SMALL AMPLITUDE SOLUTIONS OF THE MAXWELL-DIRAC EQUATIONS

We prove the existence of a global solution of the Cauchy problem for the non-linear Maxwell-Dirac system in 3 + 1-dimensional space-time in Lorentz gauge. The space of initial data consists of functions sufficiently small with respect to suitable Sobolev weighted norms and satisfying the well-known constraint conditions. The essential novelty in the definition of the space of initial data is its explicit form and the possibility to choose initial data with noncompact support. The proof is based on the application of a gauge transformation leading to suitable gauge condition playing the role of the null condition of Klainerman and the "compatible form" condition of Bachelot.