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

Abstract

The Dimensional Regularization of Bollini and Giambiags (Phys. Lett. {\bf B 40}, 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 Distributions Explicitly Lorentz Invariant (STDELI) ${\mathcal{S}}_L^{{}^{\prime }}$. In this paper we overcome here 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 in [Int. J. of Theor. Phys. {\bf 43}, 1019 (2004)] 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 [Math. Ann. {\bf 136}, 38 (1958)], also known as Ultrahyperfunctions, obtained by Bollini et al. [Int. J. of Theor. Phys. {\bf 38}, 2315 (1999), {\bf 43}, 1019 (2004), {\bf 43}, 59 (2004),{\bf 46}, 3030 (2007)]. Using the Inverse Fourier Transform we get the ring with zero divisors ${\mathcal{S}}_{LA}^{{}^{\prime }}$, defined as ${\mathcal{S}}_{LA}^{{}^{\prime }}={\mathcal{F}}^{-1}\{{\mathcal{S}}_L^{{}^{\prime }}\}$, where ${\mathcal{F}}^{-1}$ denotes the Inverse Fourier Transform. In this manner we effect a dimensional regularization in momentum space (the ring ${\mathcal{S}}_L^{{}^{\prime }}$) via convolution, and a product of distributions in the corresponding configuration space (the ring ${\mathcal{S}}_{LA}^{{}^{\prime }})$. This generalizes the results obtained by Bollini and Giambiagi for Euclidean space in [Phys. Rev. {\bf D 53}, 5761 (1996)]. We provide several examples of the application of our new results in Quantum Field Theory. In particular, the convolution of $n$ massless Feynman propagators and the convolution of n massless Wheeler propagators in Minkowskian space.

26 pages. No figures.Text has changed. Contents index added. Open access

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