New Approximations to the Fradkin representation for Green's functions

A new variant of the exact Fradkin representation of the Green's function $G_c(x,y|gU)$, defined for arbitrary external potential $U$, is presented. Although this new approach is very similar in spirit to that previously derived by Fried and Gabellini, for certain calculations this specific variant, with its prescribed approximations, is more readily utilizable. Application of the simplest of these forms is made to the $\lambda\Phi^4$ theory in four dimensions. As an independent check of these approximate forms, an improved version of the Schwinger-DeWitt asymptotic expansion of parametrix function is derived.