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 ϵ that involve terms growing at spatial infinity linearly and logarithmically in r, ϵ ~ a(θ, φ)r + b(θ, φ) ln r + c(θ, φ). The charges of the logarithmic u(1) transformations are found to be conjugate to those of the O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(1) transformations (abelian algebra with invertible central term) while those of the O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(r) transformations are conjugate to those of the subleading O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(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.