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

TL;DR

This paper proves the global existence, asymptotic completeness, and infrared behavior analysis for solutions of the Maxwell-Dirac equations, establishing a nonlinear Poincaré group representation and addressing the Cauchy problem.

Contribution

It introduces a global nonlinear representation of the Poincaré group for Maxwell-Dirac equations and solves the Cauchy problem with a cohomological interpretation of infrared effects.

Findings

Existence of global solutions for small initial data.

Construction of modified wave operators and proof of asymptotic completeness.

Nonlinear asymptotic representations linked to infrared tail phenomena.

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 Poincar\'e 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 $\Omega_+$ and $\Omega_-$ and asymptotic completeness is proved. The asymptotic representations $U^{(\epsilon)}_g = \Omega^{-1}_\epsilon \circ U_g \circ \Omega_\epsilon$, $\epsilon = \pm$, $g \in {\cal P}_0$, 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.