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The Dimensional Regularization of Bollini and Giambiags (Phys. Lett. {\bf B40}, 566 (1972), Il Nuovo Cim. {\bf B 12}, 20 (1972). Phys. Rev. {\bf D 53},5761 (1996)) can not be defined for all Schwartz Tempered DistributionsExplicitly Lorentz Invariant (STDELI) ${\cal S}^{'}_L$. In this paper weovercome here such limitation and show that it can be generalized to allaforementioned STDELI and obtain a product in a ring with zero divisors.  For this purpose, we resort to a formula obtained in [Int. J. of Theor. Phys.{\bf 43}, 1019 (2004)] and demonstrate the existence of the convolution (inMinkowskian space) of such distributions. This is done by following a proceduresimilar to that used so as to define a general convolution between theUltradistributions of J. Sebastiao e Silva [Math. Ann. {\bf 136}, 38 (1958)],also known as Ultrahyperfunctions, obtained by Bollini et al. [Int. J. ofTheor. Phys. {\bf 38}, 2315 (1999), {\bf 43}, 1019 (2004), {\bf 43}, 59(2004),{\bf 46}, 3030 (2007)]. Using the Inverse Fourier Transform we get thering with zero divisors ${\cal S}^{'}_{LA}$, defined as ${\calS}^{'}_{LA}={\cal F}^{-1}\{{\cal S}^{'}_L\}$, where ${\cal F}^{-1}$ denotes theInverse Fourier Transform. In this manner we effect a dimensionalregularization in momentum space (the ring ${\cal S}^{'}_{L}$) via convolution,and a product of distributions in the corresponding configuration space (thering ${\cal S}^{'}_{LA})$. This generalizes the results obtained by Bollini andGiambiagi for Euclidean space in [Phys. Rev. {\bf D 53}, 5761 (1996)].  We provide several examples of the application of our new results in QuantumField Theory. In particular, the convolution of $n$ massless Feynmanpropagators and the convolution of n massless Wheeler propagators inMinkowskian space.

journal of physics communications

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