Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations

Abstract

In this monograph we prove that the nonlinear Lie algebra representation given by the manifestly covariant Maxwell-Dirac (M-D) equations is integrable to a global nonlinear representation $U$ of the Poincaré group ${\cal P}_0$ on a differentiable manifold ${\cal U}_\infty$ of small initial conditions for the M-D equations. This solves, in particular, the Cauchy problem for the M-D equations, namely existence of global solutions for initial data in ${\cal U}_\infty$ at $t=0$. The existence of modified wave operators $Ω_+$ and $Ω_-$ and asymptotic completeness is proved. The asymptotic representations $U^{(ε)}_g = Ω^{-1}_ε\circ U_g \circ Ω_ε$, $ε= \pm$, $g \in {\cal P}_0$, turn out to be nonlinear. A cohomological interpretation of the results in the spirit of nonlinear representation theory and its connection to the infrared tail of the electron is given.