Higher-Order Kinetic Term for Controlling Photon Mass in Off-Shell   Electrodynamics
  In a relativistic classical and quantum mechanics with Poincare-invariant parameter, particle worldlines are traced out by the evolution of spacetime events. In pre-Maxwell electrodynamics -- the local gauge theory associated with this framework -- events induce five local off-shell fields, which mediate interactions between instantaneous events, not between the worldlines which represent entire particle histories. The fifth field, required to compensate for dependence of gauge transformations on the evolution parameter, enables the exchange of mass between particles and fields. In the equilibrium limit, these pre-Maxwell fields are pushed onto the zero-mass shell, but during interactions there is no mechanism regulating the mass that photons may acquire, even when event trajectories evolve far into the spacelike region. This feature of the off-shell formalism requires the application of some ad hoc mechanism for controlling the photon mass in low energy classical Coulomb scattering of charged events, and in the renormalization of off-shell quantum electrodynamics. We discuss a nonlocal, higher derivative correction to the photon kinetic term, which provides regulation of the photon mass in a manner that preserves the gauge invariance and Poincare covariance of the original theory. We demonstrate that the inclusion of this term is equivalent to an earlier solution to the classical Coulomb problem, and that the resulting quantum field theory is renormalized. 
Martin Land