80 Hence yx e p Q7Pxdp 3 86 where QX 2DXhXDXZX PX 3 87 It will be evident that

80 hence yx e p q7pxdp 3 86 where qx 2dxhxdxzx px 3

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80) Hence, y(x) =e' «p) -Q(7)P(x')d&p', (3. 86) where Q(X) =2[D(X)h'»(X)+D&»(X)Z(X)] P(X). (3. 87) It will be evident that bus is the electromagnetic mass of the electron. The identity The explicit construction of P(X) and Q(X) proceeds analogously to that of G(X) in the (3 8l) second section. We 6rst note that ax, J" E ax, enables p(x) to be written: y(x) =e' I yg D(x x') 5&»(— x x') Bxy BP(X) and 1 expf iX(n+p)+i f- (2') 4 & 4n) 1(n P) Xn-f + fdndP, 2( fnf fPf) +D~»(x x') A(x x') Bx), +2& [D(x x')5&'&(x x') + D'"(x Y')Z(Y x')] P(x')dp&' (3 82) Z p ( vo'~ PP) = ~ expl ix(nyP)+i (2 )' " 4nJ n 1(n P) X -l + fdndP. (3. 38) n+P2 &fnl Zp') Q(». )= ~ expf iz( yp)+i (2pr) 4 4nJ ar P, ) as&»(X) ai(7) = D(7) +D~'&(Z) (3 83) 8A, 8X n )1(n Pg Xf 2- I-f + ld dP. (3. 89) n+&» & lnl The utility of this quantity stems from the rela- tion In terms of the variables v and m, defined by (2. 34), this reads 8D'" (x) 86(x) D(x) + D'" (x) BXp Xp, 3 v I. " dw 4(2~) 4 ~-& (1 e')' BP(X) BP(X) = 2x„=, (3. 84) Hence, We now define P(X), a function of X = (x„— x„') p according to which permits the first term of (3. 82) to be & v XKO' &(exp iw +i— , (3. 90) 2 w(1 -. i')
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QUANTUM ELECTRODYNAMICS which in turn becomes Q = (dk) exp(ik„(x„— x„')) 16(2or)' & 1 " dm X, I' (3-. )d. t' 1 v k„' Xexp~ i + " (1 v) w ~, (391) 2 4Ko' ) on using the integral representation (2. 36). We now remark that f(x) satisfies the second order differential equation ( ' Ko'-)P(x) =0, (3. 92) which implies that a Fourier decomposition of 33 (x) into plane waves of the form e'"i*~ involves only such propagation vectors that obey K p ~ (3. 93) '1 herefore, in evaluating the integral t Q(X) P(x')doi', Q(P, ) = I' exp(ik„(x„— x„')) F(k„') (dk), (3. 94) (2m. ) 4 with Q(li) expressed as a Fourier integral in- volving e'~~( ~ *~', multiplied by a function of k„: To complete the evaluation, it is convenient to integrate by parts, according to dL(v 5) (1 v) 7 mp 16m ~ t "dm f'1 v)' X (I cos/ ) w ~o w ( 2 n r cos'w I' (1 vp = 6 II dtv+ II (5 8K ~o w & i 4 2 ~oo ($ p) 2 X I sin( ) udw &2) 3+1 1 5 = log +- 2%' 2 pK'p 6 (3. 97) whereby we obtain a logarithmically divergent result for the electromagnetic mass of the electron or positron. An alternative evaluation can be given, which permits comparison with previous treatments, ~ by employing directly the Fourier integral representations for the functions D(x), h(x), D"'(x) and 6"'(x), in (3. 82). One thus obtains the electromagnetic mass as an integral over the momenta of the virtual quanta involved in the self energy process, with the result the latter quantity may be replaced by Kp2. 'fhus 6ns 3n X+Ep 1 = log mp 2m Kp 6~ (3. 98 j Q(li) P(x') doi' 1 ) (dk) tdio' (2~) ' X exp (ik„(x„— x„') ) F( Ko') P(x') = F( Ko') 3t (x), (3. 95) which confirms the statement that g(x) is pro- portioned to P(x), and yields as the value of the constant: where Ko= (13. '+Koo) &. Evidently 1/wo (&/Ko)'. To justify the identification of bm with the electromagnetic mass, we must show that it is possible to remove the term Ki, o(x) from (3. 71) and thereby alter the equation of motion for the matter field into that of a particle of mass ss =ssp+8m, thus demonstrating the unity of the two contributions to the actual electron mass.
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