We currently understand electromagnetism as a U(1) gauge theory. If you take a point out the space manifold (base space) you can create magnetic monopoles with integral charges by making non-trivial bundle. However, to me this "monopole" is just creating a background electromagnetic field in which other charges move via the Lorentz force law. But how is the monopole itself affected by other charges?

For example, it would seem that two monopoles should attract each other in exactly the same way that two electric charges would. This seems like a very difficult thing to understand using non-trivial bundles. It seems as though the duality between electric charges and magnetic charges should be more obvious. What's going on with the dynamics of these excised points?


The starting point would be the invariance of Maxwell's equations in vacuum under duality transformation: $\mathbf{E} \to \mathbf{B}$, $\mathbf{B} \to - \mathbf{E}$ (in Gaussian cgs units). In order to extend this symmetry to include field sources we must introduce in addition to (electric) charge density and current ($\rho_\mathrm{e}$ and $\mathbf{j}_\mathrm{e}$) also magnetic charge density and current, that transform under duality as $\rho_\mathrm{e}\to \rho_\mathrm{m}$ and $\mathbf{j}_\mathrm{e} \to \mathbf{j}_\mathrm{m}$. With that, the extended Maxwell's equations take the following form: \begin{eqnarray} \mathbf{\nabla} \cdot \mathbf{E} &=& 4 \pi \rho_{\mathrm{e}}, \\ -\mathbf{\nabla} \times \mathbf{E} &=& \frac1c \left(4 \pi \mathbf{j}_{\mathrm{m}} + \frac{ \partial\mathbf{B} }{ \partial t} \right), \\ \mathbf{\nabla} \cdot \mathbf{B} &=& 4 \pi \rho_{\mathrm{m}} , \\ \mathbf{\nabla} \times \mathbf{B} &=& \frac1c \left( 4\pi \mathbf{j}_{\mathrm{e}} + \frac{\partial\mathbf{E}} { \partial t}\right) . \end{eqnarray}

Similarly we could generalize the Lorentz force that is acting on a point particle (a dyon) with electric charge $q_\text{e}$ and magnetic charge $q_\text{m}$: $$ \textbf{F}= q_\text{e} \left(\textbf{E}+\frac1c \textbf{v}\times \textbf{B}\right)+q_\text{m} \left(\textbf{B}-\frac1c \textbf{v}\times \textbf{E}\right). $$ Of course if a particle is truly point-like, the total energy of field would be diverging. This could be handled just like for a point electric charge by an appropriate mass renormalization, so that bare mass and EM field energy combined to give a finite mass value. So, for example, dynamics of nonrelativistic monopole of mass $m$ would be governed by the equation: $$ m \frac{d\mathbf{v}}{dt}=q_\text{m} \left(\textbf{B}-\frac1c \textbf{v}\times \textbf{E}\right). $$ One could see that the motion of a monopole in the field of an immobile electric charge is equivalent to the motion of a charge in the field of a static monopole. Also, two monopoles would repel each other, but monopole and anti-monopole would attract each other and could form a bound system, and their classical motion is given by an ordinary solution of a Kepler problem.

The motion of two particles with different type of charges (electric and magnetic) or two particles each carrying both kinds of charges also could be solved analytically, see for example these papers for pedagogical treatment of a problem:

One notable feature of this two body problem is that angular momentum of point particles is no longer conserved (since forces are no longer radial), but still there is a conserved vector, Poincaré integral of motion, which could be seen as a total angular momentum of point particles and electromagnetic field. Therefore the motion is confined to a Poincaré cone, see figure for an example trajectory:

image from Sivardière's paper


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