A note on the asymptotic symmetries of electromagnetism

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 ~ 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 $$ \mathcal{O} $$ O (1) transformations (abelian algebra with invertible central term) while those of the $$ \mathcal{O} $$ O (r) transformations are conjugate to those of the subleading $$ \mathcal{O} $$ 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.

Asymptotic $$ \mathcal{O} $$(r) gauge symmetries and gauge-invariant Poincaré generators in higher spacetime dimensions

“…This has been shown to be useful in previous treatments of the problem in the context of electromagnetism, see e.g. [2,16,20]. The extended Hamiltonian action principle in d spacetime dimensions on a Minkowski background reads…”

“…In light of the very recent results for gravity and electromagnetism [1,2], where it was possible to extend the asymptotic symmetries of the theories by consistently accommodating logarithmic branches and a linear u(1) symmetry (namely, generated by an O(r) gauge parameter) at spatial infinity, we address the direct question about whether similar extensions can be found in the case of electromagnetism in higher spacetime dimensions d > 4.…”

“…Following the approach introduced in [4, 5] (see also [2,16] for the Maxwell theory in d = 4), we will deform the usual decay of the gauge potential ( Ã0 , Ãi ) ∼ (r −1 , r −d+3 ) (obtained from the fall-off of the electric and magnetic fields at spatial infinity ∼ r −d+2 ), by improper gauge transformations, namely,…”

“…Analogue enhancements were explored in electromagnetism. Concretely, a relaxation of the asymptotic conditions of [16] was performed in [2]. As a result of this analysis, it was consistently accommodated not only a logarithmic but a linear growth (associated to the subleading soft photon theorems [17][18][19]) for the gauge potential at infinity.…”

“…The corresponding canonical generators are shown to form a centrally extended abelian algebra. It must be mentioned that the role of the logarithmic gauge transformations in d = 4 found in [2] is now played by the improper subleading O(r −d+4 ) gauge transformations in all higher dimensions d > 4 (note that these subleading u(1) symmetries turn out to be analogue to the subleading supertranslations in higher dimensions found in [26,27]).…”