G 0 exp I mei 4v4 w1 v 11v 1 v dv dw x i 2JIvf JI v 21 v w dv f p4 I 4r4 g 1 v

# G 0 exp i mei 4v4 w1 v 11v 1 v dv dw x i 2jivf ji v

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G()&, ) = «0' exp I me+i (4v)4 ~ „& w(1 v')) 1(1+v 1 v ) dv dw x-i + 2(JI+vf JI v) 21 v' w' dv f ~p4 I' (4&r)4 " g 1 v' ~ w' 2i ( Xexp~ iw+i (. (2. 35) w(1 v') ) = i(4&r) '«04 ( «09, w(1 v') ) w)w( (1 v2)2 (2. 36) The integral representation ( k„' ' (dk) exp(ik„(x„— x„')) exp~ i w(1 v') ~ ~ J ( 4~g'
J UI. I AN SCH W I N GER then transforms (2. 35) into G(li) = (dk) exp(ik„(x„— x„')) 8 (42r) ' dzo X l (1 v')dv l 0 ~n k„2 Xcosi 1+ (1-") iw (2 37) 4~02 The Fourier integral contained in the second term of (2. 39) can be recognized as related to that of the function h(x) (Eq. (A. 10)): f exp(ik„x„) ~(x) = P il (dk). (2. 41) (22r) ' ~ k„'+ a22 Indeed, 1 f exp (ik»x») (dk) (2 )' " 1+(k. '/4 o')(1 v') which can be followed by an integration by parts with respect to v, according to ( v ) f dw ( " dj v ) „— cosi ly (1 v') )w 3) &2 w E 4&22 so that 16~22 (2. 42) (1 v')' ((1 v2)& ) 2 f cosw f' ( v dw- I 1 iv dv 3 "2 w 2222 "o & 3) G(li) = 1 log 8('x x') 48m' ym 0 ( k„' dw sini 1+ (1 v') iw. (2. 38) 0 4a22 The first I integral is logarithmically divergent at the origin. On introducing a lower limit, mo, we obtain 1 (v'/3) 2 1 log P ~ n2dv 3 yw2 2~22 & 2 1+ (k„2/42 ') (1 v') as the value of the integral (2. 38), where y =1. 781. The insertion of this result into (2. 37) yields G(li) = 1 1 log -8(x x') 48m"- ymo 4 1 l' ( V') ( 1 iv'dvP (42r)' ~22 &, E 3 ) exp (ik„(x„— x„') ) X ~' (dk) (2. 39) 1+ (k„2/4K22) (1 v'} in which it has been noticed that the operator ' is equivalent to multiplication by k„' in its effect on exp(ik„x„}, and that exp(ik„x„) (dk) = b(x) (22r)' ~ = b(x2) 8(xi) ii(x2) b(x2). (2. 40) 1 f' ( 2 (x x') i 42r2 ~2 E(1 v2)& p w 9 v'dv. (2. 43) (1 v2) 2 T& expression for the induced current that is obtained from this form for G(li) is n 1 4 ()„(x))= log - J„(x) u l d~' 3% +MD f'-( X t Zi (x x') ( ( (1 v-') & 1. -'v2 3 X v'd v J„(x'), (— 2. 44) (1 v')' where u=e2/42rkc is the fine structure constant. The current induced at a given point is thus exhibited in two parts: a logarithmically diver- gent multiple of the external current at that point, and a finite contribution involving the external current in the vicinity of the given point. The first part reduces the strength of the external current by a constant factor and hence produces an unobservable charge renormaliza- tion, as discussed in I. The second part of (2. 44) is therefore the physically significant induced current.
QUANTUM ELECTRODYNAMICS 663 An alternative form for the latter is' t' (j„(x)}= a— d(a' Z~ (x x') I vr ~ &0 ((I v')& 1 -'v' v'dv "J„(x'). (2. 45) (1 v2) 2 If the external current varies sufhciently slowly, the relevant unit of length being 1/~0=k/mac, one can obtain a series in ascending powers of ' applied to J„(x). This may be done by con- tinued application of the relation: 2 ) (1 v')' k(1 v')& j 16~0' where d v' here denotes a three-dimensional volume element. Now G(r) = &(r, xo)dxo (2. 49) obeys the differential equation (j„(x)) = - a dr' ' dxo' J, jt' 2 2 XZ( (r r), (x, x, ') ) E (1 v') (1 v') 1 v v'dv V" J„(r'), (2. 48) (1 v2) 2 (V- ~02)G(r) = 8(r) (2.

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