A note on the asymptotic symmetries of electromagnetism

We extend the asymptotic symmetries of electromagnetism in order to consistently include angle-dependent $u(1)$ gauge transformations $\epsilon$ that involve terms growing at spatial infinity linearly and logarithmically in $r$, $\epsilon \sim a(\theta ,\phi )r+b(\theta ,\phi )\ln r+c(\theta ,\phi )$. The charges of the logarithmic $u(1)$ transformations are found to be conjugate to those of the $\mathcal{O}(1)$ transformations (abelian algebra with invertible central term) while those of the $\mathcal{O}(r)$ transformations are conjugate to those of the subleading $\mathcal{O}(r^{-1})$ transformations. Because of this structure, one can decouple the angle-dependent $u(1)$ asymptotic symmetry from the Poincaré algebra, just as in the case of gravity: the generators of these internal transformations are Lorentz scalars in the redefined algebra. This implies in particular that one can give a definition of the angular momentum which is free from $u(1)$ gauge ambiguities. The change of generators that brings the asymptotic symmetry algebra to a direct sum form involves non linear redefinitions of the charges. Our analysis is Hamiltonian throughout and carried at spatial infinity.