Convolution of n-dimensional Tempered Ultradistributions and Field Theory

TL;DR

This paper defines a general convolution for n-dimensional Tempered Ultradistributions, extending existing concepts, and applies it to quantum field theory Green functions, including practical examples like Wheeler's propagators.

Contribution

It introduces a new general convolution definition for Tempered Ultradistributions applicable in multiple dimensions and quantum field theory, with explicit formulas for practical cases.

Findings

Convolution of two massless Wheeler's propagators calculated.

Convolution of two complex mass Wheeler's propagators derived.

New framework for singular products of Green Functions in QFT established.

Abstract

In this work, a general definition of convolution between two arbitrary Tempered Ultradistributions is given. When one of the Tempered Ultradistributions is rapidly decreasing this definition coincides with the definition of J. Sebastiao e Silva. In the four-dimensional case, when the Tempered Ultradistributions are even in the variables $k^0$ and $\rho$ (see Section 5) we obtain an expression for the convolution, which is more suitable for practical applications. The product of two arbitrary even (in the variables $x^0$ and $r$) four dimensional distributions of exponential type is defined via the convolution of its corresponding Fourier Transforms. With this definition of convolution, we treat the problem of singular products of Green Functions in Quantum Field Theory. (For Renormalizable as well as for Nonrenormalizable Theories). Several examples of convolution of two Tempered Ultradistributions are given. In particular we calculate the convolution of two massless Wheeeler's propagators and the convolution of two complex mass Wheeler's propagators.