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The outgoing fields are given by formulas differing from (5,13) only by the fact that all inequalities involving x'Q, x'Q', or x^' should be omitted. In order that the indexes should remain specifically connected with time ordering, it is convenient then to suppress the index 0 and to write v,out(„) {χ„Ί = f d { i d { f . . ά ξ Λ . . [ [ / » ( * » ),H1],H2},·- Hn]. (5,14) The domains of integrations are now independent of the points at which the field functions are being computed. This gives the 103 Nr.

The generalization of these formulas to higher orders is obvious. Let A(x) be any field function; the term of order n in its expansion is given by A<"> (ore) = \ dxxdxt ■ · ■ dxn {■ ■ ■ [[Ain O o ) , # i . ] . H J , · · · # „ ] , (5,6) where Hn has been written for f/} n (x n ), and xf > xf for x;· > xf. Formula (5,6) is well known in field theory, and is usually deduced from the Schrödinger equation. It can also be obtained from the field equations (4,3) by induction (see appendix III). The next step is the calculation of the outgoing fields.

5,2) where L1/1 is equal to Lj with the field functions replaced by the corresponding incoming field functions. It is well known that the function S occurring in the commutation relations is connected to the Green's functions bv S(x) = S_(x)-S+(x). As S+(x) vanishes if x 4 < 0 , and S_(x) vanishes if x 4 > 0 , it follows that | S,+ W (x) = — S(x) if x 4 > 0, (5,3) = 0 if x* < 0 . J Similarly S(x) = 0 if x 4 > 0, 1 (5,4) 4 J = S(x) if x < 0 . Similar relations hold for the D functions. The relations (5,3) show that the expression (5,2) can also be written OLVn or still ψ(1> (x) = \ dx' [y>in (x), Η} η ( x ' ) ] .

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