Relativistic electrons coupled with Newtonian nuclear dynamics

In the study of the electronic structure of heavy atoms, relativistic effects cannot be neglected and the Dirac operator naturally appears in place of the Schr\"odinger operator, raising a number of additional difficulties. The complexity of these systems has been addressed by various approximations. We consider a model consisting of differential equations coupling the time evolution of a finite number of relativistic electrons with the Newtonian dynamics of finitely many nuclei. The electrons are described by the Bogoliubov-Dirac-Fock (BDF) equation of quantum electrodynamics (QED). This system can be seen as a first step in the study of molecular dynamics phenomena, when both relativistic and quantum effects have to be taken into account for the electrons. We address the Cauchy problem for this model and prove global well-posedness.