# Conservation of magnetic charge

It is well known that the electric charge of a system can be thought of as the Noether charge associated with isotropic large gauge transformations. That is, given Einstein-Maxwell theory

$$S=\frac{1}{2}\int_{\mathcal{M}}\left(\star R - F\wedge\star F\right),$$

the Noether charge associated with the guage transformation $$\delta A=d\chi$$ is given by

$$Q_{\chi}=\int_{\partial\Sigma_t}\chi\,\star F$$

where $$\partial\Sigma_t$$ is the boundary of some spacelike codimension 1 hypersurface (used to define a time variable) (see, for instance, 1). Taking $$\chi$$ constant gives a charge proportional to the electric charge bounded by $$\partial\Sigma_t$$.

However, in configurations with a monopole source, the magnetic charge

$$Q_m\propto\int_{\partial\Sigma_t}F$$

is also conserved. Is there a corresponding symmetry which leads to the conservation of this charge?

1. B. Julia, S. Silva, On Covariant Phase Space Methods, hep-th/0205072

Note: I'm aware that Dirac's quantization condition $$Q_eQ_m=2\pi n$$ implies that, if the electric field is conserved, then the magnetic charge must also be conserved. This, however, is a quantum mechanical result, and my question lies purely within classical mechanics.

• FWIW: Without an action principle, one cannot apply Noether's theorem. – Qmechanic Mar 21 '19 at 10:10
• @Qmechanic Is that strictly true? For instance, Noether’s theorem has a pretty simple interpretation in the Hamiltonian framework. – Bob Knighton Mar 21 '19 at 10:31
• Well, I have discussed this elsewhere on this site. – Qmechanic Mar 21 '19 at 10:39