Continuous non-perturbative regularization of QED

TL;DR

This paper introduces a non-perturbative, gauge-invariant regularization method for QED using a continuous, non-local action constructed via regularized delta functions, addressing divergences while maintaining gauge symmetry.

Contribution

It presents a novel continuous non-perturbative regularization scheme for QED that preserves gauge invariance and analyzes the associated divergences and obstructions.

Findings

The regularized theory exhibits a quadratic divergence in the polarization operator.

The divergence can be removed by redefining the regularized action.

An obstruction exists to constructing a fully continuous, non-ambiguous regularization in four dimensions.

Abstract

We regularize in a continuous manner the path integral of QED by construction of a non-local version of its action by means of a regularized form of Dirac's $\delta$ functions. Since the action and the measure are both invariant under the gauge group, this regularization scheme is intrinsically non-perturbative. Despite the fact that the non-local action converges formally to the local one as the cutoff goes to infinity, the regularized theory keeps trace of the non-locality through the appearance of a quadratic divergence in the transverse part of the polarization operator. This term which is uniquely defined by the choice of the cutoff functions can be removed by a redefinition of the regularized action. We notice that as for chiral fermions on the lattice, there is an obstruction to construct a continuous and non ambiguous regularization in four dimensions. With the help of the regularized equations of motion, we calculate the one particle irreducible functions which are known to be divergent by naive power counting at the one loop order.