Two Mathematically Equivalent Versions of Maxwell's Equations

TL;DR

This paper reviews a new classical electrodynamics approach based on two equivalent Maxwell's equations versions, leading to a source-clock fixed framework, altered light speed invariance, and implications for relativistic quantum theory.

Contribution

It introduces a novel version of Maxwell's equations fixing the source clock, resulting in a non-invariant light speed and a new invariance group, advancing classical and quantum relativistic theories.

Findings

Two mathematically equivalent Maxwell's equations versions.

A non-invariant effective speed of light depending on source motion.

Distinct spin-1/2 particle equations: Dirac and square-root forms.

Abstract

This paper is a review of the canonical proper-time approach to relativistic mechanics and classical electrodynamics. The purpose is to provide a physically complete classical background for a new approach to relativistic quantum theory. Here, we first show that there are two versions of Maxwell's equations. The new version fixes the clock of the field source for all inertial observers. However now, the (natural definition of the effective) speed of light is no longer an invariant for all observers, but depends on the motion of the source. This approach allows us to account for radiation reaction without the Lorentz-Dirac equation, self-energy (divergence), advanced potentials or any assumptions about the structure of the source. The theory provides a new invariance group which, in general, is a nonlinear and nonlocal representation of the Lorentz group. This approach also provides a natural (and unique) definition of simultaneity for all observers. The corresponding particle theory is independent of particle number, noninvariant under time reversal (arrow of time), compatible with quantum mechanics and has a corresponding positive definite canonical Hamiltonian associated with the clock of the source. We also provide a brief review of our work on the foundational aspects of the corresponding relativistic quantum theory. Here, we show that the standard square-root and the Dirac equations are actually two distinct spin-$\tfrac{1}{2}$ particle equations.