# Maxwell Electrodynamics in terms of Physical Potentials

## TL;DR

This paper presents a covariant, gauge-invariant formulation of Maxwell's electrodynamics using physical potentials, linking it to spacetime symmetries and demonstrating its consistency with standard field equations and quantum effects.

## Contribution

It introduces a fully relativistic, gauge-invariant potential-based formulation of Maxwell's equations, connecting classical electromagnetism with spacetime symmetries and quantum phenomena.

## Key findings

- Physical potentials satisfy Maxwell equations with Minkowski spacetime symmetry.
- The formulation is consistent with standard Maxwell equations for fields.
- The approach relates to the Aharonov-Bohm effect and the Abelian Higgs model.

## Abstract

A fully relativistically covariant and manifestly gauge invariant formulation of classical Maxwell electrodynamics is presented, purely in terms of gauge invariant potentials without entailing any gauge fixing. We show that the inhomogeneous equations satisfied by the physical scalar and vector potentials (originally discovered by Maxwell) have the same symmetry as the isometry of Minkowski spacetime, thereby reproducing Einstein's incipient approach leading to his discovery of special relativity as a spacetime symmetry. To arrive at this conclusion, we show how the Maxwell equations for the potentials follow from stationary electromagnetism by replacing the Laplacian operator by the d'Alembertian operator, while making all variables dependent on space and time. We also establish consistency of these equations by deriving them from the standard Maxwell equations for the field strengths, showing that there is a unique projection operator which projects onto the physical potentials. Properties of the physical potentials are elaborated through their iterative N\"other coupling to a charged scalar field leading to the Abelian Higgs model, and through a sketch of the Aharonov-Bohm effect, where dependence of the Aharonov-Bohm phase on the physical vector potential is highlighted.