A note on the asymptotic symmetries of electromagnetism

TL;DR

This work extends the asymptotic symmetry structure of electromagnetism by admitting angle-dependent $u(1)$ gauge transformations that grow as $\epsilon \sim a(\theta,\varphi) r + b(\theta,\varphi) \ln r + c(\theta,\varphi)$ at spatial infinity. Using a Hamiltonian framework with extended boundary conditions, the authors derive a rich set of improper gauge charges with central extensions, enabling the internal $u(1)$ sector to decouple from the Poincaré algebra via nonlinear, field-dependent redefinitions. The resulting algebra is a direct sum of the Poincaré algebra and the enlarged Abelian $u(1)$ sector, allowing a Lorentz-covariant, gauge-ambiguity-free definition of angular momentum. The analysis hinges on carefully chosen parity conditions and a finite boundary term that preserves integrability of the charges. The work paves the way for analogous treatments at null infinity and potential higher-dimensional generalizations, while highlighting the intricate role of central extensions in decoupling internal symmetries from spacetime symmetries.

Abstract

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 $r$, $ε\sim a(θ, \varphi) r + b(θ, \varphi) \ln r + c(θ, \varphi)$. 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.