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) S'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 S'LA, defined as S'LA = F-1 {S'L}, where F-1 denotes the Inverse Fourier Transform. In this manner we effect a dimensional regularization in momentum space (the ring S'L) via convolution, and a product of distributions in the corresponding configuration space (the ring S'LA). This generalizes the results obtained by BG for Euclidean space. We provide several examples of the application of our new results in Quantum Field Theory (QFT). In particular, the convolution of n massless Feynman’s propagators and the convolution of n massless Wheeler’s propagators in Minkowskian space. The results obtained in this work have already allowed us to calculate the classical partition function of Newtonian gravity, for the first time ever, in the Gibbs’ formulation and in the Tsallis’ one. It is our hope that this convolution will allow one to quantize non-renormalizable Quantum Field Theories (QFT’s).