Point Splitting and U(1) Gauge Invariance

Abstract

It is well known that, in local quantum field theory, one encounters di­vergences which arise from taking products of field operators at the same space-time point. As a result, these products do not have a well-defined meaning. Quite some time ago, Dirac [] suggested point splitting as a remedy for this difficulty: instead of taking all the field operators at the space-time point x a fixed four-vector e is introduced so that only field operators with different arguments (x,x±ε,...) appear in their products. As long as E is taken to be different from 0, the products of field operators are well defined and the theory can be expected to be free of divergences, i.e., to be regularized. At the end of the calculation, the limit ε → 0 is taken, in order to recover the original theory.