Gaussian as test functions in Operator Valued Distribution formulation of QED

TL;DR

This paper demonstrates that Gaussian functions can serve as test functions within the operator valued distribution formalism of QED, leading to divergence-free calculations and new insights into quantum field regularization.

Contribution

It introduces the use of Gaussian test functions in the OPVD formalism for QED, providing a divergence-free approach and analyzing key quantum phenomena.

Findings

Gaussian functions effectively regularize loop integrals

The method reproduces known QED anomalies and identities

Loop convergence is improved with Gaussian test functions

Abstract

As shown by Epstein and Glaser, the operator valued distribution (OPVD) formalism permits to obtain a non-standard regularization scheme which leads to a divergences-free quantum field theory. We show, with the example of a scalar quantum electrodynamics theory, that Gaussian functions may be used as test functions in this approach. After a short recall about the OPVD formalism in 3+1-dimensions, Gaussian functions and Harmonic Hermite-Gaussian functions are used as test functions. The vacuum fluctuation, Feynman propagators and a study about loop convergence with the example of the tadpole diagram are given. The approach is extended to Quantum Electrodynamics. Calculations concerning triangle anomaly and Ward-Takahashi identity are performed in the framework of the method.