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The Hartree-Fock approximation of Quantum Electrodynamics provides a rigorousframework for the description of relativistic electrons in external fields.This nonlinear model takes into account the infinitely many virtual electronsof Dirac's vacuum as well as the Coulomb interactions between all theparticles. The state of the system is an infinite-rank projection satisfying anonlinear equation. In this paper, we construct solutions to this equation, inthe regime of weak interactions (that is, small coupling constant $\alpha$),and strong external fields (that is, large atomic charge $Z$ such that $\alphaZ:=\kappa$ stays fixed). In this regime, we are able to remove the ultravioletcut-off $\Lambda$ as soon as $\alpha\log\Lambda$ stays fixed. As an applicationof this result, we compare the critical strength $\kappa_c(\alpha)$ of theexternal potential needed to produce an additional particle in the vacuum, when$\alpha=0$ or $\alpha>0$. We prove that$\lim_{\alpha\to0}\kappa_c(\alpha)/\kappa_c(0)> 1$, and we identify the limitexactly. Because of the dielectric behavior of Dirac's vacuum, staticelectron/positron pair production occurs in the interacting case for a strongerfield that in the non-interacting case, which is a mere consequence of chargerenormalization.

Charge renormalization and static electron/positron pair production for a nonlinear Dirac model with weak interactions