Off-Shell Quantum Electrodynamics

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.