Renormalization of the Vector Current in QED

TL;DR

This paper demonstrates that the electromagnetic current in QED is renormalized, provides a method to compute its renormalization to all orders, and clarifies the correct definition of the electron number operator consistent with gauge invariance.

Contribution

It reveals the non-conservation and renormalization of the electromagnetic current in QED and offers a systematic way to define the electron number operator.

Findings

The electromagnetic current in QED is not conserved and undergoes renormalization.

The current mixes with the divergence of the electromagnetic field tensor.

The correct electron number operator aligns with Gauss's law when the current is properly renormalized.

Abstract

It is commonly asserted that the electromagnetic current is conserved and therefore is not renormalized. Within QED we show (a) that this statement is false, (b) how to obtain the renormalization of the current to all orders of perturbation theory, and (c) how to correctly define an electron number operator. The current mixes with the four-divergence of the electromagnetic field-strength tensor. The true electron number operator is the integral of the time component of the electron number density, but only when the current differs from the MSbar-renormalized current by a definite finite renormalization. This happens in such a way that Gauss's law holds: the charge operator is the surface integral of the electric field at infinity. The theorem extends naturally to any gauge theory.