Renormalization and Short Distance Singular Structure

Abstract

The relation between renormalization and short distance singular divergencies in quantum field theory is studied. As a consequence a finite theory is presented. It is shown that these divergencies originate from the multiplication of distributions (and worse-defined mathematical objects). Some of them are eliminated when the multiplication is defined based on dimensional regularization, while others disappear when the states are considered as functionals over the observables space. Nonrenormalizable theories turn to be finite, but anyhow they are endowed with infinite arbitrary constants.