Generalized gauge transformation: Abelian case

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 generalized the gauge transformation by using different space-time points in the spirit of Diracʹs point splitting. It is found that such a generalization indeed exists and the resulting generalized infinitesimal gauge transformation takes the form of an infinite series in the coupling constant.