Exact and approximate fermion Green's functions in QED and QCD.

Abstract

That special variant of the Fradkin representation, previously defined for scalar Green's functions ${\mathit{G}}_{\mathit{c}}$(x,y\ensuremath{\Vert}A) in an arbitrary potential A(z), is here extended to the case of vector interactions and spinor Green's functions of QED and QCD. An exact representation is given which may again be approximated by a finite number N of quadratures, with the order of magnitude of the errors generated specified in advance, and decreasing with increasing N. A feature appears for both exact and approximate ${\mathit{G}}_{\mathit{c}}$[A]: the possibility of chaotic behavior of a function central to the representation, which in turn generates chaotic behavior in ${\mathit{G}}_{\mathit{c}}$[A] for certain A(z). An example is given to show how the general criterion specified here works for a known case of ``quantum chaos,'' in a potential theory context of first quantization. When the full, nonperturbative, radiative corrections of quantum field theory are included, such chaotic effects are removed.