Quantum field theory, Feynman-, Wheeler propagators, dimensional regularization in configuration space and convolution of Lorentz Invariant Tempered Distributions

Abstract

The Dimensional Regularization (DR) of Bollini and Giambiagi (BG) can not be defined for all Schwartz Tempered Distributions Explicitly Lorentz Invariant (STDELI) L . In this paper we overcome such limitation and show that it can be generalized to all aforementioned STDELI and obtain a product in a ring with zero divisors. For this purpose, we resort to a formula obtained by Bollini and Rocca and demonstrate the existence of the convolution (in Minkowskian space) of such distributions. This is done by following a procedure similar to that used so as to define a general convolution between the Ultradistributions of J Sebastiao e Silva (JSS), also known as Ultrahyperfunctions, obtained by Bollini et al. Using the Inverse Fourier Transform we get the ring with zero divisors LA , defined as