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

TL;DR

This paper introduces a classical deformation method using universal enveloping algebras to connect symmetries like Galilei, Poincare, and (anti)de Sitter, revealing how their representations relate through curvature modifications.

Contribution

It presents a novel classical deformation approach to derive and relate representations of fundamental spacetime symmetries from Galilei to Poincare and (anti)de Sitter.

Findings

Derived Poincare algebra from Galilei with zero Casimirs.

Expressed Poincare operators in terms of Galilean operators.

Connected Newton-Hooke and (anti)de Sitter algebras via curvature deformations.

Abstract

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 Poincare 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 Poincare algebra with both Casimirs necessarily equal to zero. The Poincare 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 Poincare algebra provides representations of the (anti)de Sitter as Poincare deformations.