SMALL AMPLITUDE SOLITARY WAVES IN THE DIRAC MAXWELL SYSTEM

We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system, proving the existence of solutions in which the Dirac wave function is of the form phi(x, omega)e(-j omega t) with omega is an element of (-rn, omega(*)) for some omega(*) > -m. The solutions satisfy phi(., omega) is an element of H-1(R-3, C-4), and are small amplitude in the sense that parallel to phi(., omega)parallel to(2)(L2) = O(root m vertical bar omega) and parallel to phi(., omega)parallel to(L infinity) = O(m vertical bar omega). The method of proof is an implicit function theorem argument based on the identification of the nonrelativistic limit as the ground state of the Choquard equation. This identification is in some ways unexpected on account of the repulsive nature of the electrostatic interaction between electrons, and arises as a manifestation of certain peculiarities (Klein paradox) which result from attempts to interpret the Dirac equation as a single particle quantum mechanical wave equation.