Recently, I read a review on magnetic monopole published in late 1970s, wherein some conjectures of properties possibly possessed by a longingly desired quantum field theory of monopoles are stated.
My question is what our contemporary understanding of the quantum field theory of monopoles is. Do we have a fully developed one? Any useful ref. is also helpful.
This answer is based on David Tong's lectures on solitons - Chapter 2 - Monopoles.
The general answer to the question is that it is known how to construct a quantum mechanical theory of magnetic monopoles acting as individual particles among themselves and also perturbatively in the background of the standard model fields.
t' Hooft - Polyakov monopoles appear as solitons in non-Abelian gauge theories, i.e. as stable static solutions of the classical Yang-Mills-Higgs equations. These solutions depend on some free parameters called moduli. For exmple the center of mass vector of the monopole is a modulus, since monopoles centered around any point in space are solutions since the basic theory is translation invariant. The full moduli space in this case is:
$\mathcal{M_1} = \mathbb{R}^3 \times S^1$.
The first factor is the monopole center of mass, the second factor $S^1$ will provide after quantization an electric charge to the monopole by means of its winding number.
A two monopole solution will have apart of its geometric coordinates an and charge another compact manifold giving it more internal dynamics. This part is called the Atiyah-Hitchin manifold after Atiyah and Hitchin who were the first to investigate the monopole moduli spaces and compute many of their characteristics:
$\mathcal{M_2} = \mathbb{R}^3 \times \frac{S^1 \times \mathcal{M_{AH}}}{\mathbb{Z}_2}$.
The knowledge about the arbitrary Atiyah-Hitchin manifolds is not complete. We can compute its metric and its symplectic structure. It is known thta they are HyperKaehler, which suggests that they can be quantized in a supersymmetric theory. Also, some topological invariants are also known.
These moduli spaces can be quantized (i.e., associated with Hilbert spaces on which the relevant operators can act), and the resulting theory will be a quantum mechanical theory of the monopoles. For example the for the charge 2 monopole one can in principle find the solutions representing the scattering of the two monopoles. It should be emphasized that this is a quantum mechanical theory and not a quantum field theory.
One way to understand that is to let the moduli vary very slowly (although strictly speaking the solutions are only for constant moduli). Then the resulting solutions will correspond to the classical scattering of the monopoles.
Basically, one can find the interaction of the monopoles with the usual fields of the theory by expanding the Yang-Mills theory around the monopole solution, then quantize the moduli space. In particular, the Dirac equation in the monopole background has zero modes which can be viewed as particles in the infrared limit.
This is almost, but not quite, a duplicate of What tree-level Feynman diagrams are added to QED if magnetic monopoles exist?.
In principle quantum electrodynamics includes magnetic monopoles as well as electrons, so yes we do have a theory to describe them. However we expect monopoles to be many orders of magnitude heavier than electrons, and that causes problems trying to describe both with a perturbative calculation.
