Gaussian as Test Functions in Operator Valued Distribution Formulation of QED

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. The present formulation gives a tool to calculate finite scattering matrix without adding counter-terms. 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 field resulting from this approach obeys the Klein-Gordon equation. The vacuum fluctuation calculation then gives a result regularized by the Gaussian factor. The example of scalar quantum electrodynamics theory shows that Gaussian functions may be used as test functions in this approach. The result shows that the scale of the theory is described by a factor arising from Heisenberg uncertainties. The Feynman propagators and a study on loop convergence with the example of the tadpole diagram are given. The formulation is extended to Quantum Electrodynamics. Triangle diagrams anomaly are calculated efficiently. In the same approach, using Lagrange formulae to regularize the singular distribution involved in the scattering amplitude gives a good way to avoid infinities. After applying Lagrange formulae, the test function could be reduced to unity since the amplitude is regular. Ward-Takahashi identity is calculated with this method too and this shows that the symmetries of the theory are unbroken.