Particles and events in classical off-shell 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 τ , off-shell electrodynamics constitutes a dynamical theory of spacetime events, interacting through five τdependent pre-Maxwell potentials. We present a straightforward solution of the classical equations of motion, for a test event traversing the field induced by a "fixed" event (an event moving uniformly along the time axis at a fixed point in space). This solution 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 λ 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 λ > 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.

Green functions for wave propagation on a five-dimensional manifold and the associated gauge fields generated by a uniformly moving point source

“…In [12], M.C. Land studied the equations of motion of a test particle in a field with similar singular support behavior.…”

Electrostatics in Stueckelberg-Horwitz electrodynamics

“…Since the τ-translation invariance of the theory leaves the Coulomb interaction invariant under shift of the worldline origin x (τ) = (τ, 0) → (τ + δτ, 0), it was suggested in [15] to relax the synchronization of interacting events by taking the induced current to be…”

“…Because the formalism contains a 4-vector potential and a fifth scalar potential, it provides a framework for relativistic generalizations of the classical central force problems [10,11,12,13,14]. Possible experimental signatures have been found in classical and quantum scattering [15,16]. Nevertheless, relatively little work has been done in the areas of classical electrostatics and electrodynamics that first led to the Maxwell theory.…”

Discrete Symmetries of Off-Shell Electromagnetism

“…where j α is given in (19) and (20). Although λ and e 0 must be dimensional constants, the dimensionless ratio e 0 /λ is the Maxwell charge e. In four-vector component form, (24) becomes…”

“…Third, the τ -translation symmetry of the asymptotic events leads to a strong dependence on initial conditions that, under concatenation, should not be observable; that is, scattering in the gauge field induced by the event x µ (τ − a) will depend qualitatively on the value of a, even though the concatenated Coulomb field does not. A reasonable approach was found [20] by smoothing the field-inducing current as…”