Special variant of the Fradkin representation

The exact Fradkin representation of a Green’s function ${\mathit{G}}_{\mathit{c}}$(x,y‖A) defined in the presence of an arbitrary external field A(z), is converted into a practical sequence of approximations whose first few terms should lead to a better-than-qualitative representation of those ${\mathit{G}}_{\mathit{c}}$[A] and closed-loop functionals L[A] needed in both potential theory and quantum field theory applications. The zeroth term of this approximation is the ‘‘phase averaged’’ (〈ph〉) form previously described, which generates a symmetrized version of the no-recoil (or eikonal) approximation; additional corrections correspond, in effect, to a proper-time rescaling of the 〈ph〉 approximation. An independent criterion exists to estimate the quality of the approximation obtained by retaining any finite number of corrections. The general construction is presented for scalar ${\mathit{G}}_{\mathit{c}}$[A] in 3+1 dimensions, and comparisons are made to the simplest, nontrivial, soluble examples in potential theory and in quantum field theory.