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


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.

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

Magnetic charge is a topological charge that is conserved due to the topology of the manifold of solutions to the field equations --- these are distinct from Noether charges like electric charge that arise from symmetries of the action.

From https://arxiv.org/pdf/1411.3099.pdf:

"Unlike the above mentioned Noether currents, there exist another class of currents called topological currents, which are conserved due to topological reasons. Their conservation is not in reference to any particular action or Hamiltonian under consideration, and holds identically in general"


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