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In this paper, the authors prove global well-posedness of the
massless Maxwell-Dirac equation in the Coulomb gauge on
$\mathbb{R}^{1+d} (d\geq 4)$ for data with small scale-critical
Sobolev norm, as well as modified scattering of the solutions. Main
components of the authors' proof are A) uncovering null structure
of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability
of the underlying covariant Dirac equation. A key step for
achieving both is to exploit (and justify) a deep analogy between
Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous
result was proved earlier by Krieger-Sterbenz-Tataru, which says
that the most difficult part of Maxwell-Dirac takes essentially the
same form as Maxwell-Klein-Gordon.
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