Vector potentials in gauge theories in flat spacetime

TL;DR

The paper clarifies the transformation properties of vector potentials in electrodynamics, emphasizing the importance of gauge choices and Lorentz covariance, and discusses the limitations of extending these concepts to non-Abelian gauge theories.

Contribution

It refutes the idea that vector potentials are non-tensorial under 4D rotations and redefines gauge-independent separation of potentials, highlighting differences between Abelian and non-Abelian theories.

Findings

Vector potentials transform homogeneously under 4D rotations in electrodynamics.

Gauge-independent separation of A into dynamical and non-dynamical parts is redefined.

Abelian quantizations are valid in linearized perturbation theory despite nonlinear complications.

Abstract

A recent suggestion that vector potentials in electrodynamics (ED) are nontensorial objects under 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: one for rotation and one for translation. For such a gauge, it is much more important to preserve {\it explicit} homogeneous Lorentz covariance by simply skipping the troublesome gauge-restoration step. A gauge-independent separation of $A$ into a dynamical term and a non-dynamical term in Abelian gauge theories is re-defined 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 {\it 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.