Convolution of Ultradistributions, Field Theory, Lorentz Invariance and Resonances

TL;DR

This paper develops a comprehensive framework for convolving ultradistributions of exponential type, extending Fourier analysis and regularization techniques, with applications to resonance normalization in quantum mechanics.

Contribution

It introduces a general definition of convolution for ultradistributions of exponential type and extends dimensional regularization to these distributions in Euclidean and Minkowskian spaces.

Findings

Defined convolution for arbitrary ultradistributions of exponential type

Derived Fourier transforms of spherically symmetric and Lorentz invariant ultradistributions

Provided solutions for resonance normalization in quantum mechanics

Abstract

In this work, a general definition of convolution between two arbitrary Ultradistributions of Exponential type (UET) is given. The product of two arbitrary UET is defined via the convolution of its corresponding Fourier Transforms. Some examples of convolution of two UET are given. Expressions for the Fourier Transform of spherically symmetric (in Euclidean space) and Lorentz invariant (in Minkowskian space) UET in term of modified Bessel distributions are obtained (Generalization of Bochner's theorem). The generalization to UET of dimensional regularization in configuration space is obtained in both, Euclidean and Minkowskian spaces As an application of our formalism, we give a solution to the question of normalization of resonances in Quantum Mechanics. General formulae for convolution of even, spherically symmetric and Lorentz invariant UET are obtained and several examples of application are given.