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It is well known that Maxwell equations can be made symmetric w.r.t. $E$ and $B$ by introducing non-zero magnetic charge density/flux.

In this case we have $div B = \rho_m$, where $\rho_m$ is a magnetic charge density.

But this means that $B$ cannot be expressed as the curl of vector potential anymore.

Does it mean that it is impossible to develop electrodynamics of non-zero electric and magnetic charges in terms of vector and scalar potentials? What happens to $U(1)$ gauge invariance then?

P.S.: I know that for the point-like magnetic charge vector potential can still be introduced, but not in the entire space. My question is related to non point-like magnetic charge densities.

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Although in the presence of magnetic monopoles $\mathbf{B}$ can no longer be expressed as the curl of a vector potential, it can still be written as the sum of the curl of a vector potential and the gradient of a scalar potential:

$\mathbf{B} = \nabla\Xi + \nabla\times \mathbf{A}$

This is a consequence of Helmholtz's theorem.

So, electrodynamics can still be developed in terms of vector and scalar potentials, most straightforwardly by the introduction of an additional, magnetic scalar potential. Furthermore, the U(1) symmetry of electrodynamics is not broken by the introduction of monopoles; see this review by Milstead and Weinberg for a discussion of monopoles and electrodynamic symmetries.

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