In Gauss's law for electricity it is required to consider a closed surface, which $\mathbb{R}^n$ is not. Therefore if a uniform charge distribution $\rho$ occupies all of space, the classical theory for electromagnetism can't be applied to obtain the electric field. Intuitively, this electric field must have a value of zero.
I understand that quantum electrodynamics is a more complete theory to describe electromagnetism, therefore I thought that it might be possible to prove with it that the electric field is zero for a uniform charge distribution through all of space. From what I understand, in quantum mechanics charge is in a continuum rather than being discrete. Therefore it is more fitting for the uniform charge distribution. The question therefore is: Is the electric field produced by $\rho$ zero in all $\mathbb{R}^n$ using quantum electrodynamics ?
P.S. My motivation for this question is that there is also an issue in newtonian gravity by using a uniform distribution of mass through all of space, and can only be understood by using GR. I thought that it would be very elegant if the same issue occured in electromagnetism where quantum mechanics must be involved, since GR and quantum mechanics are famously complicated to join.
