Null structure and almost optimal local well-posedness of the Maxwell-Dirac system

We uncover the full null structure of the Maxwell-Dirac system in Lorenz gauge. This structure, which cannot be seen in the individual component equations, but only when considering the system as a whole, is expressed in terms of tri- and quadrilinear integral forms with cancellations measured by the angles between spatial frequencies. In the 3D case, we prove frequency-localized L^2 space-time estimates for these integral forms at the scale invariant regularity uo to a logarithmic loss, hence we obtain almost optimal local well-posedness of the system by iteration.