Canonical Realization of (2+1)-dimensional Bondi-Metzner-Sachs symmetry

We construct canonical realizations of the ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_3$ algebra as symmetry algebras of a free Klein-Gordon (KG) field in $2+1$ dimensions, for both the massive and massless case. We consider two types of realizations, one on-shell, written in terms of the Fourier modes of the scalar field, and the other one off-shell with non-local transformations written in terms of the KG field and its momenta. These realizations contain both supertranslations and superrotations, for which we construct the corresponding Noether charges.