  More than twenty years have passed since the threads of the `proper time
formalism' in covariant classical and quantum mechanics were brought together
to construct a canonical formalism for the relativistic mechanics of many
particles. Drawing on the work of Fock, Stueckelberg, Nambu, Schwinger, and
Feynman, the formalism was raised from the status of a purely formal
mathematical technique to a covariant evolution theory for interacting
particles. In the context of this theory, solutions have been found for the
relativistic bound state problem, classical and quantum scattering in
relativistic potentials, as well as applications in statistical mechanics.
  It has been shown that a generalization of the Maxwell theory is required in
order that the electromagnetic interaction be well-posed in the theory. The
resulting theory of electromagnetism involves a fifth gauge field introduced to
compensate for the dependence of the gauge transformation on the invariant time
parameter; permitting such dependence relaxes the requirement that individual
particles be on fixed mass shells and allows exchange of mass during
scattering. In this paper, we develop the quantum field theory of off-shell
electromagnetism, and use it to calculate certain elementary processes,
including Compton scattering and M{\o}ller scattering. These calculations lead
to {\em qualitative} deviations from the usual scattering cross-sections, which
are, however, small effects, but may be visible at small angles near the
forward direction. The familiar IR divergence of the M{\o}ller scattering is,
moreover, completely regularized.
