  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.
