A Celestial Kinematical Interpretation for an Extended BMS$_4$

Abstract

Motivated by the work of Longhi and Materassi, who constructed a realisation of the (centreless) BMS$_4$ algebra for the massive Klein-Gordon field in $3+1$, we build a realisation of the (centreless) massless BMS$_4$ algebra including super-rotations. This realisation depends only on the momenta in the lightcone expressed in celestial coordinates without any reference to the Klein--Gordon field. The quadratic Casimir of the Lorentz algebra is written in terms of a second order differential operator and the volume form plays an essential role in this construction. The BMS$_4$ algebra in terms of vector fields shows its kinematical nature, like the Poincaré algebra. We also construct a dynamical realisation of BMS$_4$ from the symplectic structure on the solutions of the massless four-dimensional Klein--Gordon field in terms of quadratic expressions of the Fourier modes and plane waves invariant under translations. Using the Mellin transform, we rewrite the Klein--Gordon field in terms of the boost invariant basis, and write down the corresponding BMS$_4$ realization. We also provide the relation with spherical harmonics, linking our results with the solutions of Longhi-Materassi, which are in fact a subset of ours.