In Is it possible to elevate the electric-magnetic duality discrete symmetry to a continuous one? the OP asks about how to derive the Noether current associated with the electromagnetic duality transformation. Michael Seifert points out that the Lagrangian involves the potentials $A_\mu$ and the duality transformation is not a transformation of $A_\mu$, which prevents us from being able to apply Noether's theorem.
Clearly, the duality transformation is only physical in free space (i.e. no sources) since otherwise it introduces unphysical magnetic charges and currents. In free space, how can we prove that the electromagnetic duality transformation $\vec E \to \vec E \, \cos \theta + \vec B \, \sin \theta; \quad \vec B \to \vec B \, \cos \theta - \vec E \, \sin \theta$$$\vec E \to \vec E \, \cos \theta + \vec B \, \sin \theta; \quad \vec B \to \vec B \, \cos \theta - \vec E \, \sin \theta$$ can't be expressed as a continuous, local transformation of the potentials $\varphi$ and $\vec A$?
