Some Variations on Maxwell’s Equations

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.