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

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.

arXiv comment19 pages, LaTeX.7. Comments and references added

References in corpus (8)

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• Non-Relativistic Spacetimes with Cosmological Constant
• Casimir invariants for the complete family of quasi-simple orthogonal algebras
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Cited by in corpus (9)

• The kappa-(A)dS quantum algebra in (3+1) dimensions
• On kappa-deformation and triangular quasibialgebra structure
• The $\kappa$-Newtonian and $\kappa$-Carrollian algebras and their noncommutative spacetimes
• Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups
• Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry
• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• Universal constants and natural systems of units in a spacetime of arbitrary dimension
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups