  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.
