  Despite the many successes of the relativistic quantum theory developed by
Horwitz, et. al., certain difficulties persist in the associated covariant
classical mechanics. In this paper, we explore these difficulties through an
examination of the classical Coulomb problem in the framework of off-shell
electrodynamics. As the local gauge theory of a covariant quantum mechanics
with evolution parameter $\tau$, off-shell electrodynamics constitutes a
dynamical theory of spacetime events, interacting through five $\tau$-dependent
pre-Maxwell potentials. We present a straightforward solution of the classical
equations of motion, which is seen to be unsatisfactory, and reveals the
essential difficulties in the formalism at the classical level. We then offer a
new model of the particle current -- as a certain distribution of the event
currents on the worldline -- which eliminates these difficulties and permits
comparison of classical off-shell electrodynamics with the standard Maxwell
theory. In this model, the ``fixed'' event induces a Yukawa-type potential,
permitting a semi-classical identification of the pre-Maxwell time scale
$\lambda$ with the inverse mass of the intervening photon. Numerical solutions
to the equations of motion are compared with the standard Maxwell solutions,
and are seen to coincide when $\lambda \gtrsim 10^{-6}$ seconds, providing an
initial estimate of this parameter. It is also demonstrated that the proposed
model provides a natural interpretation for the photon mass cut-off required
for the renormalizability of the off-shell quantum electrodynamics.
