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An analysis of the self-energy problem for the electron in quantum electrodynamics Daykin, Philip Norman 1952

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T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A F A C U L T Y OF G R A D U A T E STUDIES P R O G R A M M E OF T H E . F I N A L O R A L E X A M I N A T I O N F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y of P H I L I P N O R M A N D A Y K 1 N B.A. (University of Britisli Columbia) 1947 M . A . (University of British Columbia) 19.49 WEDNESDAY, APRIL 23rd, 1(152, at 3:00 P.M. IN R O O M 303, PHYSICS BUILDING C.OMMrnr.r. I N CIIARCK: Dean H. F. Angus, Chairman Professor F. A. Kacmpii'cr Professor W. A. Clemens Professor C. M. Shrum Professor R. W. Welhvood Professor R. D. James Professor R. Daniells Professor \V. Opccliowski Professor 13. G. Laird PUBLISHED PAPERS Radiations from Zinc"-'', Physical Review 76, 1719, 1949 (Co-authors K. C. Mann and D. Rankin) An Analysis of the Self-Energy Problem lor the Free Resting Electron, Physical Review 83, 895 (A) , 1951 (Co-author F. A. Kaemplfer) (Read to the American Physical Society) Conservation Laws in Feynman's Modified Electrodynamics,. Physical Review 83, 986(A), 1951. (Read to the American Physical Society) Conservation Laws in Feynman's Modified Electrodynamics, Canadian Journal of Physics, 29, 459, 1952 An Analysis of the Self-Energy Problem for the Electron in Quani'um Electro-dynamics, Canadian Journal of Physics, 30, 70, 1952 T H E S I S A N ANALYSIS OF T H E SELF-ENERGY P R O B L E M FOR T H E E L E C T R O N IN Q U A N T U M E L E C T R O D Y N A M I C S The self-energy of the free electron at rest is evaluated without the restriction that the self-interaction be a purely retarded interaction. Both the one-electron theory and the hole theory of the positron are treated. It is shown that in the one-electron theory the normally quadratically divergent transverse part of the self-energy vanishes if the self-interaction is assumed to be one half retarded plus one half advanced, the remaining Coulomb part of the self-energy being only linearly divergent. A similar theorem does not hold for the hole" theory. A particular type of self-interaction leads to a vanishing self-energy in 'one-electron theory. However this does not solve the self-energy problem, as in this case radiation corrections to scattering will vanish as well. The self-energy of a bound electron is evaluated in a similar manner. The decay probability of an excited state is calculated as the imaginary part of the self-energv; the correct value is obtained only for a purely retarded self-interaction in hole theory. In the special case in which the external field -is a uniform magnetic field, again only this interaction in hole1 theory gives the correct value for the anomalous magnetic moment. , It is therefore concluded that any solution of the self-energy problem by introducing advanced self-interactions is to be ruled out. G R A D U A T E STUDIES Field of Study: Physics Spectroscopy—I'rof. A. M. Crooker Relativity—Prof. F. J . Belinfante Wave Mechanics—Prof.-F. J. Belinfante Beta Ray Spectroscopy—Prof. K . C . Mann Electromagnetic Theory—Prof. G. L. Pickard Nuclear Physics—Prof. K. C. Mann Electronics—Prof. A. van der Ziel f Chemical Physics—Prof. A. J. Dekker Quantum Theory of Molecules and Solids—Prof. H . M. James Statistical Mechanics—Prof: H.^Frohlich , Quantum Theoryof Radiation—Prof. F. A. Kaempfler Theoretical Physics Seminar—Prof. W. Opechowski Other Studies: Advanced Dilfercntial Equations—Dcnn W. 11. Gage " Tensor Calculus—Dean-W. H. Gage • . . Theory of Functions of a Real Variable— Prof. S. A. Jennings Group Theory—Prof. S. A. Jennings Integral Equations—Prof. T . E. Hull Modern Algebra—Prof. D. Deny AN ANALYSIS. OF THE SELF-ENERGY PROBLEM FOR THE ELECTRON IN QUANTUM ELECTRODYNAMICS by PHILIP NORMAN DAYKIN A THESIS SUBMITTED IN PARTIAL FULFILMENT- OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN PHYSICS We accept t h i s thesis as conforming to the standard required from candidates for the degree of DOCTOR OF PHILOSOPHY Members of the Department of Physics THE UNIVERSITY OF BRITISH COLUMBIA APRIL,.1952 ABSTRACT The self-energy of the free electron at rest Is evalu-ated without the r e s t r i c t i o n that the s e l f - i n t e r a c t i o n be a purely retarded i n t e r a c t i o n . Both the one-electron theory and the hole theory of the positron are treated. It i s shown that i n the one-electron theory the normally quadratically divergent transverse"part of the self-energy vanishes i f the s e l f - i n t e r -action i s assumed to be one half retarded plus one half advanced, the remaining Coulomb part of the self-energy being only l i n e a r l y divergent. A similar theorem does not hold f o r the hole theory. A p a r t i c u l a r type of s e l f - i n t e r a c t i o n leads to a vanishing self-energy i n one-electron theory. However th i s does not solve the self-energy problem, as i n t h i s case radia-t i o n corrections to scattering w i l l vanish as w e l l . The self-energy of a bound electron i s evaluated i n a similar manner. The decay p r o b a b i l i t y of an excited state i s calculated as the imaginary part of the self-energy; the correct value i s obtained only for a purely retarded s e l f -i n t e r a c t i o n i n hole theory. In the special case i n which the external f i e l d i s a uniform magnetic f i e l d , again only t h i s i n t e r a c t i o n i n hole theory gives the correct value for the anomalous magnetic moment. It i s therefore concluded that any solution of the s e l f -energy problem by introducing advanced s e l f - i n t e r a c t i o n s i s to be ruled out. ACKNOWLEDGEMENTS This research was carried out with the aid of a Fellow-ship from the National Research Council. The author wishes to thank Professor F.A. Kaempffer for suggesting the problem and for numerous hel p f u l discussions during the invest i g a t i o n . CONTENTS Introduction SECTION I SECTION II Retarded and Advanced Interactions Between Two Dirac P a r t i c l e s 11 SECTION III Evaluation of the Free Electron Self-Energy Matrix 21 SECTION IV Self-Energy Matrix For An Electron In An External F i e l d 26 SECTION V The Anomalous Magnetic Moment of the Electron 32 SECTION VI The Retarded F i e l d of a Free Rest Electron 35 A. Number of Photons i n the V i r t u a l F i e l d 36 B. The E l e c t r i c and Magnetic F i e l d s of a Rest E l e c t r o n 3 8 SECTION VII Appendix - A. The Anomalous Magnetic Moment V3 B. The Photon Number *+7 C. The Transverse F i e l d s 50 D. The Coulomb F i e l d 5*+ SECTION VIII References 56 - 1 -I, INTRODUCTION The self-energy problem for the electron i n quantum e l e c t r o -dynamics arises from the coupling between one electron and the t o t a l r a d i a t i o n f i e l d . According to the formalism of perturba-t i o n theory the electron can emit and reabsorb photons. Thus, the system consisting of one electron and the t o t a l r a d i a t i o n f i e l d can. exist p a r t l y i n intermediate states which contain v i r t u a l photons and i n which the electron has an additional r e c o i l momentum. The self-energy of the electron i s the average value over the intermediate states of three terms:; the k i n e t i c energy of the electron, the energy of the v i r t u a l photons and the energy of the coulomb f i e l d of the electron. The f i r s t two terms com-bined give the transverse self-energy; the t h i r d the l o n g i t u d i n a l or coulomb self-energy. Both parts, of the self-energy are divergent: the transverse part from the large contribution from intermediate states with high k i n e t i c energy;; the coulomb part, which i s analogous to the c l a s s i c a l self-energy, e s s e n t i a l l y because the electron must be treated as a point charge. The self-energy f o r a free electron was f i r s t calculated r e l a t i v i s t i c a l l y by Waller (1, 1930) using the one-electron theory of Dirac (2 , 1928) and the quantum theory of r a d i a t i o n developed by Dirac (Q, 1927) and by Heisenberg and Pauli (h, 1929). The coulomb self-energy was shown to be l i n e a r l y divergent as for a c l a s s i c a l point charge but the transverse self-energy was - 2 -quadratically divergent. The reason for the quadratic divergence i s connected with the d i f f i c u l t y of the negative energy states i n the one-electron theory. For t r a n s i t i o n s to the low l y i n g states energy i s very nearly conserved and so these states contribute l a r g e l y to the self-energy. In f a c t , as shown i n Section ¥1 the number of v i r t u a l photons i s quadratically divergent i n the one-electron theory. Oppenheimer (5 , 1930) extended the r a d i a t i o n theory to the problem of a bound electron. The transverse self-energy i n second order was shown to have the form where b£*is the matrix element for emission of a photon of f r e -quency vr and p o l a r i z a t i o n A with corresponding t r a n s i t i o n of the electron from the state m to state n. The integration extends over a l l s o l i d angle dwr and frequence V,. . The improper i n t e g r a l over ^ i s resolved by deforming the path of integration about the pole. This adds an imaginary term to the self-energy for E£' < E'^ which c o r r e c t l y accounts f o r the half width of the spectral l i n e s . However the center of the spectral l i n e , i t was shown, would be shifted toward the red by the difference between the self-energies and E*j? and t h i s d i f -ference would i n general also be i n f i n i t e . The hole theory of the- positron formulated by Dirac (6,1929) gave r i s e to some hopes that the degree of divergence could be reduced. Weisskopf (7, 193^; 8, 1939) showed i n f a c t that the self-energy r e l a t i v e to the vacuum was only logarithmically diver-gent. B r i e f l y , the reasons for the divergence being only l o g a r i t h -mic i n hole theory are two-fold. F i r s t l y , the quadratic divergence of the one-electronr; theory i s reduced to a l i n e a r divergence by exclusion of t r a n s i t i o n s to the negative energy states. Secondly, i n the neighborhood of the electron v i r t u a l pairs are created which i n e f f e c t produce a spread of charge over a region of radius equal to the Compton wavelength. This spread of charge i s s u f f i c i e n t to make the self-energy logarithmically divergent. However, the conclusion of Oppenheimer regarding the predicted i n f i n i t e s h i f t i n the frequencies of atomic spectral l i n e s remained v a l i d . The self-energy i n quantum electrodynamics was therefore discarded, as i n the c l a s s i c a l theory, as meaningless. During the three years 1937 to 19kO some spectroscopic evidence was accumulating which indicated that the Dirac equation did not predict exactly the correct f i n e structure of atomic states (9))« Pasternack (10) showed that the observed f i n e structure of the H* l i n e could be accounted f o r by a s h i f t of only the 2S^ l e v e l by .03 cm' from the value predicted by the Dirac equation. A. number of authors discussed the effects of a postulated departure from the coulom:b f i e l d . In a l l cases the necessary departure was much too large to be reasonable. These doubts concerning Dirac's f i n e structure formula were confirmed by the b r i l l i a n t experiments of Lamb and Retherford (11, 19^7 > 19^9). which showed conclusively - k, -that the 2P§- and 2S§ spectroscopic states of the hydrogen atom were not degenerate as predicted by the Dirac equation but were separated by some 1000 megacycles per second. It was shown by a number of authors that t h i s discrepancy could be accounted for i n the following way by the self-energy terms which had e a r l i e r been discarded. The s e l f ^ -energy of a free electron i s proportional to the mass and may be regarded as a contribution to the electron's k i n e t i c energy from addition of electromagnetic mass to the mechanical mass. However t h i s e l e c t r o -magnetic mass, Am, must already be included i n the t o t a l observed mass, m, and should formally be put equal to zero. This argument i s equivalent to assuming that the o r i g i n a l mass appearing i n the Dirac equation was the mechanical mass and that t h i s should now be replaced by the t o t a l mass. The same renormalization of mass must now be performed on the self-energy of a bound electron. A. term equal to the average value of (dH/dm)Am for the atomic state should be subtracted. The remaining f i n i t e part should give the observed term s h i f t . I t should be emphasized that this procedure can only be j u s t i f i e d on the basis that future modifications of the theory for high energy photons w i l l make the electromagnetic mass a small correction. A. preliminary n o n - r e l a t i v i s t i c c a l c u l a t i o n by Bethe (12,19^7) gave nearly the observed Lamb s h i f t . R e l a t i v i s t i c corrections f o r the intermediate states with high k i n e t i c energy by a number of authors (13, 19^9> 1950) gave results i n good agreement with the more accurate measurements of Lamb and Retherford (19^9). It was pointed out by Schwinger (1^, 19^8) that one of the terms i n the r e l a t i v i s t i c expression for the Lamb S h i f t could be explained on the basis of a small correction to the magnetic moment of the electron of magnitude «0rr Bohr magnetons. The existence of t h i s anomalous magnetic moment had e a r l i e r been suggested by Br e i t (15, 191+7) on the basis of spectroscopic data. A. number of physically equivalent Lorentz-covariant forma-lisms were developed to deal with the problem of radiatio n corrections to scattering by Tomonaga, Schwinger, Dyson and Peynman (16-21, 19^6-50). These had the advantage over the older perturbation theory that the terms equivalent to a renormaliza-t i o n of mass can be separated out from observable e f f e c t s unam-biguously. Feynman's formalism has an added advantage of sim-p l i c i t y i n application; Feynman has derived a set of rules which enable the matrix elements to be written down at once unam-biguously from the Feynman scattering diagrams. Feynman's method d i f f e r s from others i n another respect: a r e l a t i v i s t i c cut-off i s introduced to make the integrals f i n i t e . It has been shown (12) that t h i s i s equivalent to modifying the Maxwell equations i n the manner described by Podolsky, Bopp and others. After renormalization the observable r a d i a t i o n effects are only s l i g h t l y sensitive to the cutoff. A. confirmation of the correctness of the new formalismssfor fourth order has recently appeared i n the measurement of the anomalous magnetic moment of the electron. Koenig, Prodell and - 6 -Kusch (18, 195D found experimentally for the g-factor of the electron the value 2(1.0011^5 0.000013) which agrees with the t h e o r e t i c a l value of 0011^5^) calculated by Karplus and K r o l l (19, 1950) using Dyson's S-matrix. These well known calculations are based.on the assumption, either e x p l i c i t or i m p l i c i t , that the i n t e r a c t i o n between charges (which includes the s e l f - i n t e r a c t i o n of a charge as a special case) i s a retarded i n t e r a c t i o n . In quantum electrodynamics a retarded i n t e r a c t i o n Is described by the exchange of p o s i t i v e energy pho-tons, as Feynman (21) has shown by elimination of the f i e l d o s c i l l a t o r s . However, there i s up to now no conclusive experi-mental evidence that this assumption i s correct. In f a c t Wheeler and Feynman (25) have shown i n t h e i r absorber theory of r a d i a t i o n that a consistent c l a s s i c a l ' description of electromagnetic phen-omena including r a d i a t i o n damping can be obtained i f the r a d i a t i o n f i e l d of a charge i s assumed to be one half of the retarded plus one half of the advanced f i e l d . Dirac (26) has proposed a r e l a t i v i s t i c quantum theory of r a d i a t i o n which .treats positive and negative energy photons sym-me t r i c a l l y . It w i l l be shown i n Section II that t h i s treatment i s equivalent to the assumption that the i n t e r a c t i o n between electrons i s one half retarded plus one half advanced, correspond-ing to the c l a s s i c a l Wheeler-Feynman in t e r a c t i o n between charges. P a u l i (27) has shown that this treatment leads to a vanishing transverse part of the free electron self-energy i n one-electron theory, but i f the hole theory of the positron i s adopted a similar - 7 -theorem does not hold. Addition of the effects of positive and negative energy photons i n f a c t destroys the logarithmic character of the divergence, leaving quadratically divergent terms. Muto and Inone (28) report, however, that the divergence i n the fourth order s e l f energy i n one-electron theory i s not,eliminated by introducing negative energy photons. In r e f e r r i n g to these r e s u l t s the eff e c t s of the A-limiting process, which i s also i n -cluded i n Dirac's method of f i e l d quantization, have been neglected. This process r e s u l t s i n introducing a factor cosAk into the i n -tegrals over photon energy, k, and leads to a vanishing coulomb part of the s e l f energy i n one-electron theory but does not im-prove the re s u l t i n hole theory. Mbto and Inone have shown that the energy l e v e l s h i f t of the 2S-J state of the hydrogen atom van-ishes i f both the V l i m i t i n g process and the negative -energy pho-tons are included. It i s the main purpose of t h i s thesis to investigate the self-energy problem without the r e s t r i c t i o n that the i n t e r a c t i o n of a charge with i t s e l f be purely retarded. For this i n v e s t i -gation use has been made of the sim p l i f i e d form of quantum e l e c t r o -dynamics described by Feynman (19» 2 0 ). It i s shown that the d i s t i n c t i o n between one-electron theory and hole theory and between the d i f f e r e n t types of int e r a c t i o n can be made by d e f i n -ing d i f f e r e n t contours for evaluating the Fourier transforms of the singular solutions of the Maxwell equations and Dirac equations, i n terms of which the in t e r a c t i o n between electrons i s defined. The meaning of the d i f f e r e n t types of in t e r a c t i o n i s i l l u s t r a t e d 6 8"-by a discussion of the second-order i n t e r a c t i o n between two elec-trons. A Hamiltonian can be obtained for each type of i n t e r a c t i o n as an expansion i n powers of v/c up to the second. This i s com-pared with.the B r e i t i n t e r a c t i o n (29). Higher order terms i n v/c are i n general not hermitian, the antihermitian parts correspond-ing to the f i n i t e l i f e time of two-electron states without r a d i a -t i o n . However, i f the in t e r a c t i o n i s assumed to be one-half retarded plus one-half advanced, the matrix of the in t e r a c t i o n i s hermitian to a l l powers i n v/c i n accordance with the Wheeler-Feynman theory. In the self-energy problem for a free electron Pauli's r e s u l t s are v e r i f i e d for the half retarded plus half advanced s e l f i n t e r a c t i o n . In addition i t i s found that there exists a pa r t i c u l a r type of int e r a c t i o n corresponding to a certa i n singular solution of the Maxwell equations discussed by S'tueekelberg (30) which leads to a vanishing self-energy i n one electron theory. However, t h i s does not represent a sat i s f a c t o r y solution of the self-energy problem as i n t h i s case the rad i a t i o n corrections to scattering would vanish as well. Thus i t would be impossible to obtain t h e o r e t i c a l l y expressions for experimentally ""observed e f f e c t s l i k e the Lamb s h i f t and anomalous magnetic moment of the electron. In the case of a bound electron the modified Dirac equation fo r an electron interacting with i t s e l f i s obtained which i s similar i n form to that given by Schwinger (3D• The self-energy i s then evaluated for the .same contours as used i n the free - 9 -electron case. The self-energy i n general i s not r e a l ; the neg-ative of the Imaginary part gives the decay p r o b a b i l i t y per unit time for the i n i t i a l bound state. It i s found that the correct decay pro b a b i l i t y arises from only one contour, the contour which corresponds to hole theory and a retarded i n t e r a c t i o n . The self-energy i s evaluated for the ease- of an electron i n a weak homogeneous magnetic f i e l d following the method of Lut-tinger (32). Again i t i s only the retarded i n t e r a c t i o n in.hole theory that gives the correct value of the anomalous magnetic moment. In Section VI the v i r t u a l f i e l d of a free rest electron i s investigated for the purpose of determining the character of the divergence of the f i e l d strengths and of the t o t a l energy of the f i e l d . Only the retarded f i e l d of the electron i s c a l -culated. It i s shown that i n one-electron theory the e l e c t r i c f i e l d i s just the Coulomb f i e l d of a point charge and the mag-netic f i e l d i s the f i e l d of a "point" magnetic dipole. This r e s u l t reveals the o r i g i n of the divergence of the self-energy. In hole theory the vacuum p o l a r i z a t i o n causes a spread i n the , charge and dipole densities over a region whose dimension i s of the order' of the Compton wavelength. This spread i s s u f f i c i e n t to reduce the order of divergence of the f i e l d strengths at the center of the charge d i s t r i b u t i o n but not to eliminate the di v -ergence completely. The average number of photons i n the f i e l d i s evaluated i n one-electron theory and i n hole theory. In one-electron theory the number i s quadratically divergent while i n - - 10 -hole theory i t vanishes. These results are i n agreement with those of Weisskopf (8) who calculated the transverse e l e c t r i c and magnetic f i e l d energies separately. It was shown that i n one electron theory these energies were equal and c u b i c a l l y diver-gent while i n hole theory they were equal i n magnitude but with opposite sign. - 11 -I I . RETARDED AND ADVANCED INTERACTIONS BETWEEN TWO DIRAC PARTICLES The notation of Feynman (20) w i l l be used throughout t h i s section. Feynman has shown that the kernel or Green's function for two intera c t i n g electrons a and b may be approximated to f i r s t order i n e* by , <$Jsk) K*b (M) d+Xs- d \ where K +(2,l) i s the kernel for the propagation of a free e l e c -tron from space time point 1 to point 2 which s a t i s f i e s the inhomogeneous Dirac equation & t-tn) K+(z,t) = I6(x,i) ; = IYH* o**i* (2) and <S+(Si,) i s the kernel for the propagation of a (positive energy)photon between 1 and 2 which s a t i s f i e s the unhomogeneous Maxwell equation •1 <5*(si.) * -4n-6f>,/) ; s»x, = «»-0* -(^-n)x (3) These kernels can be defined by the Fourier i n t e g r a l s where e i s a r e a l i n f i n i t e s i m a l which i s introduced to locate the poles of the integrand. It i s assumed that the integration w i l l always be carried out f i r s t over p f or k^. Thus, equation (1).describes the propagation of two electrons with an exchange of one (positive energy) photon. As w i l l be shown below the second term i n (1) corresponds to a retarded i n t e r a c t i o n between a and b. However, t h i s i s not the only i n t e r a c t i o n which i s consistent with the Maxwell equations. One could substitute for the 6+ function any other solution of (3); that i s , any function whose fo u r i e r transform i s k"\ The d i s t i n c t i o n between such functions l i e s i n the convention for loc a t i n g the poles of k~* r e l a t i v e to the r e a l k + -axis. There exis t four p o s s i b i l i t i e s for locating the poles k^ =-K and kJJf'+K which are l i s t e d i n Table I together with the corresponding photon kernels and the type of i n t e r a c t i o n between charges. The term "photon kernel" was introduced i n analogy with Feynman's '^electron kernel". - 13 -TABLE I Pos i t i o n of poles k and k r e l a t i v e to the r e a l k axis Corresponding photon kernel Type of i n t e r -action between charges k above k below k below k above 6+(Ssb) - I S^Lt^-tsi.) + 6+(-tft, -/-«,) j Retarded action *rv"« of a on b plus - -i /dK -cne^-Ml r ^ a r d e d action r ^ t J l r ) e + e f of b on a 4.(s^ ,) = _L {61 (tS(.-rru) ^ i ( • f J t - f f t ) | Advanced action ^ * ' of a on b plus i / - > . ^ / j*r+ f \ • v^  advanced action = ^ r ^ l e e ^ ^ - ^ > J of b on a k below k below 14- 6(ts<f-rn) ; tr>t(, 1 0 ; *r< tt, Retarded action of a on b plus advanced .action of b on a k above k above 0 » *>> Advanced action of a on b plus retarded action of b on a - Ih -The following r e l a t i o n s hold between the kernels It' can be seen from Table I that the Fourier i n t e g r a l represen-tations of the <£+(s;<,) and d(sjt) photon kernels d i f f e r only i n the sign of the photon energy K. Thus, D i r a c 1 s suggestion that the positive and negative energy photons be treated symmetrically corresponds to the assumption of a Wheeler-Feynman 6{s^)- i n t e r -action between charges which, as shown below, i s a half retarded plus half advanced i n t e r a c t i o n . This point w i l l be i l l u s t r a t e d again i n the self-energy problem. The entries in-Column 3 w i l l now be explained. Following a suggestion of Feynman* the second term of (1) may be written as the scattering of b: K'b (+,X) = -I | Ktb (+,(,) ^ A* M K,b (6,2.) d*Xb (7) by the potential plus the scattering of a: Ka(V) - -I /•<•«, (3,*-) yAf4. A^iS) K+a(S,l) d+xs (9) by the potential I t w i l l be shown that AJ(6) gives the retarded f i e l d at 6 a r i s i n g from the positive frequency part of the "current" *Ref. 20 Foot note page 773 " - 15 -j£(s) = exK+a(3,s) Y^ Knts-,1) produced by a i n going from 1 to 3« An analogous statement holds for A£(5)» From (8) and Table I we have If - M\~A« e " " ( t f c ~ ^ J-^L) ^ (12) ' o where I f J£V(5) i s defined by j ; t ( 5 ) = ^ f r d K e " i K t r j ^ ( r ' ) Then ah) A^(fc) = j JhV (>y, t 4 -r r t) d 3 ^ (15a) S i m i l a r l y (15b) Thus A^ , and A£ give the retarded f i e l d s emitted from the p o s i t -ive frequency parts of J£ and . It can be seen from the d e f i n i t i o n of the current,Equation (11), i n terms of eigenfunc-tions j ;<*> = 4>ni*>) t^r) Y^ <p(*r) $ ix,) e * l M for fr3 > tT> t, that the positive frequency part of the current i s the part produced by t r a n s i t i o n s which lose k i n e t i c energy. The r e s u l t i s e s s e n t i a l l y independent of the manner i n which the i n t e r a c t i o n term of (1) i s s p l i t into two. Thus, i f one interchanges the functions i n the d e f i n i t i o n s of A£ and , Equations (8) and (10) , one would f i n d instead of (15) > the advanced f i e l d s a r i s i n g . from the negative frequency parts of J * and j£ respectively. However, an advanced f i e l d t r a v e l l i n g outward from a p a r t i c l e which i s gaining k i n e t i c energy must be interpreted as a retarded f i e l d t r a v e l l i n g inward to be absorbed by the p a r t i c l e . These re s u l t s are v a l i d also i f one of the p a r t i c l e s i s a positron. Suppose f o r definiteness that p a r t i c l e a i s a positron. According to Feynman's theory of positrons (16), i t i s described as an electron t r a v e l l i n g backwards i n time with negative k i n e t i c energy. In t h i s case the order i n time of events i n Equations (1) and (8) i s t 3<t r<t, and the current of a defined by Equation -(11) i s the sum over negative energy states: - fn - 6 M . Thus the positive frequency part of the current defined by Equa-t i o n (ih) i s the part for which |E*)<|Enl ; and, since the order of events i s reversed, t h i s part corresponds to t r a n s i t i o n s |EJ —i | E j i n which the positron loses k i n e t i c energy. The other three types of interactions can be worked out i n a similar way. For the <£(§£)••' photon kernel one finds = j J£» (*>, ft, -t-rst.) <H3xf. - 17 -which give the advanced f i e l d s emitted by a and b respectively. The D + and D_ kernels can be s p l i t into two terms each, since 6(tr(,-rr») - ^ | 6t(tFL- I-st.) + S.(ts-(,- n-b)\ and the r e s u l t s given i n Table I follow e a s i l y . These kernels correspond to the singular solutions of the Maxwell equations described by Stueckelberg ( 3 ) . The d i f f i c u l t y mentioned by Wheeler and Feynman (25), that a retarded i n t e r a c t i o n of a on b corresponds to an advanced i n -teraction of b on a, i s v a l i d for the Situeckelberg type of i n -teractions but not the purely retarded (5+-type of i n t e r a c t i o n . The c l a s s i c a l derivation by Darwin (33) of the Hamiltonian for two charges was based i m p l i c i t l y on a Stueckelberg type .of i n -teraction and not the retarded i n t e r a c t i o n . However as w i l l be shown fo r the quantum mechanical derivation of the Hamilto-nian for two electrons i t makes no difference up to the second order i n v/c which of the four possible interactions i s assumed. Let us consider the matrix elements of the i n t e r a c t i o n term i n Equation (1) between the two-electron states 9>>> 1 W e ^ » f ' a n d f^, f<"**«*> which can be assumed for s i m p l i c i t y to be p o s i t i v e energy states, The volume integ r a l s can be carried out by using the general r e l a t i o n - 18 -¥e then obtain for the matrix element The time integrals d ts , dt f c extend over time T which must be taken very long. Now i f we consider only t r a n s i t i o n s which con-serve energy then the integrand i s proportional to the time difference ty t, except for small effects at the end points which vanish as T becomes large. The integration over t f c then gives a r e s u l t proportional to the time. Thus where EA-Ei ~ -(£b-£l). Substituting for the photon kernel from Table I and completing the time integration gives the r e s u l t rrt, .p.p. M " -ie>-TJj d**i g^v) % (*,) *bH, fac*s) f ^ ) rs'l •By using the r e l a t i o n s § = ej« (3 and Xiu- - (*^#* ^ h i s can be written as K - -iVT , where The same expression can be obtained for electrons moving <Ln a - 19_ -common external f i e l d , A.; i t i s only necessary to replace the free electron kernels appearing i n Equation (1) by kernels f o r the propagation of electrons i n the f i e l d . The analysis follow-ing Equation (1) remains v a l i d . As Feynman has pointed out, an amplitude M proportional to the time may be regarded as the f i r s t order term of the trans-formation exp(-iVt). In general V i s not hermitian, the a n t i -hermitian part a r i s i n g from the sine factor i n the integrand. The hermitian part can be expressed as the matrix element of an operator by expanding the cosine factor i n a power se r i e s . This i s the method used by.Bethe and Fermi (3*0 . Up to the second order i n v/c the operator may be written e*{ oCa^  dbhi. _ [H a, rab, Hb]] = _el ( / - ± ~ ±- K * rat,)(<4b> tat)) which i s the well known Brei t formula (29). The antihermitian part of V apparently cannot be written as the matrix element of an operator because of the absolute value sign on the argument. The f i r s t term from the power series f o r the sine function vanishes by orthogonality; the second term gives an antihermitian part to V' which i n (exp(-iVT)| x produces an exponental decay with time. This i s related to the f a c t that the t o t a l p r o b a b i l i t y for e l a s t i c scattering i s not i n general unity. For zero external f i e l d the antihermitian part i n fac t vanishes. This i s to be expected, because r a d i a t i o n processes for two electron scattering are of t h i r d order and higher. For non zero external f i e l d the antihermitian part of V, when - 20 -diagnonalized by choice of correct zero order wave-functions, would give part of the decay pro b a b i l i t y per u n i t time of the stationary two-electron states. The remaining contribution comes from the„self-energy of each electron i n d i v i d u a l l y ; since, as Oppenheimer has shown (5)» i t i s the imaginary part of the t o t a l s e l f energy of atomic electrons which gives the half width of the spectral l i n e s . The advanced i n t e r a c t i o n gives a similar expression f o r V but with the sign i n front of |E a-E»|reversed. Thus the assump-t i o n that the i n t e r a c t i o n i s half retarded plus half advanced leads to cancellation of the sine term i n the expression for V, giving an hermitian matrix with r e a l eigenvalues. This r e s u l t i s i n agreement with the Wheeler-Feynman absorber theory (25) i n which radiati o n e f f e c t s are accounted f o r by the action of the absorber. The interactions of the Stueckelberg type give expressions f o r V without the absolute value sign on (E^-E/). The sine term i s therefore not symmetric i n the p a r t i c l e s and cannot be ex-pressed as the matrix element of an operator which i s symmetric. The cosine term i s exactly the same as for the retarded and advanced interactions and gives the B r e i t formula. - 21 -I I I . EVALUATION OF THE FREE ELECTRON SELF-ENERGY MATRIX The notation introduced by Feynman w i l l be used throughout t h i s section and the next. According to Feynman (20) the s e l f -energy of the free electron can be written i n momentum space ' as the single i n t e g r a l A E = u JY^ (p-k-pty' u k-x4+k (1) where d*k ='(xnY*Jk+J*K , (£-*-•»;"' = (p_ -k*m)(jf-io*>- *»*;-*• and u i s a solution of the Dirac equation (£-h*)u =c , ot*u = / } £ - u.*(z (2) The l e f t hand side of (1) can therefore be written as Aftt 'ctor , since EAE=mAm and u*u=u E u, as Am uu. It i s convenient to m. carry out the summation o v e r i n . ( 1 ) right away by using Xjtfp, ^ + ) P U> > MU. (3) thus removing the y-matrices and obtain IT l ~ with It i s now the integration over k^'which exhibits f u l l y the advantage of treating the problem i n momentum space, as t h e d i s -t i n c t i o n s between one-electron theory and hole theory and the (5) - 22 d i f f e r e n t kinds of s e l f - i n t e r a c t i o n can most competently be made by assigning appropriate conventions regarding the positions of the- poles of the integrals (5) . As we are interested only i n the case of the resting electron (P*s4>, we need calculate only J, and because J* vanishes i n t h i s c a s e by symmetry. If the integration over the variable k + i s performed by completing the contour with a semicircle above the r e a l k^ . -axis the integrals (5) can be written where Z has to be carried out over the respective residues at the poles positioned above the r e a l k^ -axis. Each of the i n -tegrals (5) has four poles along the k + -axis which are tabulated together with the corresponding residues of J, and J x . i n Table I I . C6) TABLE II Poles Residues of J Residues of J k; = -K £ __£ 4 K* k;' = K £ R >4 4 >»x K T 4 *»!*-« / - 23 -In the computation of this table extensive use has been made of the r e l a t i o n f o r the free rest electron. The d i s t i n c t i o n between one-electron theory and hole theory i s now made i n the usual way by assigning for one-electron theory an i n f i n i t e s i m a l l p o s i t i v e imaginary part to E, thus bringing kX* and k" above the r e a l k^ -axis, while assignment of an i n f i n i t e -simal negative imaginary part to m leads to the hole theory, bringing k^ " above and k" below the r e a l k + -axis. There remain then four p o s s i b i l i t i e s for locating the poles kj. and k^.' eafych corresponding to a di f f e r e n t kind of s e l f - i n t e r a c t i o n as l i s t e d i n Table I. The photon kernels of main Interest from the point of view of t h i s thesis are the <S+ and <£. functions which correspond to the purely retarded or purely advanced s e l f - i n t e r a c t i o n s . The i n t e r -actions of the Stueckelberg type are included for the sake of completeness. Thus the Integration over k + i n (5) as outlined above y i e l d s the following complete self-energy Table III for the free resting electron.. - 2h -TABLE' III Photon Kernel £ R Am One Electron Theory 6¥ R! t R" i- R" e pin <• 1~K<4K } 6- *• R'" *- R* £ jp* - p<<f/r J 2>_ R'*• R1'r fi?"'i-zero Hole Theory n i l / / ( i f m 1 *tV « l f » > 1 Wi* J o <5- /?'• r/?'" i i a f t K l /-^  - K 1 p . /?'/• Rn*-R'" ^.tdKi i - e. - * x ? It can be seen that i n the one-electron theory the assump-t i o n that the i n t e r a c t i o n i s half retarded plus half advanced, which corresponds to Dirac's assumption that the positive and negative energy photons be treated symmetrically, leads to can-c e l l a t i o n of the quadratically divergent transverse part of the s e l f energy. It i s inter e s t i n g to note that an i n t e r a c t i o n of the Stueckelberg type D-, leads to vanishing self-energy i n one-electron theory. Unfortunately t h i s does not solve the - 25 -self-energy problem, as i n t h i s case r a d i a t i o n c o r r e c t i o n s to s c a t t e r i n g c h a r a c t e r i z e d by i n t e g r a l s of the type e * f lpx-b -*»;" <* (Pi -jt-"*f >*KJ+k 711 j — ~ w i l l vanish as w e l l , because a l l poles of such i n t e g r a l s too are then l o c a t e d above the r e a l k + - a x i s . This would c o n t r a d i c t experimental evidence l i k e the Lamb S h i f t and the anomalous magnetic moment of the e l e c t r o n . In the hole theory i t i s only the retarded i n t e r a c t i o n which makes the divergence l o g a r i t h m i c , a l l other i n t e r a c t i o n s l e a d i n g to q u a d r a t i c a l l y divergent expres-sions f o r Am. Thus the assumption that the i n t e r a c t i o n i s h a l f retarded plus h a l f advanced would remove the term K/m*" i n the integrand of Am and destroy the l o g a r i t h m i c character of the divergence l e a v i n g a q u a d r a t i c a l l y divergent expression f o r Am. - 26 -IV. SELF ENERGY MATRIX FOR AN ELECTRON IN AN EXTERNAL FIELD Feynman's k e r n e l f o r the f r e e e l e c t r o n i n t e r a c t i n g w i t h i t -s e l f may be gen e r a l i z e d f o r an e l e c t r o n i n an e x t e r n a l f i e l d , A.? by r e p l a c i n g the f r e e e l e c t r o n k e r n e l K + ( 2 , l ) by K +(2,l)., the k e r n e l i n hole theory f o r the Di r a c equation The k e r n e l f o r an e l e c t r o n propagating i n the e x t e r n a l f i e l d . a n d i n t e r a c t i n g w i t h i t s e l f may then be w r i t t e n to f i r s t order i n e* as The second term a r i s e s because the e l e c t r o n may emit a photon at 3 and reabsorb i t at k thus spending part of i t s time i n an intermediate s t a t e . The wave f u n c t i o n at 2 may be obtained i n terms of the wave f u n c t i o n at 1 i n the usual way according to the r e l a t i o n TU) = ( KU,I) /w, vcodv,, (3) where V, i s a hypersurface e n c l o s i n g the point 2 and N, i s the inward drawn normal to V, . Operating on the l e f t w i t h the D i r a c operator ( i Vx-A(:2)-m) and using Equations (1) and (2) then gives the modified D i r a c equation - 27 - > 1 This i s s i m i l a r i n form to the equation given by Schwinger (31), but i s obtained here by a r e l a t i v e l y simple c a l c u l a t i o n . The self-energy may be c a l c u l a t e d as a p e r t u r b a t i o n by rep-l a c i n g y by an e i g e n f u n c t i o n , ^ ( x ) exp(-iE t ) , of the unperturbed D i r a c equation. The self-energy i s then the diagonal matrix element* f *A K f ' ^ s ) ^ &jxs) €cEmtxilu*h) *n j^i*!***. ] (5) This may be evaluated i f the kernels are replaced by t h e i r F o u r i e r r e p r e s e n t a t i o n s *f<*A) ^-jpfM*k e-"**" k'x , (6) £ <t>x(^) k^) -t t.>r3 (7) £ &,<x*) e10"^ i tx<ts. I n one-electron theory K*f(2,3) i s replaced by the one-electron k e r n e l K e ( 2 , 3 ) given by 0 ,/ (8) The d i s t i n c t i o n between the d i f f e r e n t types of i n t e r a c t i o n i s now made as i n Pa r t s I I and I I I by a s s i g n i n g d i f f e r e n t con-ventions f o r e v a l u a t i n g the poles of k x i n the k -complex plane. *Provided e i t h e r the unperturbed sta t e m i s non degenerate or W i s diagonal by choice of c o r r e c t zero order wave f u n c t i o n s . - 28 -The necessary s p l i t t i n g of the t 3 i n t e g r a t i o n i n t o two ranges t a ) and (tKf*°) gives r i s e to an imaginary part to the s e l f -energy according to the r e l a t i o n 'A o(9) The i n t e g r a t i o n over it f o r the imaginary part can most conveniently he completed by making the d i p o l e approximation and using the formula (10) I f we w r i t e WmsAE^-i fl/2 then Tm produces i n Jexp (-iWmt)I* an e x p o n e n t i a l l y decreasing amplitude w i t h time; so Tm i s j u s t the decay p r o b a b i l i t y per u n i t time f o r the s t a t e m. F i n a l l y the r e l a t i o n s j£ = <f>* p **M = are used to s i m p l i f y the expressions f o r AE* and Pm which are l i s t e d i n Table IV. - 2 9 -TABLE IV Photon Kernel A E M T m One-Electron Theory 4 6. ^ (d3K r - |Vf* 4-n*J K ^ E B-E m-K D + lo<. fn.< C <^ E«. / D- Z e r o Z e r o 6V c£ Hole Theory K IV 5"-^*^ - En-E„-K ) +n*l K ( V E„-EIB-K +L- -Em*K J B* ^ I r t r l 1YIX IVi" ) +nzJ K ^ - l e n - E m ^ K £ n -e„-K ) P_ - 30 - ^ '• , An i n s p e c t i o n of Column 3 shows that only the 6+ - i n t e r a c t i o n i n the hole theory gives the c o r r e c t r a t e of r a d i a t i o n from an e x c i t e d s t a t e , as shown below. For s i m p l i c i t y t h i s t a b l e i s r e s t r i c t e d to the case i n which Em. i s p o s i t i v e . / In-the.'one-e l e c t r o n theory the ^ - i n t e r a c t i o n allows the e l e c t r o n to f a l l i n t o the negative energy s t a t e s w i t h emission of a p o s i t i v e energy photon and the <5_ - i n t e r a c t i o n allows t r a n s i t i o n s upward w i t h emis-s i o n of a negative energy photon. The D + - i n t e r a c t i o n gives the same r e s u l t as the +«?-) - i n t e r a c t i o n . The D _ - i n t e r a c t i o n gives a v a n i s h i n g s e l f - e n e r g y as for. the r a d i a t i o n c o r r e c t i o n s to s c a t -t e r i n g r e f e r r e d to i n Sec t i o n I I I . In the hole theory the <5--i n t e r a c t i o n contains a second set of terms a l l o w i n g t r a n s i t i o n s to negative energy s t a t e s . These terms must be associated w i t h the p o s s i b i l i t y of p a i r c r e a t i o n out of the vacuum along w i t h emission of a negative energy photon, but they w i l l not be d i s -cussed f u r t h e r here. The Stueckelberg i n t e r a c t i o n s a l l o w t r a n -s i t i o n s both up and down to occur. I t appears that the l o n g i t u d i n a l and s c a l a r photons as w e l l as the transverse photons c o n t r i b u t e to T, but the f i r s t two c o n t r i b u t i o n s e x a c t l y c a n c e l . For proof, the matrix element can be w r i t t e n i n momentum space as VC = \ fa {?-*) <Pn(?) *~ «n(l*))d>P , i n which u n(p) i s the Di r a c spinor and 0 x(p) i s the c o e f f i c i e n t i n the expansion of &(x) i n momentum ei g e n f u n c t i o n s . I t f o l l o w s from the Dirac equation and i t s conjugate that -•31 For a f r e e photon k+=-K so V£ - V j ^ . Thus which i s j u s t the sum over the transverse photons. F i n a l l y , averaging over a l l d i r e c t i o n s K and making the u s u a l non-r e l a t i v i s t i c approximations leads to the r a d i a t i o n formula given by H e i t l e r (35). For a second proof see Feynman (20). - 32 -V. THE ANOMALOUS MAGNETIC MOMENT OF THE ELECTRON Luttinger.(32) has shown that the anomalous magnetic moment may be obtained from the self-e n e r g y of the e l e c t r o n i n a u n i -form magnetic f i e l d by tak i n g advantage of the existence of a p a r t i c u l a r unperturbed non-degenerate sta t e f o r which E^m. For t h i s s t a t e the c o n t r i b u t i o n to AE due to the change i n mass of the e l e c t r o n i s j u s t Am so that the mass r e n o r m a l i z a t i o n r e q u i r e s only the d i r e c t s u b t r a c t i o n of Am. The method of L u t t i n g e r i s foll o w e d i n c a l c u l a t i n g the anomalous magnetic moment pr e d i c t e d by the self-energy formulas given i n Column 2 of Table IV. How-ever the f a c t o r |V|* contains the sum over a l l f o u r p o l a r i z a t i o n d i r e c t i o n s , / * , thus i n c l u d i n g the Coulomb i n t e r a c t i o n d i r e c t l y as the e f f e c t of i n t e r a c t i o n w i t h l o n g i t u d i n a l photons. This s i m p l i f i e s somewhat the c a l c u l a t i o n of the m a t r i x elements which i s given i n Appendix A. The r e s u l t i n g expression f o r AE i s ex-panded i n powers of the magnetic f i e l d strength H e. up to the f i r s t . The term independent of H 0 i n each case i s j u s t the cor-responding expression f o r Am from Table I I I (with E=m). I t was found that the renormalized s e l f - e n e r g i e s could be expressed i n terms of a s i n g l e divergent i n t e g r a l F(m,K):: +• + The r e s u l t s are l i s t e d i n Table V. - 33 -TABLE V Photon Kernel A F - A m - A E - A m . One-Electron Theory t+ F(m,K) + F(-m,-K) Q u a d r a t i c a l l y divergent 6. F'(m,-K) +F(-m,K) Q u a d r a t i c a l l y divergent D+- 6* + &- Q u a d r a t i c a l l y divergent D- Zero Zero Hole Theory s+ F(m,K) - F(-m,K) _ ex etf» 6. F(m,-K) -F(-m,-K) Q u a d r a t i c a l l y divergent D- F(m,K) + F(m,-K) Q u a d r a t i c a l l y divergent D- F(-m,-K) + F(-m,K) Q u a d r a t i c a l l y divergent - 3^ -Table V shows that the c o r r e c t expression f o r the anomalous magnetic moment given by L u t t i n g e r and Schwinger i s obtained only from the o"+ - i n t e r a c t i o n i n hole theory. The renormalized self-energy f o r t h i s i n t e r a c t i o n i s i n ordinary u n i t s ! e± et « which can be i n t e r p r e t e d as an a d d i t i o n to the magnetic moment of «</«./rBohr magnetons. The remaining s i x non-vanishing expres-sions l e ad to a q u a d r a t i c a l l y divergent anomalous magnetic moment. These r e s u l t s show c o n c l u s i v e l y that there i s no s o l u t i o n to the self-energy problem p o s s i b l e by i n t r o d u c i n g advanced s e l f -i n t e r a c t i o n s or the corresponding negative energy v i r t u a l photons of D i r a c . I t must be assumed that the e l e c t r o n i n t e r a c t s w i t h i t s e l f v i a i t s retarded f i e l d . The one-electron theory must be r e j e c t e d f o r the same reasons. - 35 -V I . THE RETARDED FIELD OF A FREE REST ELECTRON In t h i s s e c t i o n the v i r t u a l f i e l d of a f r e e r e s t e l e c t r o n i s i n v e s t i g a t e d f o r the purpose of determining the character of the divergence of the f i e l d strengths and of the t o t a l energy of the f i e l d both i n one-electron theory and i n hole theory. "Since the r e s u l t s of the previous s e c t i o n i n d i c a t e that only the r e t a r -ded f i e l d of the e l e c t r o n has a p h y s i c a l meaning t h i s f i e l d only w i l l be c a l c u l a t e d . This means using the conventional quantum electrodynamics w i t h p o s i t i v e energy photons. The f o l l o w i n g r e s u l t s from p e r t u r b a t i o n theory w i l l be r e -q u i r e d . I f i s a s t a t i o n a r y s t a t e s o l u t i o n of the S'chrodin-ger equation where H can be w r i t t e n as H„ plus a small p e r t u r b a t i o n A V (with of the eigenfunctions yZ of the unperturbed Hamiltonian H e up to second order i n X by ( i ) A l a t e r put equal to u n i t y ) , then c a n b e expressed i n terms ft - i A 1 Z (2) where The expectation value of any Hermitian operator A may then be w r i t t e n up to second order as - 36 -For the i n t e r a c t i o n of a s i n g l e e l e c t r o n w i t h the e l e c t r o -magnetic f i e l d > and the operators < y^ have the matrix elements (*0 'n,^, » = (5) A. NUMBER OF PHOTONS IN THE VIRTUAL FIELD The operator of photon number i s given according to (5) by (6) and the average number of photons i s given by *'r (7) where m i s the s t a t e which i n zero order approximation contains no photons and e i t h e r one e l e c t r o n or one e l e c t r o n plus the D i r a c - 37 -sea. When t h i s expression i s s u b s t i t u t e d i n t o equation (3) the only non-vanishing terms are " (8) This expression i s evaluated i n Appendix B. In one-electron theory the r e s u l t i s (9) which i s q u a d r a t i c a l l y d i v e r g e n t . The energy of the f i e l d i s th e r e f o r e c u b i c a l l y d i v e rgent. This divergence i s p a r t l y com-pensated, as Weisskopf (8) has shown, by the k i n e t i c energy of the e l e c t r o n i n the intermediate s t a t e s l e a v i n g a q u a d r a t i c a l l y divergent self-energy f o r the e l e c t r o n . In hole theory the value of N r e l a t i v e to the vacuum i s zero. Thus, i n hole theory the self-energy i s j u s t the k i n e t i c energy of the e l e c t r o n i n the i n -termediate s t a t e s which i s only l o g a r i t h m i c a l l y d i v ergent. - 38 -B. THE ELECTRIC AND MAGNETIC FIELDS OF A REST ELECTRON The f i e l d s about a Di r a c e l e c t r o n at r e s t may be described i n terms of a c o r r e l a t i o n f u n c t i o n which gives the products, averaged over the e l e c t r o n coordinates, of the p r o b a b i l i t y d e n s i t y f o r the e l e c t r o n at ? and the f i e l d i n t e n s i t y at ( r % R ) . This method i s an extension of the method used by Weisskopf (8) f o r the charge and dip o l e d e n s i t i e s i n the neighbourhood of an e l e c t r o n , ( a ) . The Transverse F i e l d -» The transverse f i e l d strengths at a d i s t r a n c e R from an e l e c t r o n are defined i n accordance w i t h the idea given above, as the expectation values: EiCR) * \VL?>4) E ^ X ? * 8) Y(n<t) 4?Jd_, (io) = JyY?j«?; HoP(F7t#) Ytr,4) <Xf 4<j, ( i i ) The i n t e g r a t i o n i s to be c a r r i e d out over a l l coordinates q of the f i e l d o s c i l l a t o r s and over the coordinates r of the e l e c t r o n . The operators f o r the f i e l d strengths are obtained from equation (*f): - t'W> = ^ * * Sk'"- C **rr * > (12) If) - * * Air) = ^ Z % v - ^ ?k'r. U3> The f i e l d strengths defined by (11) and (12) may now be evaluated f o r a r e s t e l e c t r o n with, s p i n i n the p o s i t i v e z d i r e c t i o n , using equations (3) to ( 6 ) . (See Appendix C). The r e s u l t s are: ( i ) one-electron theory: EiCR) - o (llf) - -fa \ A i ft-fa % + ifi*\:R,*fin d5) where R, i s a u n i t vector i n the d i r e c t i o n R and M i s the e l e c -t r o n d i p o l e moment eV2mc. These are j u s t the f i e l d s a s s o c i a t e d w i t h a magnetic d i p o l e of strength M. ( i i ) hole theory: £<"/?) = 0 , (16) . / > h H t o i * ( & \ * ' & « > * ; \ ' ( 1 7 ) where A> i s the Compton wavelength tymc and K y i s the Hankel func-t i o n represented by K W z ) = [ ° e - t w U c ^ r f 4t. -» The asymptotic values of H f o r small R and l a r g e R can be e a s i l y worked out w i t h t h i s r e p r e s e n t a t i o n . For l a r g e z p l x - ± IT At /- rr j . _ / i n * «-For small z 0 K>(*) —» f* J t = - tog? , i'd*. —> /-/,32 Thus H has the asymptotic forms /? » Xo (18) For l a r g e values of R the magnetic f i e l d goes over i n t o the value found i n one-electron theory. For small R the s i n g -theory. This form of the f i e l d i n hole theory r e s u l t s from the p o l a r i z a t i o n of the vacuum i n the neighbourhood of the e l e c t r o n . The mean ra d i u s of the p o l a r i z a t i o n i s of the order of the Compton wavelength \. (b) The Coulomb F i e l d The Coulomb f i e l d i s most e a s i l y c a l c u l a t e d by using second q u a n t i z a t i o n f o r the e l e c t r o n s . The operator f o r the Coulomb f i e l d i s defined by the c o r r e l a t i o n f u n c t i o n u l a r i t y i n the f i e l d i s only R"-' compared to R"3 f o r one-electron = - V* J PC?) <t>4P ( r + * ; d? ; (19) where (20) and ? i r ) = Jk *S a* (*J u*) e (21) I where the i % are D i r a c spinors and a£ and a^ are the operators f o r c r e a t i o n and a n n i h i l a t i o n r e s p e c t i v e l y of an e l e c t r o n i n the s t a t e K. The operators s a t i s f y - t h e ariticommutation r e l a t i o n s < *j + «, < ~ s*j • . . (22) The Coulomb f i e l d i s given t o f i r s t order i n e by the expectation value of E n(R); i n the unperturbed s t a t e . When t h i s i s evaluated w i t h the help of the anticommutation r e l a t i o n s (22) (See Appendix D), the r e s u l t s a r e: ( i ) one-electron theory ( i i ) hole theory *«<"> - K°(%) ' 0 which has the asymptotic values The r e s u l t obtained f o r one-electron theory shows that the e l e c t r o n must be tr e a t e d as a point charge. In hole theory however the p o l a r i z a t i o n of the vacuum reduces the s i n g u l a r i t y of the p o t e n t i a l to l o g a r i t h m i c . I t i s i n t e r e s t i n g to note that the charge d e n s i t y of a s i n g l e e l e c t r o n i n the vacuum found by Weisskopf (8) (23) (2*0 _ 1+2 -e (26) which has the asymptotic values r « >> gives the above Coulomb field (21*) i f this density is substituted into the definition (19) , A similar relation should hold for the magnetic field in hole theory, since the dipole density is also given by (26), but the calculation is involved and has not been done here. - h3 -. V I I . APPENDIX A. THE ANOMALOUS MAGNETIC MOMENT The matrix elements V^ M w i l l be evaluated f i r s t . An a d d i t i o n s u b s c r i p t , i , i s used to l a b e l the four c l a s s e s of eig e n f u n c t i o n yhi which are l i s t e d by L u t t i n g e r ; the double s u b s c r i p t s , i stands f o r the s i n g l e s u b s c r i p t n used by Lu t -t i n g e r . cK-r 6 ^ 0 where 3 S{ because rfiV*. = - ^  - 1 * -•I *3 FsW (5) 1 JSA. ( u 5 m ) ; t /• *e» ,<» , i j \i , [x J r" I 9sW\X ((£>-»./ - - iSX? \?>i«>\X (6) (7) - -In s i m p l i f y i n g Equation (7) frequent use has been made of the Equations (3) . For example 2 « « , N I - £ ( S M ) = r ^ l * The s e l f - e n e r g i e s l i s t e d i n Table IV can now a l l be ex-pressed i n terms of a simple f u n c t i o n F(m,k): -(o) where (Note that E n i s negative f o r the negative energy s t a t e s but E s i s always p o s i t i v e ) . Replacing (s -t 1) by s i n the f i r s t term of F(m,k) gives ; +n* ) IT ? sT (10) where * * ) s J ^ * ' * * ) " f j Ce.-""?" ( I D F o l l o w i n g L u t t i n g e r we can replace the sum over s to f i r s t order i n e Ee by ? ^s; a? s(*)+ * ( 1 2 ) SO?) - (13) - 1+6 -which i s independent of eH and gives the s e l f energy of the f r e e r e s t e l e c t r o n as l i s t e d i n Table I I I . According to the method of mass r e n o r m a l i z a t i o n t h i s term i s to be subtracted. I t remains to c a l c u l a t e g"(>j)/2. I f s i s w r i t t e n as t->i + e) theog"(^)/2 i s the c o e f f i c e n t of € x i n the expansion of g(s) i n powers of e. When t h i s has been found and r ^ replaced by i t s average value over the sphere, f t h e observable part of FCm,K) i s I t should be noted that the l a s t term alone i n the integrand s p o i l s the convergence of the i n t e g r a l . The convergence of the terms i n square brackets together however i s c o n d i t i o n a l on the signs of K and m. + 6 t»<*1-Kx) *'» »>»« (•>».* (1^) - V7 -B. THE PHOTON NUMBER The matrix element V n m i s the emission element and I n one-electron theory the sum extends over a l l spins i n the intermediate s t a t e n. The expression f o r N i s s i m p l i f i e d hy ta k i n g a f a c t o r fcc out of the energies. This gives *.» U to ^ — *Z (u\ jo) **it Uyl-AyXuy&h) uYJo) ) (5) where Each sum over V can he extended over a l l four s p i n s t a t e s a f t e r i n t r o d u c i n g the p r o j e c t i o n operators C t(-k) defined hy The value of N i s averaged over both spins v„ f o r the r e s t e l e c t r o n . E v a l u a t i o n of N~ then r e q u i r e s the sum (1) (2) SCO* k Z spi*** c*t-*)4kt9 cuo)) f S p c-(_k)dMyct(0)) y (H —ji/ftltMx -h)x (M-*\^oThT - k)* (5) - h-8 -This i s e a s i l y evaluated using the f a c t t h a t spurs c o n t a i n i n g an odd number of f a c t o r s *y vanish and using the commut-a t i o n r e l a t i o n s f o r «< and £ . The sum over p o l a r i z a t i o n s gives simply a f a c t o r 2 . With the sum over k replaced by (L*Axnp){d*k , N reduces to To make the t r a n s i t i o n to hole theory, the sum over i n t e r -mediate s t a t e s i s f i r s t r e s t r i c t e d by the e x c l u s i o n p r i n c i p l e to p o s i t i v e energy s t a t e s . C o n t r i b u t i o n s to N from e l e c t r o n s i n the Dirac sea would now have to be added. However since we are i n t e r e s t e d i n the value of N r e l a t i v e to the vacuum we need only t o subtract the c o n t r i b u t i o n s from the sea which are d i s -allowed by the presence of the one p o s i t i v e energy r e s t e l e c t r o n . These terms, which a r i s e from a negative energy e l e c t r o n w i t h i n i t i a l momentum k* which emits a photon of momentum -ic, give the sum 2g ^1 Z. £ £ (ukk)^kiY uvJo)Ht&0 *.klf uvCk)) . Replacing k by -k and combining w i t h the f i r s t term of N i n (3) gives. xn c i £ i f ^ K,y C(-k) **,Y C+M) (9) I t i s worth n o t i n g that the t r a n s i t i o n to hole theory i n e f f e c t simply changes the si g n on the second term of B(k) i n (5) and changes the s i g n of the photon energy k i n the denominator of t h i s term. This f a c t w i l l be used again i n Appendices C and D. The new value of B(k) i s now obtained d i r e c t l y from Equation (6). I t reduces to &(k) = fi (fit V/*x r/kx ) Kit** Oh^tH1-) - D ) do): and N vanishes i n hole theory r e l a t i v e to the vacuum - 50 -C. THE TRANSVERSE FIELDS -» -» -» The average values of the operators E X (R) and H(R) from Equations (10), (11) and (3) are required up t o second order i n e. The only non-vanishing terms up to t h i s order are the f i r s t order terms H a?) = r . fvg,'^f> ».r wtfcf) *?m fc»*]. ( I D S u b s t i t u t i n g f o r V and E\ o p from (if) and (1) and c a r r y i n g out the i n t e g r a t i o n gives i n one-electron theory (12) w i t h C r(k) defined by Equation (*+). A s i m i l a r expression holds f o r H but i s replaced by k**eV/k. The sum over r gives £ = ^ - r/*) , £ 4 ? V ^ y = ^ (13) For a r e s t e l e c t r o n w i t h s p i n up r»i 0 O o WJLiJLi Uo) -'; l=i,*,3 . (Ik) ' (15) Using these we f i n d •(uf^'k) 2 u.) = $ f i -Myi,) -Wk-er-e Z, gx «»-e (16) 'the u n i t v e c t o r s of the x,y,z coordinate system; • ) ? = . (17) (18) With these r e s u l t s the matrix elements f o r E and H are e a s i l y evaluated (19) -» -» C l e a r l y E vanishes by symmetry, and H reduces to - 52 -(R) = «_ / (£ 3 - *, ) COS *./? (20) To perform the i n t e g r a t i o n over the angles take p o l a r on R and two a u x i l i a r y axes S,T to complete the coordinate sys-tem, as shown i n the diagram. For any vector which can be chosen i n the R,S plane without l o s s of g e n e r a l i t y , we have Expressing the components of k i n terms of the \ S^^TF^^ T angles and c a r r y i n g out the (0 - i n t e g r a t i o n then gives (22) This r e s u l t i s now used i n Equation (20) but w i t h (k>e 3) f o r k 3, jj3k = An- j*Vw* (23) and I I kR * L kR {FR)K 0*R)3 J - ' - W W " * * * * ! ] } . - 53 -+ [£2p - * - ^ ] [ ^ x £ e 3 * ^ 7 ] - (25) The i n t e g r a l s , appearing in.(25) have the values * '* : 4 y ( 26) and H(R*) becomes The t r a n s i t i o n to hole theory i s made as i n Appendix B s t a r t -i n g w i t h Equation (19). The e l e c t r i c f i e l d vanishes as i n one-electron theory while This r e s u l t d i f f e r s from the corresponding^Equation (20) by a f a c t o r /*( /*VA1)"* , so that Equation (25) can be used w i t h the integrand m u l t i p l i e d by t h i s f a c t o r . = w - - ^  ](€>•*,;*; r [ c ^ _ 1#,k&* ?.]] • (29) - 5h -The f o l l o w i n g i n t e g r a l s are re q u i r e d 3 H s"*y = , f f ^ H * Kt(A) (30) 4 4 4 i**1*?1 * i (31) where K y(a) i s the Hankel f u n c t i o n of the t h i r d k i n d . In the l a s t i n t e g r a l parametric i n t e g r a t i o n over x was used to removed the f a c t o r y~' and the order of. int e g r a t i o n " was then reversed. With these i n t e g r a l s s u b s t i t u t e d i n t o (29) Equation (17) of S e c t i o n VI i s obtained. D. THE COULOMB FIELD The expectation value of E;,(R) i s »- 3 J ' H ' jk ) |r%-f?- r'j In one-electron theory .(&', ^ a„ <, aK> X/f*) = ( V , «; by using the anticommutation r e l a t i o n s (22) of S e c t i o n V I . Since there i s only one e l e c t r o n to be a n n i h i l a t e d we have - ft -ecr,«/ w) s«y - *jv * and The transition to hole theory is made as in Appendix B: = - 7 e ( d i ^ r ' A J C*(K>", < ^ r ; > - a," C'(*> « . ? , 2. f<£/T S m _ K / ? _ Formula (3D can now be used: I? I\?+R-r>\ -» = - 7 I - 56 -V I I I . REFERENCES 1. I . W a l l e r , Z e l t s . f . P h y s . 63, 673,(1930) 2 . . P.A.M. D i r a c , Proc.Roy.Soc.Lon. A11Z, 610,(1928) 3 . P.A.M. D i r a c , Proc .Roy.Soc .Lon. .All 1!-. 2^-3,(1927) k. W. Heisenberg and W.Pauli, Z e i t s . f . P h y s . 56, 1,(1929) 527 168,(1930) 5. R. Oppenheimer, Phys.Rev. ^1, ^61,(1930) 6. P.A.M. D i r a c , Proc.Roy.Soc.Lon., A126, 360,(1929) 7. V. Weisskopf, Z e i t s . f . P h y s i k , 82, 27,(193*0 8. V. Weisskopf, Phys.Rev., 56j. 72,(1939) 9. W.E. Lamb J r . , Reports of Progress i n P h y s i c s , 1^,20,(1951) (Review a r t i c l e and bi b l i o g r a p h y ) 10. S. Pasternack, Phys. Rev. ^ f , 113,(1938) 11. W.E. Lamb J r . and R.C. Rethe r f o r d , Phys.Rev. £2, 2^1, (19^7).; 25, 1325A, ( 1 W ) 12. H.A. Bethe, Phys.Rev. 7g± 339, (19^7) 13. H.A. Bethe et a l . , Phys.Rev. 2Z, 370,(1950) N. M. K r o l l and W.E. Lamb J r . Phys.Rev. 25, 388, (19*4-9) J.B. French and V. Weisskopf,Phys .Rev. 25> 121+0, (19^9) l h 0 J . Schwinger, Phys.Rev. 2^,^16, ( 1 9 W 15. G. B r e i t , Phys.Rev. 22, 98>+, (19^7) 16. S. Tomonaga., Prog.Theor.Phys. 1, 27 (19^6); 2,101,198, (191+7); i , l , 101,(19^8) 17. J . Schwinger, Phys. Rev. 2li,l!+39 (19>+8); 21, 651,(19^9) 18. F . J . Dyson, Phys.Rev. 25? ^86(19^9) and 25,1736,(19^9) 19. R.P. Feynman, Phys.Rev. 26, 7^9,(19^9) 20. R. P. Feynman, Phys.Rev. 26, 769,(19^9) 21. R.P. Feynman, Phys.Rev. 80, M+0,(1950) - 57 -22. P.N. Daykin, Can.Jour.of Phys. 22, *+59, (195D 23. S. Koenig, A.G. P r o d e l l and P.Kusch, Phys.Rev. 83. 687,(195D 2h, Karplus a n d . K r o l l , Phys.Rev. 2Z» 536,(1950) 25. J.A."Wheeler and R.P. Fey'riman,: Rev.Mod.Phys. 17 157,. (19^5) 26. P.A.M. D i r a c , Proc.Roy.Soc.Lon. Al67. 1,(19^2) 27. W.Pauli, Rev.Mod.Phys. l£, 175,(19^3) 28. T. Muto and K. Inone, Prog.Theor.Phys. j>, 1033,(19^9) 29. G. B r e i t , Phys.Rev. 553, (1929) 30. C. Stueckelberg, Helv.Phys.Acta. 11, 225,(1938) 31. J . Schwinger, Phys.Rev. 82, 66^,(1951) Appendix 32. J.M. L u t t i n g e r , Phys.Rev. 2^? 893,Q9>+8) '. 33. C.G. Darwin, PhilMag. 12, 537,(1920) 3*+. H.A. Bethe and E. Fermi, Zeits.F.Phys.ZZ, 296,(1933) 35. W.. H e i t l e r , Quantum Theory of R a d i a t i o n , Oxford Univ. Press, p. 106. 

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