The Particle as a Statistical Ensemble of Events in Stueckelberg–Horwitz–Piron Electrodynamics

Abstract

Stueckelberg-Horwitz-Piron (SHP) electrodynamics formalizes the distinction between coordinate time (measured by laboratory clocks) and chronology (temporal ordering) by defining 4D spacetime events xμ as functions of an external evolution parameter τ.  The spacetime evolution of classical events xμ(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.