On the asymptotic analysis of the Dirac-Maxwell system in the nonrelativistic limit

Abstract

We deal with the ``nonrelativistic limit'', i.e. the limit c to infinity, where c is the speed of light, of the nonlinear PDE system obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the ``moving charge'' of the spinor. This limit, sometimes also called ``Post-Newtonian'' limit, yields a Schroedinger-Poisson system, where the spin and the magnetic field no longer appear. However, our splitting of the 4-spinor into two 2-spinors preserves the symmetry of "electrons'' and "positrons''; the latter obeying a Schroedinger equation with ``negative mass'' in the limit. We rigorously prove that in the nonrelativistic limit solutions of the Dirac-Maxwell system converge in the energy space $C([0,T];H^{1})$ to solutions of a Schroedinger-Poisson system, under appropriate (convergence) conditions on the initial data. We also prove that the time interval of existence of local solutions of Dirac-Maxwell is bounded from below by log(c). In fact, for this result we only require uniform $H^{1}$ bounds on the initial data, not convergence. Our key technique is "null form estimates'', extending the work of Klainerman and Machedon and our previous work on the nonrelativistic limit of the Klein-Gordon-Maxwell system.