Is it possible to elevate the electric-magnetic duality discrete symmetry to a continuous one? I'm familiar with Electric-Magnetic duality, where in the absence of source fields one can exchange the $F_{\mu \nu}$ field with the dual field: $\tilde{F}_{\mu \nu}={\epsilon}_{\mu \nu \alpha \beta} 
 {F^{\alpha \beta}}  $ and the Maxwell equations remain the same.
I was wondering if it's possible to extend such discrete symmetry to a continuous one where one can principally derive a Noether current for the symmetry.
Basically I'd start with transforming the field to a new one:
$${F^{'} } _{\mu \nu} = {A}_{\mu \nu \alpha \beta} 
 {F^{\alpha \beta}}$$
Where ${A}_{\mu \nu \alpha \beta}$ is a generalized parameter dependent (an internal unknown symmetry parameters) pseudo-tensor which reduces to the Levi-Civita pseudo-tensor in some points of the parameter space.
To fix the number of parameters one might impose some normalization conditions on the field based on energy conservation, though I'm not really sure if it necessarily fixes them completely. But can perhaps reduce their number.
 A: Wald's books General Relativity and Advanced Classical Electromagnetism discuss a continuous duality, but I'm not sure whether it leads to an interesting Noether current. Namely, the transformation
$$F_{\mu\nu} \to \cos\alpha F_{\mu\nu} + \sin\alpha \tilde{F}_{\mu\nu}$$
generalizes the duality transformation you exhibited. Wald calls it a "duality rotation" and it is mentioned on Chap. 4, Ex. 2 of the GR book. Also mentions it on Eq. (5.10) of the E&M book. Neither of them have deep discussions nor get even close to the Noether theorem, but I think it could be a starting point if you want to fill in the details.
A: There's an interesting approach to the continuous electric-magnetic duality symmetry in 'Spacetime algebra as a powerful tool for electromagnetism' by Justin Dressel, Konstantin Y. Bliokh, and Franco Nori: https://arxiv.org/abs/1411.5002
They use the Geometric Algebra approach (Real Clifford algebras, interpreted geometrically) to unify the electric and magnetic components into a single 'complex' bivector field, where the pseudoscalar of the Geometric Algebra for spacetime is used as the imaginary unit. This gives the complex numbers a geometric interpretation, which fixes the reference frame-dependence problems of the Riemann-Silberstein complex vector interpretation.
The corresponding Noether current is described in section 8.8 as the 'helicity pseudocurrent'. Essentially, it results in conservation of helicity in the vaccuum case.
