Coordinate space representation for renormalization of quantum electrodynamics

We present a systematic treatment, up to order alpha, for the fundamental renormalization of quantum electrodynamics in real space. Although the standard renormalization is an old school problem for this case, it has not been completely done in position space yet. The most difference with well-known differential renormalization is that we do the whole procedure in coordinate space without needing to transform to momentum space. Specially, we directly derive the counterterms in coordinate space. This problem becomes crucial when the translational symmetry of the system breaks somehow explicitly (by nontrivial boundary condition (BC) on the fields). In this case, one is not able to move to momentum space by a simple Fourier transformation. In the context of the renormalized perturbation theory, counterterms in coordinate space will depend directly on the fields BCs (or background topology). Trivial BC or trivial background leads to the usual standard counterterms. If the counterterms are modified, then the quantum corrections of any physical quantity are different from those in free space where we have the translational invariance. We also show that, up to order alpha, our counterterms reduce to the usual standard one.