Some Variations on Maxwell’s Equations

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.