Particles and Events in Classical Off-Shell Electrodynamics

TL;DR

This paper examines classical off-shell electrodynamics, addressing its difficulties through a new particle current model, and compares its predictions with standard Maxwell theory, providing insights into the photon mass scale and renormalizability.

Contribution

It introduces a novel particle current model in off-shell electrodynamics that resolves classical difficulties and aligns with Maxwell theory under certain conditions.

Findings

The new model eliminates classical difficulties in off-shell electrodynamics.

Numerical solutions match Maxwell solutions when the time scale parameter exceeds 10^{-6} seconds.

The model offers a natural interpretation for the photon mass cut-off in quantum electrodynamics.

Abstract

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.