I have claimed now for years that the true equations of electromagnetism result from taking the square root of the wave equation on differential forms on a four sphere with radius where is Planck´s constant. If is an embedded three dimensional manifold with a local orthonormal coframe with defining , and if is the Dirac operator on forms on the ambient four-sphere and then the restriction of the square root of the wave equation is the square root of the operator .
If we are able to fully connect the observed electromagnetism with this picture, we would have given justification for the naturality of electromagnetism as a feature of the shape and other attributes of our universe and remove theories that posit any supernatural designer of the universe. An eternal stationary (unless changes) universe cannot have been created by a creator-God either.
The analysis here of the differential form version of Maxwell´s equations can be used to fill the link between the claims in the last paragraph and the way in which Maxwell´s equations are considered in actual use.