What does a magnetic monopole field "look like"?

Question

If magnetic monopoles exist, they are predicted to have large charges--or equivalently, a large coupling constant, which means that perturbative models don't converge.

While I get that that makes it very difficult to calculate things like the binding energy of monopolium, is it at least possible to make qualitative statements about what magnetostatics, the strongly-coupled analog of electrostatics, looks like? E.g., can you still reasonably approximate the strength of a monopole field as $\frac{1}{r^2}$ down to arbitrarily small lengths?

Answer 1

The magnetic monopole acts exactly like an electric monopole. In magnetostatics, the equation of interest

($\mu_0=1$):

$$\nabla \cdot \vec{B} = \rho$$

where $\rho$ is the magnetic charge density, has a solution:

$$\vec{B} = \frac{1}{4\pi}\frac{q}{r^2}$$

for point particles, i.e: $\rho = \delta(r)$, and $q$ is the magnetic charge. In fact, this is exactly what we want. We designed the magnetic monopole to act like a electric monopole because we saw it would be nice to have symmetry between them.

Answer 2

Classically there is no problem with defining Maxwell's equations for magnetic monopoles. They look like (in tensor notation): $$ \partial_\mu F^{\mu\nu} = J_e^{\nu} $$ $$ \partial_\mu* F^{\mu\nu} = J^\nu_m$$ Here $F^{\mu\nu}$ is the Faraday tensor, and $J^\nu_e$, $J^\nu_m$ are respectively the electric and magnetic four currents. Also, $*$ is the hodge star operator. These are well defined as PDE's, which means you can use standard PDE methods to analyze them. For example, we can consider the magnetostatic case of a single point charge as in Habouz answer.

However, there are also additional complications that arise in theories with magnetic monopoles. One is that you cannot write things In terms of potentials in a simple way. This is because $\partial_\mu *(\partial^{[\mu} A^{\nu]})= 0$ which means that writing $F^{\mu\nu} = \partial^{[\mu} A^{\nu]}$ would be inconsistent with our second maxwell equation in the presence of magnetic monopoles. There are ways to get around this such as the Dirac string, but they are all rather more complicated than regular electrodynamics. This matters because describing things in terms of potentials is how we get local couplings in quantum theories (consider the Aharanov Bohm effect).

In general, the quantum theory for magnetic monopoles is much more complicated. For example, we find that to have a consistent theory we must have a charge quantization condition. Many consider this a positive aspect of the theory because it gives a possible explanation for why we observe charge to be only found in integer multiples of some fundamental charge.

In particular, we find that the magnetic coupling constant is inversely proportional to the electric. So, as you mentioned, the electric coupling is very small, and the magnetic is big. This makes it very hard to use traditional quantum field theory methods because they are generally based on perturbation theory.

To summarize, as far as I know, there are no issues with the classical theory of magnetic monopoles i.e. we can analyze Maxwell's equations with magnetic currents using standard PDE methods. They are more complicated because we have added extra inhomogeneity, but they are perfectly well-defined (and lead to an inverse square law for example). It is moving to the corresponding quantum field theory where things become much more complicated.

Answer 3

I'll offer an additional subtlety to Habouz's answer. Classically, magnetic monopoles are forbidden by Maxwell's equations: we are subject to the constraint that anywhere, $\textrm{div} \, \mathbf{B} = 0$. It was the idea of Dirac that in the quantum realm, this constraint can be obeyed with a "Dirac string"; that is, the field lines look like $B \propto 1/r^2$, but an infinitely small solenoid brings in (or out, depending on your magnetic charge) a magnetic field from infinity which exactly cancels the outgoing (incoming) radial magnetic flux.

Check out Sec. 7.5 of https://www.pa.ucla.edu/faculty-websites/dhoker-lecture-notes/graduate-courses/quantum-mechanics.pdf for more detail. Dirac magnetic monopoles are closely related to the quantization of electric charge!