Minimization method for relativistic electrons in a mean-field approximation of quantum electrodynamics

We study a mean-field relativistic model which is able to describe both the behavior of finitely many spin-$1∕2$ particles such as electrons and of the Dirac sea which is self-consistently polarized in the presence of the real particles. The model is derived from the QED Hamiltonian in Coulomb gauge neglecting the photon field. All our results are nonperturbative and mathematically rigorous.

Figure 1

The free vacuum ${\mathcal{P}}_{-}^0$ fills the negative energies of the SCF Fock operator ${\mathcal{D}}^0$.

Figure 2

The polarized vacuum $P_{\mathrm{vac}}$ in the presence of $V$ fills the negative energies of the SCF Fock operator $F_{P_{\mathrm{vac}}-I∕2}$.

Figure 3

Decomposition of the system “$\text{vacuum}+N$ electrons” for the solution $P$ in the $N\text{th}$ charge sector.