Snyder-type spaces, twisted Poincar\'e algebra and addition of momenta

TL;DR

This paper explores generalized Snyder models incorporating Lorentz-invariant deformations of the Heisenberg algebra, calculating the associated deformed momentum addition, twist, and R-matrix, with exact results for the Snyder case.

Contribution

It provides a comprehensive framework for all Lorentz-invariant Snyder-like deformations, including explicit formulas for the twist and R-matrix, advancing noncommutative geometry understanding.

Findings

Deformed addition of momenta derived for all models.

Explicit first-order formulas for twist and R-matrix.

Exact twist formula obtained for Snyder realisation.

Abstract

We discuss a generalisation of the Snyder model that includes all the possible deformations of the Heisenberg algebra compatible with Lorentz invariance, in terms of realisations of the noncommutative geometry. The corresponding deformed addition of momenta, the twist and the $R$-matrix are calculated to first order in the deformation parameters for all models. In the particular case of the Snyder realisation, the exact formula for the twist is obtained.