(Anti)de Sitter/Poincaré symmetries and representations from Poincaré/Galilei through a classical deformation approach

A classical deformation procedure, based on universal enveloping algebras, Casimirs and curvatures of symmetrical homogeneous spaces, is applied to several cases of physical relevance. Starting from the (3 + 1)D Galilei algebra, we describe at the level of representations the process leading to its two physically meaningful deformed neighbours. The Poincaré algebra is obtained by introducing a negative curvature in the flat Galilean phase space (or space of worldlines), while keeping a flat spacetime. To be precise, starting from a representation of the Galilei algebra with both Casimirs different from zero, we obtain a representation of the Poincaré algebra with both Casimirs necessarily equal to zero. The Poincaré angular momentum, Pauli–Lubanski components, position and velocity operators, etc. are expressed in terms of ‘Galilean’ operators through some expressions deforming the proper Galilean ones. Similarly, the Newton–Hooke algebras appear by endowing spacetime with a non-zero curvature, while keeping a flat phase space. The same approach, starting from the (3+1)D Poincaré algebra provides representations of the (anti)de Sitter as Poincaré deformations.