Vector potentials in gauge theories in flat spacetime

Abstract

A recent suggestion that vector potentials in electrodynamics (ED) are nontensorial objects under four-dimensional (4D) frame rotations is found to be both unnecessary and confusing. As traditionally used in ED, a vector potential $A$ always transforms homogeneously under 4D rotations in spacetime, but if the gauge is changed by the rotation, one can restore the gauge back to the original gauge by adding an inhomogeneous term. It is then not a 4-vector, but two 4-vectors: one for rotation and one for translation. For such a gauge, it is much more important to preserve explicit homogeneous Lorentz covariance by simply skipping the troublesome gauge-restoration step. A gauge-independent separation of $A$ into a dynamical term and a nondynamical term in Abelian gauge theories is redefined more generally as the terms caused by the presence and absence, respectively, of the 4-current term in the inhomogeneous Maxwell equations for $A$. Such a separation cannot in general be extended to non-Abelian theories where $A$ satisfies nonlinear differential equations. However, in the linearized iterative solution that is perturbation theory, the usual Abelian quantizations in the usual gauges can be used. Some nonlinear complications are briefly reviewed.