E2 ax ax i 1 1 X jx jgxdo 40 3 30 ULIAN SCHNI NGE14 e have thereby succeeded in

E2 ax ax i 1 1 x jx jgxdo 40 3 30 ulian schni nge14 e

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E2 ax„" " ax„ i 1 1 X j„(x) jg(x')do), ' 4'[0]. (3. 30)
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ULIAN SCHNI NGE14 %'e have thereby succeeded in constructing an equation of motion for +[o] which no longer contains the electromagnetic field variables in- volved in the supplementary condition. The additional term thus introduced is evidently the covariant generalization of the Coulomb inter- action between charges. To exhibit the latter property somewhat more clearly, we must restrict the arbitrary space-like surface r to a plane surface with the normal e„. The advantage thereby acquired is the possi- bility of asserting that and i 8 i 8 V~ $(r, 0) =- D(r, 0) = B(r), (3. 35) c Bt c Bt 1 8 $(r, 0) = c Bt 4+r (3. 36) in which the latter statement involves the con- tent of the equations of definition (2. 17) as adapted to the special coordinate system. It follows from (3. 35) that of which the covariant expression is (a a) +n„n, ~n(x x') =0, &ax„" ax„) 8 1 1 n„X)(x x') =- n. (x. x. ') =0 (3. 31) ax„4 [(x„— x„')']& which enables (3. 30) to be simplified, yielding: n„(x„— x„') = 0. (3. 37) B%'[0] zoic Bo (x) 1 1 ( BX)(x x') -j„(x) e„(x) ~ n. The energy-momentum four-vector is modified by the unitary transformation (3. 19), according to 1 X n~„(x)-j), (x')da), ' 4'[0]. (3. 32) P [0]~cig'[aj~ C~]c iG' f'] (3. 38) The evaluation of the new operator P„[s] in- volves the following transformations, which we note without proof': To prove (3. 31), it is sufficient to verify it in a particular coordinate system. It is always pos- sible to construct a reference system for which the normal to a plane space-like surface is di- rected along the time axis; in other words, in this reference system, n„=(0, 0, 0, i). Equation (3.31) then states that the spatial derivatives of $(r r', t t') vanish for t=t'. This will be true if X)(r, 0) =0 for all r, that is, if $(r, t) is an odd function of t. New, in this special coordinate system, (3. 8) becomes e'c''jA (x)e-'s't' =A (x) t'( a +n„n. I Z)(x x')-j, (x')dir, ', &. &ax„ax„j c ay ay d~ 4V. kV. s "" 2J. " "a „a. ay ap ~ir, kV. kV. 2 g Bxy [email protected] t'1 a)' ~ ~ ~(r, t) = PX)(r, t) =D(r, t) (3. 33) Ec at) 1 aX' 1 as'- + ~ d&v gx d&x J) kc, c Bxg c Bx, (3. 39) and, since (2. 17) assures us that D(r, t) is an odd function of the time, the necessary property of S(r, t) is established. As a final step, we note In virtue of the supplementary condition (3. 26), that, in this special coordinate system we may write l9 1 8 n, $(x x') = S(r r', 0), QXp c 8$ BA. '(x) q A~(x)e[&] = ( &~(x) ]+[&]. (3 4o) ax„ ) n„(x„— x, ') =0 (3. 34) It follows, as a result of straightforward simpli-
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QUANTUM ELECTRODYNAMICS fication, that ~ 88' 88' P, t o]= der„ 2cs» y Bxp, Bx)g 88' 8 ' $8 8&, ) ' 8x„8x„& 8x, ) 8$ 8f + I do 1 1 8n(x x')— +- « -j. (x)~»(x)+- ~ &r g4~ f 2~g Bxt 1 1. X ng„(x) jz(x')do), ', (3. 41) C C which has been stated as an operator equation, rather than a derived supplementary condition, with the understanding that the operator A A' shall no longer appear in the theory.
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