Proof of Renormalizability of Scalar Field Theories Using the Epstein-Glaser Scheme and Techniques of Microlocal Analysis

Abstract

The renormalizability of QFT's is a vastly studied issue, and particularly the results concerning a scalar eld theory are well-known through the traditionalrenormalization approach in the literature. However, in this paper we analyze the problem through a less known approach, which justies in a more rigorousand mathematically neat manner, the heuristic arguments of standard treatments of divergencies in QFT's. This paper analyzes the renormalizability ofan arbitrary Scalar Field Theory with interaction Lagrangean L(x) =: 'm(x) : using the method of Epstein-Glaser and techniques of microlocal analysis, inparticular, the concept of scaling degree of a distribution. For a renormalizability proof of perturbative models in the Epstein-Glaser scheme one rst needs to dene an n-fold product of sub-Wick monomials of the interaction Lagrangean. This time ordering is an operator-valued distribution on R4n and the basic issue is its ill-denedness on a null set. The renormalization of a theory in this scheme amounts to the problem of extension of distributions across null sets.