Convolution of n-dimensional tempered ultradistributions and field theory

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 k0 and ρ, we obtain an expression for the convolution, which is more suitable for practical applications. The product of two arbitrary even (in the variables x0 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 Wheeler's propagators and the convolution of two complex mass Wheeler's propagators.