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

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

2006-12-19

arxiv.org/abs/math-ph/0612059v2

J.Phys.A41:015204,2008

Physics

Mathematical Physics

19 pages, LaTeX.7. Comments and references added

Scientific paper

10.1088/1751-8113/41/1/015204

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.

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