion so that the counter term is gauge invariant. as current the renormalized done by defining simply ./-_\

lu\x):

ieN2 r-,

2

, /-.\-- ,ln t t.,l L r p \ x ) , y p y \ x ) )L l u A p \ x )

czAr(x)\

b*"4I

'

(43.7)

Sec. 43

The Renormalization

of the Charqe

209

Ar(r): Af) (r) t J D"(x - x')i,@')dx' , we obtain ?irtx'\ aAp(A_ ae!)Q\ - ^ t ' ' _ "x,) o_@ o*, t ? , u +' J f ?ru 0*),

a,qtt,t ?r,

( 4 3. 8 )

, (43.9)

It follows from this that the difference ot (43.6) and (43 . 7) js of importance only for certain special matrjx elements with scalar and longitudinal photons . Since the new counter term in (43.7) contains the derivatives we must check to see if the canonical commuof the potentials, tation relations are changed by these new terms in the Lagrangian. Our complete Lagrangian is now

9:9,pl9e*

( 4 3. 1 0 )

9w ,

er:-f[rorQ***)v{i)l.tn''tt' - +t- +* v,+ * v@), ('),,t v{")l+ } a* x' l'p @)l,i

- +L W -W) e# - ffi-+ WW,(43.rz) ee: sr:

'*

N2Ar(x) l,t@),yrrp@)1.

( 4 3. i 3 )

The usual rules for quantizationgive {rt@),rp(*'))xo:xi,: nfiA1n - u'), in complete agreement with

(42.2) .

We also obtain

lAr(*),A,(x')1,":,1,:o ,

lY#,

(43.lS)

(43.16)

* n,(*)),"_,r: |r.€,,,6(r t') ,

(43.16a)

tu,:6r,-L6rt6u+, faAp(r)

b.ang'll

l- # ' - # - ) , ": ^ ' :

( 4 3. 1 4 )

2 '

L A , l" T =T \6'n;i + 6'1ar ) ae a' ) ' ( a3' r 7)

For Z :0 , (43 .15) through (43 .17) go over into (17. 8) and (17. 9) . T h e r e l a t i o n s ( 1 7 . 1 0 ) , ( L 7 . I 2 ) a n d ( 1 7 . 1 3 )a r e s t i l l v a l i d i n t h e r e normalized theory. Apart from the replacement of du, by €u, in (43.16), we see that the constant ltrplays an analogous'role for the electromagnetic field to the constant /y' for the electron field. Both appear in a similar way in the commutation relations for the renormalized fields. lf we do not consider the last term in (43.12), we can take the renormalization of charge to mean the replacement of the tield Ar(x) Ay lE=T ,1, (x) and of the charge

2I0

Sec. 43

G. Kiitt6n,'Quantum Electrodynamics

e by 1t1- a . In this way we can understand why the interaction term (43.13) ls formally not affected bV L . The exceptional character of the last term in (43.I2) has its origin in the gauge-invariant In order to obform used in the definition of the current (43.7). tain the expression (43.7) for the current, we need only multiply part of (17.3) bv 1 - L , This form of charge the gauge-invariant was first given by Gupta.l renormalization we are now ready to derive an After these formal preparations, explicit equation for L . To do this, we shall employ the same We therefore calculate method used in Sec. 42 to derive (42.I3). the vacuum expectation value of the commutator of two potentials Ar(*), .,,

(ol LA,(x), A,(x')f| 0) : (0 llAf' (x),,4t0) (r')lI0) + | 0) + (*')l A'j" (ollA,(x)A't'(r), I * | tnu. r, * ( o l l A f ) ( x ) , A , ( x ' ) - A l o ,l(or)' )+l | ( o l l i r @ " ) , * " ' ) x D n ( * ' ) d , x d " , x " D ' p ( x " ) * l! 1,@"')]loS. As before, it is useful to introduce a new notation for the vacuum expectation value in the last term. We therefore write

(P)+ (olli,@),i,@')llrr: ei,(''-x) IrL.,) *f,!d,p | ;fr [ P o< o

with

'nu''n'

a'P,n"o-') nf*)(P),

|

(P): v,.Z(oli, nL',) l,) Qliulo),

(a3.20a)

r l b\(p): v Z : Ir f,)( - p) .

( 43.zob)

p-':p

n\z)--*

From qeneral principles of invariance, functions nL, (p) must have the form

it follows

that the two

nlh)@): A(p,)6,,+ B(p')p,p,: nli) e p): nL;)(p). (43.2r) From the continuity equation for the current, which is also satisfied by the renormalized current, we have

o : ppnfl(il : LA(p,)+ p,B(p\1p,,

(43.22)

and therefore

I (P') nLi (il : (- f'6r, + PrP,) , nQ,): -+ >Qti,lz)(zti,)o ,) -|D-

4 p\z\:f

\1J.LJ)

(43.24)

-,, n (p2) .(43. 2s) o>=;# J Ape;or'' e(il ? f, d,,* f ,,.p,1
Sec. 43

The Renormalization of the Charge

ZII

By using (43.1) and (43.2s) we can now write (43.Ig) in the folinrrrina rv vv rrrv

fnrm. r vr rrr .

A , (,x ' ) ) 1:0 ) < o llA t "@) #

ap r'r,'-" )u(h) x .l '

,n,.,u, -!#) + p*p.(W- 2M6(p\)1. x lu,,(uro, )

If the two times rj and ro are equal, then the right side ot (43.26) must vanish, according to (43.15). The first term in the square brackets vanishes identically on the basis of symmetry. The secy+4, we have ond term gives an equation f.or M. For p:4,

o: dn I d'.pr,nr*'-,,p, I op,lp,l(W 2M6(p\il:

:-+6(r,--)l.f#-,*f,

\1J.2/)

(43.28) For other combinations of y and z the last term in (43.26) also vanishes identically. (43.26) with respect to the time ro and then By differentiating setting the two times equal , we can also derive l^t

l"\

'l

- - i 6 t , , l-j' r t ( 0 ) ] b ( r , - -r- /) <' ' lo l l od )no ! ' ,*A ' ",' ( r' ') ') l l, o - ' - )' , . : , . , \-/i-\-4"1 -

i

trTt-''\ '1 a ,o a lP r ar laf , (,/fr,o6 ( p ' ? 1 f -S' " 6 o i p t * ' - r r i rd - fa . d lr p a ) 1 " ' i " ' _ t u a l r i l i (" a : ' z o l

t' .t o . : * i 6(r - n'))16,,(l+ ]I(o))- 3,16,4fip)) , t 2 n fJ

I|

u(br): P ['oa',-,,o'. n*'"

'l

l ( 4 3. 3o )

I

A c o m p a r i s o no f ( 4 3 . 2 9 ) a n d ( 4 3 . 1 6 ) n o w g i v e s 1 :. ----------:1+-11 (0). l-L

( 4 3. 3 1 )

This is the desired eeuationl for the constant Z . As a by-product, t. A similar equation for the charge renormalization was first given by H. Umezawa and S. Kamefuchi, Progr. Theor. phys.6, 543 (1951). For the complete system of the equations given here for the renormalization constants, see G. K5i115n,Helv. phys. Acta 2 5 , 4 I 7 0 9 5 2 ) ; H . L e h m a n n , N u o v o C i m . I l , 3 a Z ( 1 9 5 4 ) ; t v l. G e t l Mann and F. E. Low, Phys. Rev. 95, 1300 (I9S4).

2t2

G . K a l 1 6 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec. 44

we have a similar equation in (43.28) for the M of Eq. (43.1). (43.26) with respect to r,j as well as ro and After differentiating using (43.28) and (43.31), it is not hard to showthat (43.17) is also satisfied. General Properties of the Functions 1/(22) and I;(22) In the earlier sections of this chapter we have given a formal system of equations for determining the renormalization constants of quantum electrodynamics. This system of equations is formally sufficient to determine the renormalization constants because there are the same number of equations and constants . We must stil1 discuss the consistency of this system and if possible decide whether it has any physically useful solutions at all . We are not going to be able to answer these difficult questions completely; at least we shall try to give a basis for some future treatment. In our formalism the renormalization constants are given as integrals over the weight functions Since these n(p2) and Zn(f\. functions will play an extremely important role in what follows, we shall now discuss their properties. We start with the important observation, which will be of crucial significance, that the definitions (41.19) and (43.24) contain only sums with a finite number of terms. In order to show this, we first note that if we consider only states with one in-particle there are only a finite number 44.

I/

t*d!'p 2n)'

of them which have momentum between p and p+dp.

At

-m2 most, these terms contribute to the weight functions_whenp2: and, in fact, this contribution "vanishes" becauser of the renormalization condition (41.I6) and because of a similar condition for the current operator l,(z) . we For states with two in-particles consider the weight functions in the particular coordinate system where the spatial p vanishes. Then the two particles have equal and oppositely directed spatial momenta q and if the two masses aremrand

m*

the total energy is

equation determines

lgl

pr:Vq,+*i-lll

as a function

q\*'r.

This

of. ps :

tat:!V-i-v'#tnl;@P

,

and the number of states of thistype in the sums (41.19) and(43.24) is again finite and equals

-

tf_'r

"'

enF--o-

(44.r)

In a similar fashion, we could proceed with three or more particles and show that there are a finite number of stateswith a finite number of in-particles which.have a given total energy-momentum

1. The word "vanishes" is used in quotation marks here because these terms can still give certain contributions to the integrals. C.f. the discussionfollowing Eq. (a2.15).

Sec. 44

Properties of the Functions

II(pr)

and Eo(pz)

2I3

vector p. On the other hand, in a state with a given finite p, there can be only a finite number of particles of non-zero mass. As elementary considerations show, this finite number is smaller ., lt h, n a r tL Jl g l g g than r r va l l - ! = , w h e r e m i s t h e s m a l l e c t n a q q n f rL h Psr V

*r'

which are present. In principle, an infinite number of photons (of zero mass) can occur in our states; the simplest way to avoid this difficulty is by introducing a small photon mass p . At the same time, this automatically avoids all difficulties involving infrared divergences (c.f. Sec.35). With these restrictions the definitions of our weight functions contain only finite sums, i.e., in the limit V--> o they involve only integrals over finite regions in p-space. If the matrix elements of aII the renormalized operators are finite, then the weight functions are finite quantities..t It has also been assumed here that there are at most a finite number of bound states of given binding energy. This restriction seems quite weak. According to the assumptions made about the mass spectrum of the system following Eq. (41.12), the weight functions must vanish for space-like values of pr. This fact has already been used several times. We can now go further and show that the function n@\ has no contribution from states of Iess than three photons (or one pair). This comes about for two reasons: According ro the renormalization condition, the contribution of the one-photon state vanishes. Also, from the charge symmetry of the theory, it can be shown2 that matrjx elements of the current operator between the vacuum and states of two in-photons vanish. The conclusron is that for -p,<91t2,the function II(pr) is zero. Because of the conservation of charge in our theory, we likewise conclude that l. Here, by the word "finite" we mean that the integral of one of the weight functions over a finite region is finite; i.e., that the weight functions contain no singularities stronger than delta functions. Such delta functions, if they are present at all, occur only in connection with states which are not scattering states. 2, By a calculation similar to (43.3) one finds (*')1,A(:'j(r")) lo> :
,. i, li^t,,rul, ti,(x"'), ip(41)lo)+ + Dpt' - /1v) D(r" - ,"') (olli,@Ivl, liil*"'),ip@))llDl. Because of the charge symmetry tc.f . Eq. (I4.2I), which also holds in the theory with interacting fieldsJ the right side vanishes, and hence also the matrix elements

( o l i , , Q ) l hh, ') .

2t4

G . K d l . I 5 n , Q u a n t u mE l e c t r o d y n a m i c s

Sec. 44

the states in (4I.19)must haver charge a and hence that they must contain at least one electron. The states of only one particle give no contribution here also, and the first non-zero contribution comes from states of one electron and one photon, so that the -p'<(rnld'. The lower limits of f u n c t i o n s X o ( p z )v a n i s h f o r ( 4 2 . 3 0 ) ( 4 2 . 2 0 ) t h e r e fore be set equal to a n d c a n i n integration (*lti' arA 9pt, respectively.2 Takins account of the small phot o n m a s s, i T ( q i n ( 4 3 . 3 1 ) m u s t b e r e p l a c e d t t v I I ( - p ' ) . Finally, we wish to show that the function II(Pz) is always positive. Fromthe definitions (43.20) and (43.23) we have

z I(o I i, 1z)(,lt I0): (pi- p\ n (p'). 'p\z):

(44.2)

p

The .r-component of the current operator has to have real expectation values, so that this operator must be self-adjoint, i.e.,

( e l i , l 0 ) : ( ( 0 1i " l r ) ) * ( - t ) " i " '

(44.3)

The symbol /ff;) denotes the number of scalar in-photons in the state lz) . From this we have

n @')-- n#

Ql i, lz)l"(- r)"1" rA rl

(44.4)

In the sum on the right side of (44.4) all states without scalar photons (or, more generaily, with an even number of scalar phoWe shall now show that the tons) give positive contributions. negative contribution of a state with one scalar photon is just cancelled by the contribution of a corresponding state with a longitudinal photon. From Eqs. (43.3) through (43.5) we obtain the following expression for the matrix element of the current between the.vacuum and a state la,k)z

(*')| k) d'x' . (44.5) (o| i,@)| a,k) : i'J @@-x')(o I [i (r),i, @'))| q.>
lQ,rp(,)l:-ewQ)

with

Q:

-iId3xin@).

On computing the matrix element of this operator equation between the vacuum and a state lz) with charge Q('), we find Qt')(ol,t' @)lr) : - e (oltp(x\ lz) , Thus QQt:e if this matrix element of the Dirac field does not vanish. 2. In (42.13) the singular contribution in the last term must into account, so that here one must integrate over the taken be a : nr2 aIs o. point

Saa

AA

Prnnarfiaq of the FunctiOns ili A2\ ^nd F.rrfi2,|

zls

Here a stands for all the particles present except the photon. The contribution of the last three-dimensional integral in (43.2) is a c -number and therefore the corresponding matrix element vanishes if la\ i q nnt the \/acrllm Affer takinn orrf tha .v. - d*a-n- a- n- d- -o -n. . c e , we can write (44 .5) as

(ol i,,la,k): Fu,@,k) ,

(44.6)

where F , , ( a , h:)#

[ A xe i h " @ ( - r )( o l [ 1 , ( o )i , @ ) ) l o ) .

\ z 1 r) - ' l

(aa.6a)

From the continuity equation for the current and from the vanishing matrix elements of the current commutator for of the appropriate s nacc -l i kc t . it fnl.lows th at

F u , @ , k ), : o

(44.7)

Equation (44.7) is clearly an expression of the gauge invariance of the theory even for virtual states. Together, the Eqs. (44.6) and (44.7) give the result that the absolute value of the matrix element (44.6) for scalar photons is equal to the corresponding quantity for longitudinal photons:

l ( o lt Ia , k , A :l ) l : l ( o li , l a , n , A : 4 > 1 .

(44.8)

The stated cancellation in (44.4) of the contributions of the scalar and longitudinal photons follows from (44.8). By similar arguments, this cancellation can readily be shown if more than one scalar photon is present. Thus we have shown that the function in (44.4) can be expressed as a finite sum of terms which il(p') are all posttive. A similar proof cannot be given for the functions Do(pz), since t h e o p e r a t o r / ( x ) i r , @ 1 . 1 9 ) i s n o t g a u g e i n v a r i a n t , and therefore a cancellation of longitudinal and scalar photons cannot be ex-

pecreo. Using the positive character of important identitv

0
II(pz) in (43 .31) gives the

(44.e)

The charge renormalization constant Z of quantum electrodynamics and one. From this, I:1 therefore lies betweerrl is a singular point of the theory, for ".ro in this case the highest derivatives formalIy cancel out of the equations of motion. Also, as is evident from the remarks following Eq. (43.17), the case Z:1 corresponds to an infinite charqe renormalization.

I. The inequalitV U4.9) was first shown by J. Schwinger (unp u b l i s h e d ) . S e ea l s o t h e a d d i t i o n a l r e f e r e n c e so f f o o t n o t e I , p . 2 l l .

Sec. 45

G. KA1l6n, Quantum Electrodynamics

216

The Phvsical Siqnificance of the Functions 11(22) and JI (22) . Connection with Previous Results Although the treatment of renormalization given in Sec. 41 is quite clear, that of the charge renormalization of Sec. 43 had a There is no obvious connection with the fairly abstract form. lowest approximation of perturbation theory, given in Sec. 29, In this section in an external field. for charge renormalization we wilt study the vacuum polarization by introducing a weak exAt the same time we shall obtain a ternal field into our system. deeper understanding of the physical significance of the functions 45.

il (p,) and If (p2). In the renormalized theory the interaction Hamiltonian external field ,4fiuss(r)and a system of quantized fields is

for an

( 4 s. I )

6E:*Idt*irQ)Ai",(x),

rrrhara u' /c\ 'lc fho ror19;1y16lizedcurrent. According to (43.7) , this so of the field operators, operator contains second derivatives that (45.i) is not a suitable term to add to our Lagrangian. Rather than (45.1), we shall use the expression ieNz -. 6 9 : . : ' ; l , p @ , y , , v @ ) f . 4 i , |* ( Lr )A * ( x ) i i " . ' ( ,* ) which differs formally only by a four-divergence. iauss /x') stand s for the external current, !u

In Eq . (45.2),

ii"*(,): - (r a,,- #ur.)ri*'(,). From (45 .2) , we obtain the following A

I

,

\

rt

\

/

\

|

e

equations /

\

'

(4s.2)

( 4 s. 3 )

of motion:

'

m ) v Q ) : i e vA ( r ) ' , p ( r1) - 6 n r p ( xa) i e vA/ i du c"q"/ -(-*\ ) ' tt -t.?\ )(t,r4- 5 . 4 ) ly # + ?zA-.(x\ ... \ ieNz .-, , , \r , 7l, J Ar @): - ? L ,p ? ),y,,t?)l + L (a A" ,( *) - i#;- i?;*( r ) ) .( 4s.s)

We are going to consider only the case where the external field is very weak and regular enough so that we can expand our results One can readily showl that the first two terms in in powers of it.

I. By the use of the so-called functional derivative, these equations can be written as 6tp(x\ ,\A

tv\

ffi:

- i @ ( x- x ' ) l i , @ ' ) , t p Q ) l ,

- i @ ( x - x ' ) l i " @ ' ) , A p ( r ) 7 l+. { a r , - d p n d ,6r()* - x ' ) . , _,

In a similar way,

i
(/') d iiluss

we can write Eq. (45.17) (io follow)

as

-n61) : - :t+ [ d,p etP{,-t)yIi@,)+ t n e(p)Ir(p')1. (2n)o J

This method of writing them has been suggested by several authors and has been used particularly by ]. Schwinger, Proc. Nat. Acad. Sci. U.S.A. 37, 432, 435 (1951). See also V. Fock, Phys. Z. sowjet. 6, 425 0934).

Sec. 45

Significance

such an expansion

of the Functions

il(pr)

217

and il(p,)

are

y ( x ) - Q ( ' ) - i I o ( x - x ' ) [ j , ( * ' ) , Q ( . . ) ] - 4 i " " ( x '+ ) d" x' ,' . 1 , , ( r ) A , , ( x ) i J O ( x r ' ) l j , ( * ' ) . A u ( / ) ] , 4 i("x" ' )d x '| | t | +' l . L' \ "(4u' " " - dl "t +' d' +"/'-)- /u , u $\ ' (- /x )lL " ' )

(4s.6) ( + s' 2 ] ,

- - - e r e $ ( r ) a n d A r ( x ) a r e t h e s o l u t i o n s o f ( 4 5 . 4 )a n d ( 4 5 . 5 )f o r v a n ::::ing external field, i.e., they are just those operatorsstudied -lra -rzinrrc - - - L r r s InJr a rsvrvu.

sc a L nv tLi nr n u ol l >

'

I vAV/ aE

nra-,a Pluvg

lA q \9J.

:--i.nEI them into the equations of motion :::i,rtion of (45 .6) into (45 .4) gives

A - /' l a n d

(Atr,

. ,7 \/

-h rv

-c *r r- lb. aL ^I +- r

(45.4) and (45.5). Sub-

' , ' ,; ', , = m ) t P ( x: ) i e Y A ( r )Q Q ) | 6 m Q ( r )- i I O (x* x') li,@'),i ey A(x)* (r) + 6m,]t(x)lx x A?"*(x')dx' - y, I, !' *' li, @'),tf (x)],4i""(r') .

[ ,n,.,,

I

-"rng (45.6) and (45.7) we can combine the fjrst two terms on the ::;::i side of (45.8) and the four-dimensional integral in the fol- :-,..-ingway: (Terms which are quadratic in ,4|u" (x) may be dropped. )

( ' ) * 6 m Q @ ).' Ai.v)Q - i , i O ( " - * ' ) L i , @ ' ) , ' ; ' e y A *( x( r)) + 6 m , l t ( x ) l A i " " ' ( x ' ) d x , ' :/ / 1 q : i t ^,,-t(x)rp(x) + 6my (x) - ; t *Z

o\

yu A.* @)y (x)'

-: -,'.-eeliminate the time derivatives of second order from the cur-=*' nncrator hv the rrqe of the enrrations nf mofion . fhe .osult :an be written as . v . - - v P u r 9 L v r v j l l r v 4 , L r r g l v

E,

--,t!, i- .\;

tieNz

\"

t)

-_,,

z A, .^, (*v,\1 \ ., \a , - b "u - L6pe! An(x),(4s.10) L P ( r ) ,n * Q ) ) + L

tpt:6pt-

L 6 * a 6 t 1 ',,

(45.11)

_ sing this result and the canonical commutation relations, we can -,..-crkout the commutator in the last term in (45.8) . In this way ','.-e!ind

y 4li,(x'),* (")f ,":,t - - ::3t^ r ^,Q@)6 (u - n') .

(45.r2)

S,:bstitution of (45.12) into (45.8) and application of (4S.9) gives ::-: right side of (45.4). ]n a similar way (45.7) can be verified. Substitution into (45.5) anc a transformation similar to (45.9) gives 1.

For another proof see R. E. Peierls, Proc. Roy. Soc. Lond.

) . ? r 4 ,t 4 3 ( 1 9 5 2 ) .

2tB

G . K l i 1 1 6 n ,Q u a n t u mE l e c t r o d y n a m i c s

Sec.45

zA,,(x)=-1i:! t'rli/tuq@*l qfn@,r,,t't,ll+ I d','lli,{r').o, "i

!!fflot""t' t)+|1 [ o1"- *'yli,c't, + li,<,t, ffi], ax'-yfr@r, d,,nd,n) x,4|"*(r') z.l'i",'(4- i=H#

I

r:r ras.

l

From (45.I0) it follows that

uniui';'L' tr'),u ,,,1 +,i, (' r' {f;,ix'r,e,@l 2:1"lol*"{r',1= __ t, .

,'

|

I O O' .r

l_

n:i:,,,, w \ l l ! ' |

l .rnr p duaE,qi".,(r)), itnr

224.t,,

)

x ' 't Il i' ," ,' ( r ' ) , 1 - J - 4 ] - l - i , { X u s s ( x ' \ d x ' 0xso)1|

.2

f

x ' 1^.\ 1 /^) 1 x 7 , A \ xi 'u' ) sd'x-' i -IJ : - L. \ - x ' 1LltiY, 1 :t t-,., O r ) JI O 6 \ '

I [lr. ., "t a''Jiu"''t.

r 44,

axh

'llt-"""'

Substitution of (45.14) and (45.15) into (45.13) and use of (45.7) gives the right side of (45.5). Thus the solution (45.6), (45.7) has been verified. As in Sec. 29, we shall now consider the vacuum expectation ValUe

Of the

rlsrna

t4\

l v . 4

nnar^i^r

CUffent

l4l

r / ,

Rrr diffaroniiatinr

(Atr, 7\ fr^'i^^

and

r^/a tind Y Y v

l J l

( . o l i , , ( xl 0) ) : - , iJ @ @- r ' ) ( 0 1l j , ( * ' )j,, ( r ) l l o ) x r X Aauss(x') d t' a ----:-7;".. (.r).

/4 (

16)

The first term in (45.16) contains the vacuum expectation value of t h e c u r r e n t c o m m u t a t o rf o r t h e u n p e r t u r b e dp r o b l e m , i . e . , e s s e n t i a l l y t h e f u n c t i o n I I ( p , ) o f ( 4 3 . 2 5 ) . B e c a u s eo f t h e @ - f u n c t i o n in (45.16), we do not obtain exactly the function II(p2) on gorng over to 1-space, but rather a linear combination of the functions I t ( p ' ) a n d 1 7t p ' t . See the calculation in Sec. 41, especially r q . ( 4 1 . 2 5 ) . 1 U p o n u s i n s ( 4 3 . 3 1 ) ,w e f i n d

( 0 1f , { i r l 0 , r ' , '

at'

o I dp s ' a 'y - 1 71 p z- )f l ( o l - in € t h 1 l ( p ' )i]1 " .(.p ) , ( 4 s . r 7 )

(x) ii*' @): I d* e-o,'"ifr""

(a\

17^\

T h e r e s u l t ( 4 5 . 1 7 ) i s a g e n e r a l i z a t i o no f E q s . ( 2 9 . 2 3 ) , ( 2 9 . 3 3 ) , and the functions n\0r(p2) and fi*r?r) introduced there are a first approximation in perturbation theory to the exact functions II(p2) a n d f i ( p 2 ) . J u s t a s i n S e c . 2 9 , ( 4 5 . 1 7 )c o r r e s p o n d st o a " d i e l e c t r i c constant" for the vacuum/

t ( p , ): 1 , i l \ p , )+ I I e ) - i n e ( p ) i l ( p \ ,

(4s.18)

T h i s f u n c t i o n € ( p r )i s n o r m a l i z e d t o o n e f o r p 2 : 6 b e c a u s e o f t h e l a s t t e r m i n ( 4 5 . 1 6 ) ,i . e . , b e c a u s e o f t h e c h a r g er e n o r m a l i z a t r o n . This means that the vacuum has a dielectric constant of unity for

Sec. 46

Charge Renormalization

for One-Electron

2I9

States

a photon. In this way we have shown the connection of the charge renormalization of Sec. 43 to the results of Sec. 29. At the same time we have found the desired interpretation of the Z-functions b y m e a n s o f E q . ( 4 5 . 1 8 ). As

was

alreadv

slvsv1

pntad

in

Sac

?.q ,

fL hr r aL

imrryinrrv

rrrrsvursr

I

n e r +L

y\!

^f

vr

the

dielectric constant corresponds to transfer of energy from the external field to the quantized fields, i.€., to the production of real particles. L i k e w i s e , t h e f u n c t i o n t r o ) ( p ' ? )w a s f i r s t i n t r o d u c e d i n Sec,24 in connection with the production of particles by an externa1 field. Here we can recall the calculation of Sec. 29, Eq. (29.38) and write the total energy given up by the external field as cuss/

r' 2,4 r\ u u: - I f i < o l

7 " ( .l o r );/ x : l r i u s s , 7 r 11 5 u s s ,_

:i;i'.ldPtf'tffn(r')' I

of A similar interpretation by considerations relating to an "external spinor field". sible, although it would be since "external spinor fields"

f

6,

I ,n,.,n, I

Zi(pz) couLd be given the functions the "polarization" of the vacuum by No doubt such a discussion is posof only limited interest physically, are not available in nature.

Charge Renormalization for One-Electron States The discussion of Sec. 45 has shown the relation between the charge renormalization and the problem of the vacuum polarization In this connection there is another problem in an external field. which is of major interest: the question of the charge of a state with one electron. It is not immediately evident whether the procedure given so far will yield the correct value for the charge of are necessary such a state or whether different renormalizations In Sec. 34 we have seen for an external field and for an electron. that in the lowest approximation of perturbation theory, the same solves both probiems and now we are going to renormalization generalize (34.13) so that this result does not depend upon per46.

+rrrhafinn

fLhrar nv rv rr z

I

.

In order to do this , we first take the matrix element of the curstates. We can do this rent operator between two one-electron by the same methods which we have already used several times before in working out matrix elements of various operators between states. We first compute the comthe vacuum and one-particle Using the complex mutator of the current and the (Dirac) in-field. c o n j u g a t e o f E q . ( 4 1. 7 ) , w e f i n d

i _ i r ! ) , ! ' ((o' ')) l : - N J o l x - x " ' ) s ( x , ' - x " ' ) i , , ( x \ . 1 ( x -" ' l l d . r " ' - l , / 4 6 .I ) - t N / S ( x ' - x " ' ) y 4 l i t , ( x \ . t 1 ' t (dxt"x' 1" 'f. [' zo *i": The last term can be simplified with the aid of (45.12):

s (l x) ytv (x) (46.2) )l=-rvI o tx 3)5 ( l J)[7rtr),1{rtJo r"'- J!- E,,,, [7u{x),,1,'o)(t

220

G. Kiill6n, Quantum Electrodynamics

Sec.46

a simplified notation for the different Here we have introduced x -coordlnates, which requires no further explanation. As the next step, we now compute the anticommutator of (46.2) with ,
to us,

l0): 4o1{lir@),vto)(r)1,'r1,to)(2)} - N : J o l x t ) s ( I 3 )d x " ' ( : | \ 1 u s, ,( t x )y , s ( x z ) (l)}I0)) (2)}ll0) - (0 | {Liut*1, x ((oI tl, (x),{l $t,,t'$) v(o'tzt1,l

l l(46'3) I l

The first term in the last parenthesis.can be rewritten as

u $),,p@(2)j=NI @0 4)u 0, 1@)\s (42)6*w- !-li ey A0) + 6/4 s(12)(46.4) by the use of ( 4 1 . 8 ). I t f o l l o w s t h a t

(0)(2))l (olk,(r),{lt)t,,t' l0):A/IO(J4)dxrv <01[lu('), ii tlr,/rqr]lto)xl X

! ( 4 6s. )

s ( 4 2 )#+r ^ t ( ) 2 ) < } t l i , @ ) , A ^ ( 1 ) l 1 0 ) I

The other term i n ( 4 6 . 3 ) c a n b e t r e a t e d i n a s i m i l a r f a s h i o n :

x ,1@ll li,@)'1 t6 )e)1:I,II@(x4)L,i,@) [ x S(42)dr"+ , ' ! r r { 4 y ^ S ( x z l €I1 ,

,nu.u,

and

N l @ ( x )S ) ( t | )d x " ' ( o l{ , p ( * ) , / ( r1) }0 ): /^\/,\ , : -(u' ^ , | I - ' \x\), 1t)*, \t ) r i{J s t r )) yort0) d3x"'J\ ' 'l0): t s 1r *; I:Jl v \y) /"'xo

1 . . ^_ . t+o'zt I

-l

Collecting our results, we obtain

(o | { tr) (r), eto)(1)l, rpto)(z)}| oS: 2

[r + 2 (N-{ )] €,,t S (1x) y, s (x2)-

- e I O ( x ) )S ( t ) ) y t S ( ) z d) x " ' ( o l l i , @ )A, t \ ] l o ) - N , I I d , x " ' d x rS Y ( t \ ( @ ( 4 ) @ 9 4 ) < o l l i , @ ) , { f t T , l r +tro})l - @(, )) o (x4\
(46.8)

For further calculation we express the commutator of the current and the potential in (46.8) bV means of the functions If (p2) and II(pr) . clearly we have

(ottir@)'A^(,)tto):t#!f ( 4 6 .e ) ,:):';:',,:;'^1.')^'li'|"lrof and therefore

sec. 46

charqe Renormarization for one-Electron states

: o ( x ) )( o l l i p ( x )A, ^ ( ) ) l l o > :-(zn)n "' J "v'

z2r

l

a2@(x3) (46'I0) r ' I ( P 2 ) { i n t ( i l n ( p \ l - , , r A (\ r aJ ,t \ ? x u a x 1I, J

: @(") eip, e(!)ryJ OrfI oO

( 46 . l 0 a )

Ine new function @(r) has the following properties:

+:l oxo

l:erefore

it follows

@ ( x ) 1 , " : oo: ,

(a6 . tta)

^:-;np1ap1.

(46 .lrb)

lxo:O

that

- i n @)6 u+6t06( x) ) ,( 46.r Lc) o ? , ' : : , ' ; + : u *l ' u r^@Qt) @@))) =:i

',','e obtain

- : . 1 o ( . x )s) ( t ) ) n s ( ) z ) d x " ' ( o l l i , ( * ) , A / ) ) ) l:o ) I r r p ('| 3r = z_', .l dPJ dx"' ei S(1)) y, S(t 2)Vi (pr)* i n e(f) II (pr))+ft+o.rzl -i6rt I ,t , S(tx)ynS(xz) :::

ihe expression

=:-anirl

under the double integral in (46. B) we introduce

h^+^+i^h.

I, l @( x f i o 0 4)
::-'using

t z n t Js . l ( 4 6 . I 2 ) a n d ( 4 6 . 1 3 )w e o b t a i n f r o m ( 4 6 . g ) , , ' . , " ' . , . I -n

I I

tao.rSl

J

.' - q'):(qI ilPIq'>lt

-i ne(Q) n e\ +, +=;l *I *u.'n, Ql a fr@) 'J -ie(Sl,ynlo)A,(q,q')(olrp$)lq,),

Q:q,_q. -_::s is the desired expression for our matrix element. -.'"-::biain for the charge of the state l?) ,

ktQtD:,[r

(a6.iaa) From it,

+ t) _ ; +u(q)Ao(s,d,,(q)].(46.rs)

l : - c r i e r t o c o m p l e t e t h e p r o o f, w e c o m p u t e . 4 " ( p , p , ) f r o m ( 4 6 . 1 3 ) :

t , P . P ' ) - 2 ( N - D 1 7 r e ) r n - ' ! ' ' , . d x r v e i p ( t t ) + i px' r a , ) II I

x ( o ( 4 1 o ( J 4 ) < o t { l t 0 \ , ; ; @ D+, i ( + ) } r o ) [tl l)t l

Sec. 46 G . K a t l 5 n , Q u a n t u mE l e c t r o d y n a m i c s ,p:,p' here, we encounter very singular expressions. If we put Because of this, we shall now set only the spatial components of the two vectors equal . It then follows that 222

An(P,f')p:p,: - 2(N - {)fi(o)Yn+ I tt p : ' ) ( r o - \ ;' l"at ' l c a*t | t( ^4,6 . 1 7 ) + e f d x-nJI d * ' u e - i l p o " .t |

x (o(x\ o 3 4)
By the use of the canonical commutation relations and the fact that the charge 0 is time independent,r we can easily show

I d ' * t i J * ) , 1 ( ) ) l :- e l ( ) )

lQ,l0l: -i

(46.18a)

and

, rc'fg)): el(4)

(46. r8b)

so that An(f, f')p:p, : - 2(N - t) II (o)y4+ +N',i

d . x o I d , x r ve - i l p " - p i ) @ "z [ " ) + t b ' \ t t v)

x ( @ ( x \@ ( t ++) o ( x + ) @ @ )( )0)1{ l ( j ) I, ( + ) t}o )

]*u.',,

The time integration can now be done easily and, after introducing t h e X - f u n c t i o n s o f ( 4 1 . 1 9 ) ,w e o b t a i n /o|,/)p-p':

-2(N - {)n@)Yn-

| (, n4,62' 0 )

|

f" - - p -'L l l l 4 - t p ' 1z A ( h l , | +' N ,IJ pP- o - P^o, T i n 6 ( o with

- 7 \ r t Lti. , "n "e(p) rR,,4r*t Z,.(fr) L\r t I \rt _ \ r ) _ -lr1pz1

l*i n e(p)t,(p\) + (iyh -l m)lE,(pz)

L I

( q o .z o a )

)

Equation (46.20) can be written in a formally invariant form as

(p- p')A(p, p'): - 2(N- 1)II (o)y lp- p')-i N'lzo (p)- t* (p')1.u6.2r) The derivation of (46.2L)is valid only if the vector p-p' is timelike. In (46.20) we can now Iet p'sapproach po and we obtain I. From the time independence of 0 one might conclude that the charge of the physical electron has to be the same as that of and hence that the calculation of this section is the in-electron, unnecessary. It must be remembered that the charge need not be In the a constant of the motion during the adiabatic switching. calculations of this section alt that is used is the time independence of the charge for finite time intervals.

Sec. 47

Proof that a Renormalization

Constant is Infinite

a@)An@,q)w(q)-- - 2 (N - 1)fr(o) + t{,w(q)F (q)r(q) ,

223 tAA

cc\

\.to.

LJ)

F (s)- llm ,t 1f,1 rn2* 21or)- t, (- *,) r yar Er(- *,)) : e-u

c

: ln Er( . mr)+ zg,E'r(- m21 . Because u ( q )q o u ( i l : m w ( q )y n w ( q )- n 1, Eq.

(4b.zt)

glves

, ( q ) A n @ , q t r @ ) : - 2 ( -r / r )( i 7 1 0+) r )

- ri_:

, (46.2s)

w h e r e ( 4 2 . I 3 ) h a s b e e n u s e d . f r o m t h i s a n d ( 4 6.. ff, su l i ,L fr r*s vcr rha a r! cj s ^ v^r r the electron

is

klQ\q):e.

(46.26)

This shows that ow definition of the charge renormalization not only normalizes the dieLectric constant of the vacuum for a photon to be one, but that it also makes the charge of the one-electron state equal to e.r 47.

Proof that the Theory Contains At Least One Infinite euantity Up to now our discussion has not been concerned with possible infinite values for the renormalization constants . In Chap. VI we saw that the renormalization constants were infinite in several examples using lowest order perturbation theory. Despite this, all the (renormalized) observable quantities were finite and in good agreement with the experimental measurements. In principle we can go to higher orders of perturbation theory and hence develop in powers of e 1y1. renormalization formalism siven here . One can show that we would obtain2 finite results for all observable quantities in every order of perturbation theory. If one could also show that by using this procedure the serieswe would obtain is convergent, then we would have a satisfactory theory--at least satisfactory for the observable quantities. Unfortunately, up ro now, it has not been possible to give an adequate discussion of the convergence ofthis series. lnvestigations using certain srmptified models of a quantum field theory3 have shown that the per-

1. One can also showthat a state with z electrons has a c h a r g e n . e . s e e E . K a r l s o n , P r o c . R o y . s o c . L o n d. A 2 3 0 , 3 8 2

0ess).

2. Dyson and coworkers have proved this using methods somewhat different from those developedhere. See F. J. Dyson, phys. R e v . 7 5 , 1 7 3 6 ( 1 9 4 9 ) ;8 2 , 4 2 8 ( 1 9 s 1 ) ;8 3 , 6 0 8 ( 1 9 5 I ) ;A . S a l a m , P h y s . R e v . 8 2 , 2 I 7 ( 1 9 5 1 ) J; . C . W a r d , p r o c . p h y s . S o c . L o n d . A64, 54 (1951); 3 . C . A . H u r s t , P r o c . C a m b r i d g ep h i l . S o c . 4 8 , 6 2 3 ( 1 9 5 2 ) ; W . T h i r r i n g , H e l v . P h y s . A c t a 2 6 , 3 3 ( 1 9 5 3 ) ;A . p e t e r m a n n , p h y s . R e v . 8 9 , I i 6 0 ( 1 9 5 3 ) ;A . P e t e r m a n nA , rch. Sci. Phys. Nat. 6, S (1953); R . U t i y a m a a n d T . I m a m u r a , P r o g r .T h e o r .p h y s . 9 , 4 3 I ( 1 9 S 3 ) .

224

G. K5115n,Quantum Electrodynamics

Sec. 47

for these turbation series diverges (even after renormalization) One cannot rule out the possibility that the perturbation models. We shall not also diverges. series of quantum electrodynamics discuss these calculations further, except to note that even if we were able to prove the divergence of the perturbation series, not very much would be settled. Indeed, it is quite possible that there is a solution with finite renormalization constants, but one which has such a form that the renormalization constants would appear It is therefore as formal power series with infinite coefficients. of some interest to discuss these questions without using perturbation theory. We shall show in this section that the last mentioned possibi.lity That is, we shail show that if there is any can be disregarded. solution at aII of the equations given here, this solution contains at least one infinite renormalization constant. In fact, it will turn out that among the constants which are infinite is either N 1 or (l*L\-1 or both. In order to show this,we start with the assumpThis implies certhere is a completely finite solution. that tion II(p2) and 2o(p2) for tain assumptions about the weight functions With these assumptions, we shall then large vaiues of. -p2. and so our asshow that the formalism contains a contradiction, p r o v e d . i s t h e n sertion II(p2) which The principal tool in our method is the function was defined ln(43.24), tn Sec.44 it was shown that this function could be written as a sum of a finite number of positive terms, (43 .24) contains certain terms with even though the definition nooaiirre sions- We therefore obtain a lower bound for the function r r v Y s u r v v

n(pr) write

it we neglect a few terms in (43.24). In particular, we can

n@') > --

n,.|::oli,lq'

q')(q" qli'lo)'

( 4 7. r )

The state lq,q') is a state with an in-pair and the matrjx elements of the current operatorwhich enter (47.1) can be taken from the calculation of Sec. 46. From Eqs. (46.8), (46.I2) and (46.13) it foliows that

q'):(oli't'lt,t'>lt- 11Q')+ n @)-irn Q')-zLi *I*r., A ,q ( *' , q ) ( o l ' p @ ),l q )

) ( 4 7, 2 a )

Q:s-lq'. B y u s i n g c o n s i d e r a t i o n s o f invariance, be written in the following f o r m :

Au@',t)- I

the function

/1,,Q'. q) can

(iyq'lm)a

( 4 7. 3 )

o ,a / : 0 , 1

x l y , F n ' n ( q ' , q )* q ' , , G a( oq ' , q )+ q p H a ' n ( q ' , d l ( i yq * m ) n . Here the functions

F , G , and 11 are invariant

and are independ-

Sec. 47

Proof that a Renormalization Constant Is Infinite

zTs

ent of the 7-matrices. From the charge symmetry of the theory or from the explicit (46.13) it follows that the function definition Ar(S',q) must have a certain symmetry in q and, q,. The following relation can easily be shown by using the method of Sec. 41, Eq. ( I.2t): [See alsoEq. (Ia.6) forthe Z-matrices.]

- - C-lA,(* q,Atr@',q) s')C.

(47.4)

From (47.4) it foliows that the functjons F, G , and .I1 ot (a7.3) satisfy F a ' n( q " d : n n " r _ q , _ q , ) , G a '@ e , q ) : 1 7 e e( _' s , _ q , ) .

( 4 7. 5 )

( 4 7. 6 ) ( 4 7 . 3 ) i s If s u b s t i t u t e d i n t o ( 4 7. 2 ) o r ( 4 6 . 1 4 ) , o n l y t e r m s w i t h g:q':0 contribute, since the others vanish by the equation of motion for the in-field. Since g2-q,z:-mz, we have

kli, lq ' ) : ( q l i ' f ' l q ' >x

I

' I .x' l1t -. f r t O , l + 1 7 1 o.i\'n_ Lrr(\ ov \) nt t(wo -2),\ zr r N - t ' l 1 _ L 1 I ? ( Q \ ti n e ( Q ) R ( Q \ l - l t n z . z l

-

cll

t i 'n) e s (t ( J , ) ] , rt*(ruts:,)(ql'!rc)10)(01?(0)lq [ S( et O

I

with

( 4 7. 7 a )

Q:q'-q, and

i'y'i!I:*,:":;::::,(Q,) +,{-tj+R(0,) + ir(R(Q,)1-l*, , ,l -

ti )tq? ns(Q1,)1, r* r,t , - S L ) Q l r t ' atl t,)(o1 rtor1S

I

with

(47,8a)

Q:q*q'.

There are the following relations between the new functionsR(Qr), Ft?l , S ( Q r ) , S = ( 0 r ), a n d t h e o l d f u n c t i o n s F , G , H :

n-(?1 + i n e(Q)R(?') : Fuo(q,q'), -

: (q,q') : Goo (- q',- q), *, li(a\ + i n e(Q)S(Qz)f Hoo

for Q:q'-q,

q2:q'2:-m2

and

e ( g ): u ( q , ) : t .

|

I

*,.,,

Equations (47.9) give the most general form which the right side can have on the basis of invariance. From the reality properties of the current operator, it then follows that the four new functions must be real. We now returnto (46.13). Fjrst we note that this equation has a structure similar to Eq. (45.16), i.€., it contains a vacuum expectation value multipiied by @-functions. If we define a func-

G . K i i t l 5 n , Q u a n t u mE l e c t r o d y n a m i c s

226 tion

Fu(P, P') l>v

( 0 1{y 0), i" (')l,l-(4)} 10 ) .:

a

Sec. 47

r r

n dp' eino,)t i p"'4'9, (p,p'\,(47.r0) 1r,,j.J J e

then we can show as in Sec. 44 that this definition (47.10) contains a sum over onlv a finite number of intermediate states. If the re,Tr(f ,1') is finite. normalized operators exist, then the function With the aid of the integral representation

@ ( x ) ) : t2+n t

dr -

t

T -

_J

?e

(47.rr)

r t ' t " z,) "

we then obtain

I

o (x\ (ot{lt (r, i,@)), I(4)}|0): : with

+@ f

dr

, f d I '-t - t € r uP( p ,r

9 , , ( t ' ,b ' \-

, 7 r ,p ' ) : - l _ J x

-@

lTere

r r e r u

.' /f l i s

an

1c\

dx

F u ( p .x ; p ' ) . ( 4 7 . r z a )

.lpo

t€l

wifh

a ynvnv 4qLi 4t iv rv r c

-@

fime-1ikA

a r b' ri Lt rr uar rJ r z

nv vnr :nl rar Ln' 't* t ' . " ) T n a c i m i l a r

lAa

,\ 2 " 'n- ), ',. [l J[ O P , P ' e i P \ 3 ) ) t i P ' t t a t f i u Q , PJ' ) ,

-j-o

=,,

I

Way we

\zAcfnr

Obtain frOm the

fime

com-

secOnd @-funCtion,

6 t ( x \ @ ( i 4 ) ( 0 t { yi0- @ ), )),t(4)}to>:

lI r n r . r r o ,

t = [ [ a b d b , -e i p t s ) )t ,p ' , , ng, ( b , 'b ' 'It.)l " tzn)",J'-r-'r "p\('r/

-"

f

gtn.O'l-

if

I

. g ( h - t " ' r' l

t_it,"p'r

J

"r

@ 7. I Z c )

h ' - n''tr \"

r"gift the functions F , G , H in (47.3) as a In this way we " u r r Hilbert transform of the finite function gr(P, p') typq of generalized in (47.10)plus a similar contribution from the second term in (46.13). Tho real

narf nf fhcqc y s

L

v !

u r r v

r v

evnrpssions v J \ F r

v

oirzes the

functions

R

and

S

in (47.9),while the imaginary part givesthe functions R and S . ISee the similar calculationin Sec. 29, especiatly Eqs. (29.27) throush (29.32). l F r o m ( 4 6 . 2 5 )w e a l s o h a v e

-.' (' (, ,t,lpi ,, ',/?l , l i tl l z + R '(.01 ) l -Lmr-,r, <-n l , p ( o ) r 0 ) ( 0 r ? ( 0 :) l q ) SI ( 0 ) t_L -

I

lU7.I3) I

: k,l, i1 'nf\ ,' .l1t^>N -ll z ' ,, -' ; + R ( o-) s ( o ) l : o , J or

(47.r4) At this point we must recall that the functions R(p') and S(p') a r e d e f i n e d b y m e a n s o f ( 4 7 . 9 ) , ( 4 7 . 3 ) a n d ( 4 6 . 1 3 )i n t e r m s o f t h e operatorsl(x)of (41.2). These operatorscontain singular expressions of the type (42.18). We must therefore expect corresponding

Sec. 47

Proof that a Renormalization Constant Is lnfinite

227

slngular piecesl of R and S. The calculations ofSec. 46 allow us to isolate these singular pieces in a simple way. The argument that leads from (46.16) to (46.25) was originally carried through for l(x), but it can obviouslybe carried through for just the regular part of l(*). If R."e(p,) and S*c(p2) are the regular parts of the functions R(22) and S(pr) , then we conclude that in addition to (47 .14) we must also have R - " f 3 ) - S - * e ( 0, ) :

'i - \'! r.',' , ' t J " *( - ,w, 21)\|+- , ,r rr n -,r(-*')):1-_'t'(47.15) 1_ 2\-z \

On the other hand, the result (46.26) is independent of the special treatment of the singular terms, and by a transformation similar to that of (33 .2) through (33.6) we can rewrite (47 .8) as

( o j ; i q . s ' > : ( 0 1 ; ; o t t q . o ' ) lIr n Q , ) n , " ' ( Q , r) 5 . ' s 1 9 2 ; I - i.n (II (Q\ - R*c(02)1 S'"s(Q'z))l * l(47.16) + i Q , < o l * l 9 l S , S [' )S ' * ( 0 ' )- i n S ' " s ( Q \ 1 . ) So far we have not madeuse of our assumptionthat the renormalization constants are to be finite and Eq . (47 .16) has been derived using only general assumptions. Now we shail take explicit ac1 - N' the fact that count of and -1=hu.r" been assumed to be I L I- L finite. This means convergence for the integral (43.30) for the function ,F1O; ar well as for the integrals which enter the definition (47.9) according to (47.12) . With this assumption we find

l i m n ( Q ' ) : n @ - ltm -a'** -O'+o

' I

4:*:ff(o)

J 0 \ t+

0

- II(o):o U7.r7)

E)

and irm

Y"\P.f'):

lfo- fi,)- a

:

drdt.' ,. -rrlT** f f ' JJ F="V=;r'e"(P' Po r' : p''l',or') lr"

lim f f l t o p 6 l - r cJ J l . Y y

( 4 7. 1 8 )

dx dyFotp. ); p,, y) (Po p;- ir))ls'- tP6- ie'l)

I. In an earlier work these termswere overlooked. tG. t<5115n, Dan. Mat. Fys. Medd. 27, No. 12 (1953).1 This woutd give a f a c t o r o f 2 N - I i n ( 4 7 . 1 9 )r a t h e r t h a n a f a c t o r o f N z .

Sec.47

G. K6ril6n, Quantum Electrodynamics

228

ff -qz is very large, we therefore obtainl from (47.16),

(47.rs)

- #' > t i o l i 1 ! ) l q , q ' ) '
in the following Equation (47.I9) can be interpreted physically way: Tf. - qz is very large, that is, if the kinetic energy of one particle is very large inthe rest frame of the other, then the interaction of the particles does not play an essential role, but rather the current is practically the current of two free particles. regard a "physical" particle as a mixture We shall , momentarily, can p a r t i c l e s zero". T h e n t h e f a c t o r 1 9 2 1 1- t ) { a t t i m e of "free be understood as an expression for the amplitude that at very large energies the two particles can penetrate the "clouds of virtual photons and pairs" and interact with each other as "bare" partifield cles. If we introduce the unrenormalized electromagnetic tntt-\-lt

r -r

u

4 Iv\

1Lp\'\.t

,the unrenormalizedDirac fieId rp,,U):Ny(x), e

and the "bare" charge €$: ,;-, - L , then we can write (47.19)as 11 follows: ( (oJ lim I Ql Ai' @)| q, q') : i eo
-0''o

Equation (a7.lga) is identical to the first approximation to the equations of motion of the unrenormalized fields. We substitute thls result into the inequality (47.I) and find

l*"'

e',,rtili)to> :"'1 n@\= ($)'_l13*J* ,l:rrotil?)ts,q') : !''"0(+)' : (t5)'-$g rr@) (P2)

l . I n ( 4 7 . 1 9 )i t h a s a l s o b e e n a s s u m e dt h a t t h e w e i g h t f u n c t i o n s [tite nrp")f vanish for large values of. pz. The necessary assumption is only that integrals like (43.30) converge and this is a weaker condition. As an example, consider the integral f ::lYo., Ja 0

For the moment, if we denote F: l?res-sres-I/, then from(47.20) we see that the essential point is the divergence of the integral a

f dalI

.

t:t

"

.l

N2

-Lt

F

-tnfl.

a lr-L

-l'

I

0

Clearly @ @

/N2'r2f

J>\r_r)

da

J o-2T-

N2

LJ

fllO.

&

F o r l a r g e v a l u e s o f a , F v a n i s h e s a c c o r d i n g t o ( 4 7 . 1 7 )a n d ( 4 7 . L 9 ) and the integral / is divergent because of the first term. This is sufficient for the proof and the result (47.19) is to be understood in this sense.

Sec. 48

Concluding Remarks

229

We have therefore proved that if iy'-1 and (1 - L\-L are both finite, then for large values of -p2, the function n(p\ is larger than the positive

J

. 1 Y' 1' . cJ o n s e q u e n t l y t h e i n t e g r a l 12nz\,1_L) (43.30) cannot converge--in contradiction to our assumption about Z . We must therefore conclude that either ly' I or the constant 1 or both are infinite, i.e., that at least one of the renorU -L) malization constants is infinite. It should be emphasized that this result has been obtained without the use of perturbation theory. number

48. Concluding Remarks We have now come to the end of our discussion of the general theory of renormalization. In retrospect, we have succeeded in isolating the renormalization constants present in the theory and in writing them as integrals over certain weight functions. In addition, we have shown that even in a theory with infinite renormalization constants these weight functlons must be finite if only the matrix elements of the renormalized operators are finite quantities. In sec. 45 it has also been shownthat integrals over the same weight functions, but with an additional factor involvrnq a in the denominator, arise as observable quantities. for fSee, example, Eq. (45.17) , where the function

n61-fr1o): b,oil:g,-A a (a _..pz,

. 6l enters as an observable quantity. ] ff or-t,theory actually has physically usable solutions, we must therefore require that the observable integrals such as @

f II(-

I

.nn\/erfitr

F\/Fn if in1q916lg

a\ da

o' .l o11l
(48.1)

@

f II(- a) da a. .J 0

(48.2)

a:e divergent. No one has yet succeeded in giving a proof for the assertion g_A_rt). ffl" fu"t.-tn.t th" i"t"grals converge in every ap-Droximation of perturbation theory is of no great significance in ::i.s connecti.on--at Ieast as long as the convergence or divergence c- the perturbation series has not been considered. One may ask -,,,'hether it is not possible to generalize the argument of Sec. 47 sc ihat one would have to assume only the convergence of (4g.1), :aiher than that of (48.2), in order to be able to obtain asvmptotic conditions like (47.19) for certain matrix elements. Consequently, bl' considering several states in (43 .24) one might finally obtain a sharper bound for the weight functions than (47 .20). With this :-ew bound, if a contradiction to (48.1) resulted, then this would show that the theory had no physically useful solution at all. The :etails of such a program are quite complicated, and up tothe

Sec. 48

G . K a 1 1 5 n ,Q u a n t u m E l e c t r o d y n a m i c s

230

present it has not been possible to givel substantial results. Tn this connecti^on, it should be noted that T. D. Lee has recently constructedz an interesting model of a renormalizable field rhonrrz. Althorroh the model is non-relativistic, it is significant because it containsJ a renormalization of the coupling constants as well as a renormalization of the mass and yet is exactly solvable, in part. A detailed study of this model has shown+ that the does not have a sensible solution obtainecl after renormalization L r r v v r

f

.

e .

r $ e + r v s Y ! r

_

oc ynEaU ^L fl ur rr rrrm ,

srrsr9y

h l urLr f

fhaf

'rnhnqt

a

vrrvrL

q f L vt c r r rLsa

( " ny saut rhl vol vl oY rovi r - : a 1 \

state")

of negative probability appears in the theory. It is quite possible fh^t qnncthincr sirrilali s nre.sent in ofher theories with infinite charge renormalization, i.e., possibly in quantum electrodynamics. We must recall that in our discussion we made certain assumptions about the mass spectrum of the theory. ISee the remarks following Therefore our arguments are validonly for solutions Eq. (aI.l2).] ^ r *L lLl I >r ^ hl .J ri r-q r . rl l -l ^p q- - +L ri ucu r. Lrql, a r i f h a s l - r o e ns h O W ni n S e C . 4 7 t h a t t h e r e uI is no solution with a physicaliy sensible mass spectrum which has In no way can we exclude all renormalization constants finite. that there exist other unphysical solutions with the possibility rfr

rY

v

rrrr{

! \4

more arbitrary ProPerties. One can certainly say that no matter how interesting they are in a fundamental sense, the unresolved questions of quantum electrodynamics discussed here are of less significance practically. This is so because in quantum electrodynamics we already have a theory which enables us to compute observable quantities with nreaf nrecision and in .nmnarc them with experimental results. 9l

yru

sqL

Lv

vrp

!

vvrrry\4

lndeed, it is just this excellent agreement of the perturbation theory of quantum electrodynamics and the observations which hints that rL h n formalism yt ro u ve r rn L tv s I r Eo n l o f cL e P.Ls s vOf nI e

fuhr r ce n vr ]r zJ

and

vrrv

tuhr r as Lf

miahf

qane cu sar n h qrd s Jv tLhr rov r ra v .lLv1 s aa r

nrocanf

form

he fhc


tha

limit nraccnl

of some future theofy

will

be

more comfewafding

thinrr ahorrt meson theoriesb

in their

I

see G. Kiill5n, Proc. cERNSymposium, ffin, ( 1 9 5 6 ) . I B 7 2 , Geneva 2. T. D. Lee, Phys. Rev.95,1329 (1954). 3. The other solvable models which are known (e.9., the exa m p l e w o r k e d o u t i n S e c . i l ) s o m e t i m e sc o n t a i n a m a s s r e n o r m a l i zation but no charge renormalization. 4. G. Kill5n andW. Pauli, Dan. Mat. Fys. Medd. 30, No. 7

(ress). 5.

In this connection see Vol. XLilI of this handbook. (Hg!e

buch der Physik, edited by S. Fltgge, Springer-Verlag, Heidelberg.)

INDEX

-ldiabatic switching , 52 ,54,77 ,

152,r97 -:-:iabatic theorem ,52 ,54 ,I9B 29 ,li-,'anced function, -Lrjular momentum, conservation law of field theory,14 operator, 23 -bnihilation -l*:oi:alous magnetic moment of +L ht ta€

alg ou nu f r n nv I t l(r g r tJ J

-l.nticommutator,

59

3 r e m s s t r a h l u n q , 1 1 6- 1 2 0 3anonical commutation relations nf field ihenrv. 7 3anonically conjugate operators , 7 ,11 3anonically conjugate operators in Dirac theory, 57 Sanonical momentum,6 lanonical quantization, 7 lanonical transformation, time C e p e n d e n t ,B 9 - 9 3 C a u s a l i t y , 1 0 3- 1 0 6 , 1 36 ^haroe

conirraatinn

$J

3harge renormalization, J.35 , 2 0 7- 2 1 2 3aarge renormalization of the one-electron state,I57 -162, 219-223 . 3 1 o s e dl o o p , 1 0 1 3 c m m u t a t i o nr e l a t i o n s , c a n o n i c a l , nf ficld ihpnrrz 7 Ccmmutation relations, f nrmrr

l:+i

nn

Commutation relations ^a\mn6nanl'c

covariant

? A

of current

AA

Commutation relations of the electromagnetic field components , lB C ompton e ffe ct , 112-116 , il 9

Conservation law of angular momentum in field the ory , 14 , I5 Conservation law of enerqy and momentum in field theory, 8 , 9 Creation operator,23 Current operator, radiative corre ctions ,I44-I50 Delbriick scatterjng of photons,l93 D-function,27 D-function, advanced, 2 9 D-function, retarded, 29 Dirac equation for free particles,55 Dirac equation with external ele ctromagnetic field , 69-7 4 Dirac field, quantized,5B Dirac matrices,55 Divergence difficulties, 77 Electrodynamics in vacuo, 16 Electron, anomalous magn e t i c m o m e n t , 1 5 4- 1 5 7 Ele ctron-eIe ctron s cattering, I22-125,I91 Electron scattering by a Coulomb fieId,107-I10 Electron scattering by protons, 166-167 Energy conservation law of field theory,B Energy-momentum density, 15 Exchange effect in scattering , I2 5 Feynman diagrams (graphs) , 9B-103 Fine structure, positronium,lS 6

G. Kii116n,Quantum Electrodynamics

232

Free electromagnetic field, 16 Fraa

narlinlac

l9A

y - t e + v r v v , 4 v v

r r v v

Functional

derivative,

216

Gauge in vacuum polarization, 1aA

Gauge transformations, Graphs (Feynman),98

42

Hamiltonian density,3 , 9 Hamiltonian of a fieId,T Hamiltonian of the Dirac theory,57 Heisenberg picture ,3 Heisenberg picture, perturbation calculation using, 79-85 Hiibert space of free photons,20 Hilbert space of different pictures, 3-6 Hole theory,60 Hyperfine structure of hydrogen, 167 -174 Incoming particles,197 In fields (incoming fields),47, 85,197 I n d e f i n i t e m e t r i c , 3 8 - 4 3 , 4 7 , 78 I n f r a r e d p r o b l e m , 1 4 8 , 1 6 5 ,2 1 3 Interaction of Dirac field and radiation fieId,75 Interaction picture ,4 Interaction picture for natural line width ,12 7 Klein- Nishina formula, lI5 Klein-Nishina, radiative corrections ,190 T : a r a ns rnr vi a nr \ . r . , lsYr

A v

Lagrangian for interaction of radiation and eIectrons,T5 Lagrangian of Dirac theory,57 Lagrangian of electromagnetic fie1d,16 Tamf.r

ehiff

171-19.1

Lifetime of positronium,lSS Line width , natural , 125 -l3I Lorentz condition, 32, 38 Lorentz invariance, l0 -I5

Magnetic moment, anomalous , of the electron,154 Mass renormalization, 150-154, l9 8-2 03 Measurement of field strengths, 43-46 Metric operator,3 9 YldIIer s cattering , 124 Momentum conservation law of fiald

fhanrrr

NTnrm:l

R

q . r v v + J , v

l r v r v

n r nvdv rs rv nL ,+

Q 1r

v

yr

A v rnrag- a l agart sr nu hL r

cfafa

,

vn rhrasrrnYav

]r vo -

normalization,I57 -162, 2r9-223

Operators, canonically conjugate,7,11 O u t g o i n gf i e l d s , 4 8 , B 5, 8 6 Pair production by an external field,110-I12 Pair production by photons,l20-i2I D rrf

i nl a

nr rmharc

u

,

d a f ir r r rnr L i! vf r r i, n n

vv

l 9 v( ]v

Pauli spin matrices,S5 Perturbation calculations in Heisenberg picture, T9-85 Photons as a consequence of wave quantization,23 Photons scattered by Photons , 193 P h y s i c a l p a r t i c l e s , 1 97 Pictures in Hilbert space,3-6 Plane waves as solutions of the T.)ira. Anraf i nn .5 6 Polarization, longj.tudinal, 19 Polarization of radiation,IB-26 Polariz ation, scalar, 19 Polarization,vector (for photons), ur r sv

vY ssLrv..,

18,I9 Positronium,182-190 Positronium, fine structure, I86 Positronium, mean 1ife,l88 P-sirmbot for time ordering,9I Radiative corrections of the KIein-Nishina formula, 190 Regulariz ation ,I42 R e n o r m a l i z a t i o n o f c h a r g e , 1 3 2 - 1 39 ,

157-162,207-2r2,2r9-223 of mass,150-I54, Renormalization I 9B - 20 3

Index Renormalization of the Dirac fieId,203-207 Rest mass of the photon,34, l4I,I43,I48,I54,162-167, 2I3 Retarded function,29 Rutherford scattering cross section,llo Rutherford scattering cross sectionrexchangecorrection,12s

Self-energy of the photon,l43 Self-mass of the electron,I5l,203 Self-stress of the electron,IgI S-function,64-68 Singular functj.ons of field theory, 26-32 S-matrix,SS-Bg Spin matrices of Pau1i,55 Spin of a photon,ZS,26 T h o m s o nc r o s s s e c t i o n ' 1 1 6 Time ordering operator,9l Transformation properties of field theory,l0-I5 Transition probability,g7,92

Scattering of electrons by a Coulombfield,I07-110 Scattering of electrons by ele ctrons, I0 4, I22 -I25, I9l unitary,13l scattering of electrons, raunits , natural,2 diative corrections ,162-16z Scattering of light by elecVacuum polarization,l32-I39, trons,112-I16 216-219 Scattering

of Photons

htz nhn-

tons.194 Schroedinger picture e second quantizatior, fi ,n" electron field,58 Self-charge , 143

n3

Ward identity,I60,2l9 Weight functions in the renormalization formalism,2l2 Zero-point energy,2l