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

Abstract

We consider the low regularity well-posedness problem for the Maxwell-Dirac system in n+1 dimensions in the cases $n=3$ and $n=2$ : \begin{align*} \partial^μ F_{μν} & = - \langle ψ,α_ν ψ\rangle \\ -i α^μ \partial_μ ψ& = A_μ α^μ ψ\, , \end{align*} where $F_{\mu \nu }={\partial }^{\mu }{A}_{\nu }-{\partial }^{\nu }{A}_{\mu }$ , and ${\alpha }^{\mu }$ are the 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 in this gauge.

14 pages. This version is extended to cover the case of two space dimensions as well