Understanding the electromagnetic 4-potential in the tetrad bundle

TL;DR

This paper investigates the electromagnetic 4-potential's dependence on local reference frames, proposing a tetrad bundle approach to achieve gauge invariance and Lorentz covariance in describing electromagnetic fields.

Contribution

It introduces a tetrad bundle framework to reformulate the electromagnetic 4-potential, ensuring gauge invariance and Lorentz covariance by considering local reference frame transformations.

Findings

Electromagnetic 4-potential depends on local reference frames.

Tetrad bundle approach restores Lorentz covariance.

Electromagnetic vector satisfies Coulomb gauge in Lorentz transformations.

Abstract

Separation of the spin and orbital angular momenta of the electromagnetic field has been discussed frequently in recent years. The spin and orbital angular momenta cannot be made simultaneously gauge invariant and Lorentz covariant and are not conserved separately. After analyzing the source of the problem, we find that the electromagnetic 4-potential depends on the local reference frame instead of the global reference frame. The transformation of the local reference frame is the intrinsic degree of freedom of the electromagnetic field. Therefore, considering only the Lorentz transformation of the global reference frame and neglecting the Lorentz transformation of the local reference frame may lead to the noncovariance of the electromagnetic 4-potential. Accordingly, we redescribe these difficulties of the electromagnetic field from the perspective of quantum field theory. By using the behavior of the electromagnetic 4-potential that satisfies the Coulomb gauge in Lorentz coordinate transformation, we can construct the electromagnetic vector in the tetrad bundle. The various physical quantities that are induced by this electromagnetic vector satisfy Lorentz covariance in the tetrad bundle. This electromagnetic vector, which is projected onto space-time, is an electromagnetic 4-potential that satisfies the Coulomb gauge; thus, the electromagnetic vector is gauge invariant.