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{document}$ \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*} $\end{document}

where \begin{document}$ F_{\mu \nu} = \partial^{\mu} A_{\nu} - \partial^{\nu} A_{\mu} $\end{document}, and \begin{document}$ \alpha^{\mu} $\end{document} are the 4x4 Dirac matrices. We assume the temporal gauge \begin{document}$ A_0 = 0 $\end{document} 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.