Given a pseudo-Riemannian $n$-manifold and a $k$-form $F$ on the manifold, I will call its exterior derivative $J=dF$ the source of $F$ and the differential $K=dG$ the dual source of $F$, where $G={\star}F$ is the Hodge dual of $F$. Since $d\circ d=0$, the conservation laws $dJ=0$ and $dK=0$ follow. (EDIT: This is an obvious generalization of Maxwell's equations to arbitrary $n$, $k$ and metric tensor.)
Specialized to $n=4$ and $k=2$, and identifying $F$ as the electromagnetic 2-form (or rather, its dual), $J$ as the electric current 3-form and $K$ as the magnetic current 3-form, I have arrived at Maxwell's equations in general relativity as necessarily applying, including the associated conservation of electric and magnetic charge:
- $J=dF$ (the electric current 3-form is the source of the field)
- $K=d{\star}F$ (the magnetic current is the dual source of the field, though we observe that $K=0$)
- $dJ=0$ (conservation of electric charge)
- $dK=0$ (conservation of magnetic charge)
using standard mathematics from minimal assumptions:
- We assume a classical setting, as for general relativity, but generalized to any number of dimensions $n$.
- We assume that the field of interest can be described as a $k$-form.
My approach involves defining what the "source" and "dual source" of a field are, and then to show that these are what we call "charge", rather than the other way around. Assuming that I have not made an error, the answer to the question in the title is yes (with a field's source being a $(k+1)$-form, and its dual source an $(n-k+1)$-form. My question would then be:
Why do we learn about electromagnetism as the consequence of innumerable observations (four different observational laws), rather than as the only possible form of the law aside from the non-existence of magnetic charge once we establish that the electromagnetic field is a $2$-form (which presumably follows from its transformation law established by measurement of the Lorentz force and special relativity)?
I have not needed to invoke anything like gauge theory, potentials, Lagrangians, symmetry – only standard theorems of differential geometry and Hodge theory.