A note on the asymptotic symmetries of electromagnetism

TL;DR

This paper extends the understanding of electromagnetic asymptotic symmetries by including angle-dependent gauge transformations with linear and logarithmic growth, revealing a structure that decouples these symmetries from the Poincaré algebra and clarifies angular momentum definitions.

Contribution

It introduces a consistent extension of electromagnetic asymptotic symmetries to include new gauge transformations with specific growth behaviors, and demonstrates their algebraic structure and decoupling from spacetime symmetries.

Findings

Logarithmic $u(1)$ charges are conjugate to constant ones.

Linear $u(1)$ transformations are conjugate to subleading transformations.

The extended algebra allows a gauge-invariant definition of angular momentum.

Abstract

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 $r$, $\epsilon \sim a(\theta, \varphi) r + b(\theta, \varphi) \ln r + c(\theta, \varphi)$. 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 Poincar\'e 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.