Dynamics of unrenormalizable interactions in Minkowski and Euclidean spaces

Abstract

A new approach to unrenormalizable and marginally singular field theories is developed using generalized-function techniques and an unsubtracted Lehmann representation for increasing spectral functions. Solution methods for a class of unrenormalizable equations of time-ordered and nontime-ordered functions are devised in Minkowski and Euclidean spaces. It is shown that within ladder and string approximations unrenormalizable theories possess perturbative solutions in Minkowski space which are finite in every order and do not suffer from an infinite set of subtraction constants, the axioms of relativistic field theory being satisfied. Renormalization factorsZ 2,3 −1 turn out to be differential operators (depending on a scaling parameter to a change of which observable quantities are insensitive) and may be set equal to one without the theory becoming a free one. By complex extension, solutions nonanalytic in the coupling constant can be generated, representing interactions not associable with particle exchange. A mathematical formulation of peratization is given and confronted with the Minkowski-space perturbation approach. The formalism is applied to propagators, Wightman functions and Bethe-Salpeter amplitudes of relativistic four-fermion theories in string and ladder approximations and it is shown that the locality and positive-definiteness axioms are satisfied.