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.

String Chopping and Time-ordered Products of Linear String-localized Quantum Fields

“…The irreducible sub-representations of U correspond to the particle types described by ϕ. 3 We consider here only the case of bosons, and we exclude explicitly the case of Wigner's massless "infinite spin" particles [19]. It has been shown in [11] that then our string-localized free massive field ϕ(x, e) is of the form…”

“…In this x-space approach, the "UV problem" of divergences consists in the extension across the origin, which is not unique: At every order n one has a certain number of free parameters. If the short-distance scaling dimension of the interaction Lagrangian is not larger than 4, then this number does not increase with the order, and one can fix all free parameters by a finite set of normalization conditions: the model is renormalizable [4], see [3] for a review of the argument.…”