Generalized noncommutative Snyder spaces and projective geometry

TL;DR

This paper reviews the construction of Euclidean and Lorentzian noncommutative Snyder spaces using projective geometry, exploring the freedom in choosing physical momenta and deriving a quasi-canonical phase space structure.

Contribution

It introduces a generalized framework for noncommutative Snyder spaces, highlighting the role of projective geometry and deriving a new quasi-canonical phase space structure.

Findings

Derived a quasi-canonical structure for Snyder models

Explored the freedom in choosing projective coordinates

Established a diagonal phase space algebra

Abstract

Given a group of kinematical symmetry generators, one can construct a compatible noncommutative spacetime and deformed phase space by means of projective geometry. This was the main idea behind the very first model of noncommutative spacetime, proposed by H.S. Snyder in 1947. In this framework, spacetime coordinates are the translation generators over a manifold that is symmetric under the required generators, while momenta are projective coordinates on such a manifold. In these proceedings we review the construction of Euclidean and Lorentzian noncommutative Snyder spaces and investigate the freedom left by this construction in the choice of the physical momenta, because of different available choices of projective coordinates. In particular, we derive a quasi-canonical structure for both the Euclidean and Lorentzian Snyder noncommutative models such that their phase space algebra is diagonal although no longer quadratic.