Some Variations on Maxwell’s Equations

Abstract

Maxwell’s equations are among the most beautiful in physics, unifying the forces of electricity and magnetism in a classical field theory that explains electromagnetic waves [1]. Some well-known, profoundly-motivated variations on Maxwell’s equations have included the Born-Infeld theory (a nonlinear but Lorentz-covariant modification, that introduces an effective upper bound to the electric field strength), and the Yang-Mills equations (introducing non-Abelian gauge potentials) [2, 3]. These ideas go back many decades, and have deeply influenced the development of theoretical physics. Indeed, there has been a recent resurgence of interest in non-Abelian Born-Infeld Lagrangians [4], which turn out to have important application in string theory and related subjects [5, 6, 7, 8]. More recently, variations of Maxwell’s equations have been considered as “test theories,” with respect to which observations in astrophysics can provide upper bounds to deviations from the usual equations or laws of physics [9, 10]. We nevertheless seek to approach the idea of modifying Maxwell’s equations in new ways with appropriate humility. None of the variations considered in this article is ad hoc. Rather, each occurred in answer to a specific question in fundamental physics.

In the first sections of this article, we discuss two variations on Maxwell’s equations that have been introduced in earlier work—a class of nonlinear Maxwell theories with well-defined Galilean limits (and correspondingly generalized Yang-Mills equations), and a linear modification motivated by the coupling of the electromagnetic potential with a certain nonlinear Schrödinger equation. In the final section, revisiting an old idea of Lorentz, we write Maxwell’s equations for a theory in which the electrostatic force of repulsion between like charges differs fundamentally in magnitude from the electrostatic force of attraction between unlike charges. We elaborate on Lorentz’ description by means of electric and magnetic field strengths, whose governing equations separate into two fully relativistic Maxwell systems—one describing ordinary electromagnetism, and the other describing a universally attractive or repulsive long-range force. If such a force cannot be ruled out a priori by known physical principles, its magnitude should be determined or bounded experimentally. Were it to exist, interesting possibilities go beyond Lorentz’ early conjecture of a relation to (Newtonian) gravity.

It is a pleasure to dedicate this paper to Gérard Emch, whose skeptical perspective helps motivate those who know him to the pursuit of deeper scientific understandings.