Celestial kinematical interpretation for an extended BMS4 algebra

Motivated by the work of Longhi and Materassi, who constructed a realisation of the (centreless) BMS4 algebra for the massive Klein-Gordon field in 3 + 1, we build a realisation of the (centreless) massless BMS4 algebra including super-rotations. This realisation depends only on the momenta inthe light-cone 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 BMS4 algebra in terms of vector fields shows its kinematical nature, like the Poincaré algebra. We also construct a dynamical realisation of BMS4 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 BMS4 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.