Brown-Ravenhall model) requires that the electron (positron) be defined in terms of the positive (negative) spectral subspace of the Dirac operator [*with*]{} the magnetic vector potential $A(x)$, instead of the free Dirac operator without $A(x)$. This observation, that perturbation theory, if there is one, must start from the dressed electrons rather than the electrons unclothed by its magnetic field, might ultimately be important in a non-perturbative QED.

The first, humble step is to understand electrons that interact with the radiation field but which are free otherwise. In order for this model to make sense an ultraviolet cutoff has to be imposed that limits the energy of photon modes. The simplest question, which is the one we address in this paper, is the behavior of the self-energy of the electron as the cutoff tends to infinity (with the bare mass of the electron fixed). The self-energy of the electron diverges as the cutoff tends to infinity and it has to be subtracted for each electron in any interacting theory. The total energy will still depend strongly on the cutoff because of the interactions. This dependence will, hopefully, enter through an effective mass which will be set equal to the physical mass 