Maxwell's equations can be represented in a very compact form using the language of differential forms. This representation also makes it easy to see how the theory can be altered, in particular how it can admit, even as a classical theory of electromagnetism, magnetic monopoles.
Maxwell's equations can be represented in an extremely compact, efficient form, using the language of differential forms.
We work in four dimensions and start with a vector field, that is, a one-form $A$. We define a two-form as its exterior derivative:
\begin{align}F={\rm d}A.\tag{1}\end{align}
This two-form is the Maxwell field tensor. The exterior derivative is nilpotent, ${\rm d}^2=0$ identically, therefore we have the identity,
\begin{align}{\rm d}F={\rm d}^2A=0.\tag{2}\end{align}
This expression encompasses two of Maxwell's: Faraday's law and Gauss's law of magnetism. We then use the famous Hodge-dual operator, $\star$ (note that this requires a prior definition of the metric of spacetime, be it the Minkowski metric of empty space or a curved metrical tensor representing gravitation – the formalism works either way) to define the 4-current:
\begin{align}J={\star{\rm d}}{\star F}.\tag{3}\end{align}
This definition amounts to Ampère's law and Gauss's law. The 4-current is conserved, again as a result of the nilpotence of the exterior derivative and the fact that the Hodge-dual satisfies $\star^2=-1$ in four-dimensional Lorentzian spacetime:
\begin{align}{\star{\rm d}}{\star J}={\star{\rm d}}{\star}{\star{\rm d}}{\star F}=-{\star{\rm d}^2}{\star F}=0.\tag{4}\end{align}
These equations are so airtight, they leave seemingly no room to introduce any sensible modification.
That is, until we notice that at two different spots, we introduced ad hoc definitions into the theory. Notably, there is no compelling reason to define the current (3) the way we do; and even to define the Maxwell tensor (1) the way we do.
Redefinition of the current leads to the well-known Proca theory of massive electromagnetism:
\begin{align}J-\mu^2 A={\star{\rm d}}{\star F}.\tag{5}\end{align}
But what about changing the definition of $F$? Following an approach known as the Schwinger-Zwanziger formulation, we may opt to introduce a second potential $C$, but with a twist insofar as the Maxwell tensor is concerned:
\begin{align}F={\rm d}A+{\star ({\rm d}C)}.\tag{6}\end{align}
This formulation obviously works only in four dimensions, where the Hodge-dual of the two-form ${\rm d}C$ is itself a two-form, but hey, our world is four-dimensional, so there.
As a result,
\begin{align}{\rm d}F={\rm d}^2A+{\rm d}{\star ({\rm d}C)}={\rm d}{\star ({\rm d}C)},\tag{7}\end{align}
which is not zero anymore.
What about the 4-current?
\begin{align}J={\star{\rm d}}{\star F}={\star {\rm d}}{\star {\rm d}A}+{\star {\rm d}}{\star {\star ({\rm d}C)}}={\star {\rm d}}{\star {\rm d}A},\tag{8}\end{align}
which remains unchanged. However, we can now also define a magnetic current:
\begin{align}J_m={\star {\rm d}}F={\star {\rm d}^2A}+{\star {\rm d}}{\star ({\rm d}C)}={\star {\rm d}}{\star ({\rm d}C)}. \tag{9}\end{align}
This current depends on $C$ alone. Like the electric current, it is also conserved:
\begin{align}{\star{\rm d}}{\star J_m}={\star {\rm d}}{\star}{\star {\rm d}}{\star ({\rm d}C)}=0.\tag{10}\end{align}
This is the magnetic current.
Finally, some thoughts about classical gauge invariance:
- Maxwell's theory is gauge invariant: the substitution, $A\to A + {\rm d}\phi$ where $\phi$ is a scalar field, leaves the definition of $F$ unchanged.
- Proca's theory is not gauge invariant: The same substitution changes the Proca current (5).
- The dual potential $C$ introduced in the Schwinger-Zwanziger formalism is a gauge-invariant potential.
A most interesting twist, however, is when we introduce an alternative to the Schwinger-Zwanziger formulation, e.g., define the Maxwell tensor as
\begin{align}F = {\rm d}A+A\wedge A.\tag{11}\end{align}
This seemingly breaks gauge invariance. However, upon closer inspection it becomes apparent that we can restore gauge invariance by redefining the derivative operator: ${\rm D}={\rm d}+A$. This reveals that the theory behind (11) is still a gauge invariant theory. The motivation, in this case, could be the interpretation of $A\wedge A$ as the commutator of a Lie algebra. This is the classical route towards Yang-Mills type theories that, in turn, play an essential role in the quantum field theory formulation of the Standard Model of particle physics. But the mechanism is far more general: a similar introduction of a gauge covariant derivative $({\rm D}={\rm d}+ieA)$ is also used in the formulation of quantum electrodynamics.
