Comparison theorems for convergence of regularized field theories

Abstract

One of the principal obstacles in the investigation of the convergence of field theoretic perturbation expansions is the determination of the effect on the total nth order amplitude of the phase cancellations resulting from fermi statistics. A modified S-matrix is first defined involving integrations in Euclidean four-space. By including the vacuum fluctuation of the Yukawa interaction at a single point in the interaction (i.e. the interaction is not mormally ordered), the bare nth order fermion Green's function is easily bounded. If theories are regularized so that this fluctuation is finite then a simple relation between the propagators of two theories enables one to bound the total nth order amplitude of one theory by the other. From this one can infer from the convergence of the vacuum S-matrix of one theory the convergence of the other when these fluctuations are dropped. Theorems of this type are proven allowing the comparison of spinor fermion theories with scalar type fermion theories (i.e. where the propagator is the unit matrix in spin-space) and scalar fermion theories with each other. Convergence of pseudoscalar meson theory is reduced to that of scalar meson theory. The application to other than vacuum S-matrix elements is briefly outlined. Applications are presented in this and the following article.