Local well-posedness for the Maxwell-Dirac system in temporal gauge

We consider the low regularity well-posedness problem for the Maxwell-Dirac system in 3+1 dimensions:

$ \begin{align*} \partial^{\mu} F_{\mu \nu} & = - \langle \psi, \alpha_{\nu} \psi \rangle \ \\ -i \alpha^{\mu} \partial_{\mu} \psi & = A_{\mu} \alpha^{\mu} \psi \, , \end{align*} $

where $ F_{\mu \nu} = \partial^{\mu} A_{\nu} - \partial^{\nu} A_{\mu} $, and $ \alpha^{\mu} $ are the 4x4 Dirac matrices. We assume the temporal gauge $ A_0 = 0 $ and make use of the fact that some of the nonlinearities fulfill a null condition. Because we work in the temporal gauge we also apply a method, which was used by Tao for the Yang-Mills system.