Nonrelativistic limit of Klein-Gordon-Maxwell to Schrodinger-Poisson

We prove that in the nonrelativistic limit c --> infinity, where c is the speed of light, solutions of the Klein-Gordon-Maxwell system on R1+3 converge in the energy space C([0, T]; H-1) to solutions of a Schrodinger-Poisson system, under appropriate conditions on the initial data. This requires the splitting of the scalar Klein-Gordon field into a sum of two fields, corresponding, in the physical interpretation, to electrons and positrons. The proof relies on bilinear spacetime estimates related to the Klainerman-Machedon estimates, but taking into account the variation of the parameter c. A crucial fact is that the system has a null form structure in Coulomb gauge, as proved by Klainerman-Machedon.