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

Lucas Tavares Cardoso
July 19, 2017
The renormalizability of QFT’s is a vastly studied issue, and particularly the results concerning a scalar field theory are well-known through the traditional renormalization approach in the literature. However, in this paper we analyze the problem through a less known approach, which justifies in a more rigorous and mathematically neat manner, the heuristic arguments of standard treatments of divergencies in QFT’s. This paper analyzes the renormalizability of an arbitrary scalar field theory with interaction Lagrangean L(x) =:\varphi ^m (x): using the method of Epstein-Glaser and techniques of microlocal analysis, in particular, the concept of scaling degree of a distribution. For a renormalizability proof of perturbative models in the Epstein-Glaser scheme one first needs to define an n -fold product of sub-Wick monomials of the interaction Lagrangean. This time ordering is an operator-valued distribution on R^{4n} and the basic issue is its ill-definedness on a null set. The renormalization of a theory in this scheme amounts to the problem of extension of distributions across null sets.