Does existence of magnetic monopole break covariant form of Maxwell’s equations for potentials?

Question

Absence of magnetic charges is reflected in one of Maxwell's fundamental equations: $$\operatorname{div} \vec B = 0. \tag1$$ This equation allows us to introducte concept of vector potential: $$\vec B = \operatorname{rot} \vec A.$$ Using this concept, it is possible to express Maxwell's equations in a graceful symmetric form: $$\nabla^2 \vec A - \frac{1}{c^2}\frac{\partial^2 \vec A}{\partial t^2} = - \frac{\vec j}{\epsilon_0 c^2}, \tag2$$ $$ \nabla^2 \phi -\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = - \frac{\rho}{\epsilon_0}. \tag3 $$

Noticing, that vector $\vec A$ and scalar $\phi$ potentials, as well as electric current density $\vec j$ and charge density $\rho$, form a 4-vector in Minkovsky space-time. Therefore, Maxwell's equations can be expressed in a covariant form, using d'Alembertian: $$\nabla_{\mu}\nabla^{\mu} A_{\nu} = \frac{j_{\nu}}{\epsilon_0}. \tag4$$

If magnetic monopols exist, Maxwell's equation $(1)$ will look as: $$\operatorname{div} \vec B = \mu_0 c \rho_{\mathrm{magnet}}.$$

As the divergence of $\vec{B}$ isn't equal to zero, it impossible to introduce concept of vector potential. Thus, the equation in the form of $(4)$ will not be possible to express.

Answer 1

Another option, besides modifying the potential $A_\mu = (A_i, \phi)$ in some way, is to introduce another 4-potential $C_\mu = (C_i, \psi)$. Then the electric and magnetic field are given by $$E = - \nabla \times C - \frac{\partial A}{\partial t} - \nabla \phi$$ $$B = \nabla \times A - \frac{\partial C}{\partial t} - \nabla \psi$$

More on this 2-potential approach can be found here: http://arxiv.org/abs/math-ph/0203043

Answer 2

Yes, introducing a magnetic monopole into Maxwell’s equations means the existence of a vector potential that is defined everywhere and everywhere continuous is not possible anymore. In particular, this can be annoying because the vector potential representation is crucial to the prediction of the quantized value for a magnetic charge (both in the way it was historically developed, and in the way it is usually presented).

Even though the existence of the magnetic monopole hasn't been asserted, there has been quite a lot of research going on about how to go around this issue. Most formulations actually stick to some form of a vector potential, as it is the existing framework and is very convenient. This implies that such a monopole-compatible vector potential becomes a less straightforward beast. I'm not sure what is considered the best authority on this, but Wikipedia says on the issue (and it matches my own understanding):

In the mathematical theory of magnetic monopoles, A is allowed to be either undefined or multiple-valued in some places.

The topic was first approached by Dirac, and his position summarized here:

Dirac's reasoning shows that it is consistent in quantum mechanics to describe a magnetic monopole with the vector potential Equation 3, even though it has a "string" singularity