# Mathematics Colloquia and Seminars

We study the Bogolubov-Dirac-Fock (BDF) model, which is a mean-field theory deduced from QED. Contrary to usual Dirac-type theories the associated BDF-functional is bounded from below. Its variables are infinite rank projectors describing the electrons, the observable ones as well as those filling up the Dirac sea. We prove that for any ultra-violet momentum cut-off $\Lambda$ the BDF-functional attains its minimum. The minimizer fulfills a self-consistent equation representing the polarized vacuum. Moreover we show that for $\Lambda$ to infinity the theory gets "nullified" as predicted by Landau. Furthermore the BDF-functional can be minimized under the constraint of a fixed charge q allowing the description of atoms and molecules. Additionally we also state the existence of global-in-time solutions to the associated time-dependent equation. Finally, by means of the thermodaynamic limit for a first principle Hamiltonian on a torus, we derive a link between the BDF-minimizer and the stationary Schwinger-Dyson equations.