Current as statistical ensemble of events in Stueckelberg-Horwitz-Piron Electrodynamics

Stueckelberg-Horwitz-Piron (SHP) electrodynamics formalizes the distinction between coordinate time (measured by laboratory clocks) and chronology (temporal ordering) by defining 4D spacetime events x^μ as functions of an external evolution parameter τ.  The spacetime evolution of classical events x^μ(τ), as τ grows monotonically, trace out particle worldlines dynamically and induce the five U(1) gauge potentials through which events interact.  Since Lorentz invariance imposes time reversal symmetry on x⁰ but not τ, the formalism resolves grandfather paradoxes and related problems of irreversibility.  Nevertheless, the causal structure of the 5D Green's function introduces singularities in the τ-dependence of the induced fields that must be treated with care for classical interactions.  These singularities are regularized by generalizing the action to include a non-local kinetic term for the fields.  The resulting theory remains gauge and Lorentz invariant, and the related QFT becomes super-renormalizable.  The field equations are Maxwell-like but τ-dependent and sourced by a current that represents a statistical ensemble of events distributed along the worldline.  The width of the distribution defines a mass spectrum for the photons that carry the interaction.  As the width becomes very large, the photon mass goes to zero and the field equations become τ-independent Maxwell's equations.  Maxwell theory thus emerges as an equilibrium limit of SHP.  Particles and fields can exchange mass in the SHP theory, however on-shell particle mass is restored through self-interaction.