Point Splitting and U(1) Gauge Invariance

TL;DR

This paper explores a generalized approach to gauge transformations in quantum electrodynamics using point splitting, addressing the issue of ill-defined products of field operators at the same point.

Contribution

It introduces a method of generalizing gauge transformations via point splitting, providing explicit examples and formulating the transformations as infinite series.

Findings

Generalized gauge transformations can be constructed using point splitting.

The resulting transformations are expressed as infinite series in the coupling constant.

This approach offers a way to handle ill-defined products in quantum electrodynamics.

Abstract

A gauge transformation in quantum electrodynamics involves the product of field operators at the same space-time point and hence does not have a well-defined meaning. One way to avoid this difficulty is to generalize the gauge transformation by using different space-time points in the spirit of Dirac's point splitting. Such a generalization indeed exists and the resulting infinitesimal gauge transformation takes the form of an infinite series in the coupling constant. In this text I will present two examples of generalized gauge transformations.