Maxwell's equations are elementary geometric identities that apply to all thrice differentiable vector fields. This statement appears to be at odds with Proca's equation of a massive vector field, and with the concept of magnetic monopoles. This brief note explains how Proca's equation can be obtained by redefining the four-current, whereas the existence of monopoles is a consequence of topological defects.

Maxwell's equations of the electromagnetic field are, in fact, identities that apply to any thrice differentiable vector field on a smooth manifold. This statement is seen most easily using the language of differential forms. We begin with a real-valued one-form $A$ (i.e., a vector field). This field is none other than the electromagnetic four-potential, comprising the electrostatic scalar potential and the magnetic vector potential.

Forming the exterior derivative of $A$, we obtain the electromagnetic field tensor, a real-valued two-form:


As it is well known, if $F$ is written in component form in four dimensions, its six independent components are the components of the three-dimensional electric field and magnetic field vectors $\vec{E}$ and $\vec{B}$.

A basic property of the exterior derivative is that it is nilpotent, i.e., ${\rm d}^2=0$. (This can be proven easily using the explicit definition of the derivative operator, but it also follows from the fact that the algebra of differential forms is a Grassman-algebra.) From this, the following equation trivially follows:

\[{\rm d}F=0.\]

Written out in explicit form, this equation splits into two of Maxwell's equations: Faraday's law and Gauss's law for magnetism.

It is also possible to form the codifferential of $F$. Up to sign1, it reads:

\[J=\star{\rm d}{\star{F}}=\star{\rm d}G,\]

where $\star$ represents the Hodge dual of a differential form. (Note that definition of the Hodge dual requires a metric. Alternatively, we could treat $F$ and $G$—corresponding to the three-vector fields $\vec{E}$, $\vec{B}$, and $\vec{D}$, $\vec{H}$, respectively—as principal quantities and forego the requirement to define a metric. In this case, however, ${\rm d}F=0$ must be imposed, as it is no longer an identity that is automatically satisfied.) In four dimensions (and only in four dimensions!), the codifferential of the two-form $F$ is a one-form, i.e., another vector field: it is called the four-current, and the defining equation, when written out in component form, splits into the remaining two of Maxwell's equations, Gauss's law and Ampère's law.

The codifferential is also nilpotent (${\star{\rm d}\star}{\star{\rm d}\star}={\star{\rm d}}^2\star=0$), hence

\[{\star{\rm d}}{\star J}=0,\]

which is the equation of charge and current conservation.

As the above results are all trivially obtainable identities that apply to any thrice differentiable vector field $A$ on an arbitrary (curved!) smooth (or at least thrice diffentiable) manifold, there seems to be little room to generalize Maxwell's equations to the case of massive fields, as was done by Alexandru Proca. It also doesn't appear possible to introduce magnetic monopoles: their non-existence seems to be an assured fact.

Yet appearances can be deceiving. To introduce Proca's theory into the picture, all one must do is to recognize that the definition of $J$ represents a somewhat arbitrary choice. We are, in fact, free to define $J$ differently, for instance by

\[J-\mu^2A={\star{\rm d}}{\star F}.\]

This is just Proca's equation for a massive vector field with mass $\mu$. (If the Lorenz gauge is used (${\star{\rm d}}{\star A}=0$), this is just $\nabla^2A+\mu^2A=J$, and we have the theory of a massive vector field with conserved current.)

So what about magnetic monopoles? Again, the trivial identities obtained above seem to preclude their existence. However, we must not forget the underlying assumption that the vector field $A$ was everywhere thrice differentiable. If it isn't, the picture changes. In particular, Dirac's magnetic monopole can exist precisely because the vector field $A$ is not differentiable along the Dirac string even though $F$ and $G$ remain both continuous and differentiable along it. Such topological defects in the vector field or the underlying manifold can lead to many interesting consequences.

1A sign ambiguity is introduced by the signature of the metric, but this is not relevant to the present discussion.