Quantum Field Theory, Feynman-, Wheeler Propagators, Dimensional Regularization in Configuration Space and Convolution of Lorentz Invariant Tempered Distributions

M. C. ROCCA; A. PLASTINO

Journal of Physics Communications

IOP PUBLISHING LTD

Lugar: Londres; Año: 2018 vol. 2 p. 115029 - 115039

2399-6528

The Dimensional Regularization (DR) of Bollini and Giambiagi (BG) can not be defined for all Schwartz Tempered DistributionsExplicitly Lorentz Invariant (STDELI) ${cal S}^{´}_L$. In this paper we overcome such limitationand show that it can be generalized to allaforementioned STDELI and obtain a product in a ringwith zero divisors.For this purpose, we resort to a formula obtained by Bollini and Roccaand demonstrate the existenceof the convolution (in Minkowskian space) of suchdistributions. This is done by following a procedure similar tothat used so as to define a general convolution between theUltradistributions of J. Sebastiao e Silva (JSS),also known as Ultrahyperfunctions, obtained by Bollini et al..Using the Inverse FourierTransform we get the ring with zero divisors ${cal S}^{´}_{LA}$,defined as ${cal S}^{´}_{LA}={cal F}^{-1}{{cal S}^{´}_L}$,where ${cal F}^{-1}$ denotes the Inverse Fourier Transform. Inthis manner we effect a dimensional regularization in momentumspace (the ring ${cal S}^{´}_{L}$) via convolution, and a productof distributions in the corresponding configuration space (thering ${cal S}^{´}_{LA})$. This generalizes the results obtainedby BG for Euclidean spaceWe provide several examples of the application of our newresults in Quantum Field Theory (QFT). In particular, the convolution of$n$ massless Feynman´s propagators and the convolution of n masslessWheeler´s propagators in Minkowskianspace.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 quantizenon-renormalizable Quantum Field Theories (QFT´s).