Convolution of Lorentz Invariant Ultradistributions and Field Theory

TL;DR

This paper develops a comprehensive framework for convolving Lorentz invariant ultradistributions in Minkowskian and Euclidean spaces, with applications to quantum field propagators and generalizations of Fourier transform formulas.

Contribution

It introduces a general definition of convolution for Lorentz invariant ultradistributions and applies it to Feynman propagators, extending dimensional regularization to Minkowskian space.

Findings

Exact convolution of two massless Feynman propagators calculated

Derived Fourier transform expressions using modified Bessel distributions

Extended dimensional regularization to Minkowskian space for Green functions

Abstract

In this work, a general definition of convolution between two arbitrary four dimensional Lorentz invariant (fdLi) Tempered Ultradistributions is given, in both: Minkowskian and Euclidean Space (Spherically symmetric tempered ultradistributions). The product of two arbitrary fdLi distributions of exponential type is defined via the convolution of its corresponding Fourier Transforms. Several examples of convolution of two fdLi Tempered Ultradistributions are given. In particular we calculate exactly the convolution of two Feynman's massless propagators. An expression for the Fourier Transform of a Lorentz invariant Tempered Ultradistribution in terms of modified Bessel distributions is obtained in this work (Generalization of Bochner's formula to Minkowskian space). At the same time, and in a previous step used for the deduction of the convolution formula, we obtain the generalization to the Minkowskian space, of the dimensional regularization of the perturbation theory of Green Functions in the Euclidean configuration space given in ref.[12]. As an example we evaluate the convolution of two n-dimensional complex-mass Wheeler's propagators.