Celestial kinematical interpretation for an extended BMS 4 algebra

Motivated by the work of Longhi and Materassi, who constructed a realization of the (centerless) <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msub> <a:mrow> <a:mi>BMS</a:mi> </a:mrow> <a:mrow> <a:mn>4</a:mn> </a:mrow> </a:msub> </a:mrow> </a:math> algebra for the massive Klein-Gordon field in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mn>3</c:mn> <c:mo>+</c:mo> <c:mn>1</c:mn> </c:math> dimensions, we build a realization of the (centerless) massless <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:msub> <e:mrow> <e:mi>BMS</e:mi> </e:mrow> <e:mrow> <e:mn>4</e:mn> </e:mrow> </e:msub> </e:mrow> </e:math> algebra including superrotations. This realization depends only on the momenta in the 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:msub> <g:mrow> <g:mi>BMS</g:mi> </g:mrow> <g:mrow> <g:mn>4</g:mn> </g:mrow> </g:msub> </g:mrow> </g:math> algebra in terms of vector fields shows its kinematical nature, like the Poincaré algebra. We also construct a dynamical realization of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:msub> <i:mrow> <i:mi>BMS</i:mi> </i:mrow> <i:mrow> <i:mn>4</i:mn> </i:mrow> </i:msub> </i:mrow> </i:math> 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 <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:msub> <k:mrow> <k:mi>BMS</k:mi> </k:mrow> <k:mrow> <k:mn>4</k:mn> </k:mrow> </k:msub> </k:mrow> </k:math> 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.