Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry

TL;DR

This paper constructs Lorentzian Snyder spacetimes and their Galilei and Carroll limits using projective geometry, revealing new geometric structures and noncommutative features, especially in the non-relativistic limit.

Contribution

It provides a rigorous geometric framework for Snyder models and their limits, highlighting the interchange of roles between coordinates and translation operators.

Findings

Momentum space is described by projective coordinates on curved spaces.

Galilean Snyder models exhibit noncommuting time and space coordinates.

The non-relativistic limit retains space-time mixing, indicating Planck-scale effects.

Abstract

We show that the Lorentzian Snyder models, together with their Galilei and Carroll limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of such curved spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant kinematical symmetries. While the momentum space of the Lorentzian Snyder models is given by certain projective coordinates on (Anti-)de Sitter spaces, we discover that the momentum space of the Galilean (Carrollian) Snyder models is given by certain projective coordinates on curved Carroll (Newton--Hooke) spaces. This exchange between the Galilei and Carroll limits emerging in the transition from the geometric picture to the phase space picture is traced back to an interchange of the role of coordinates and translation operators. As a physically relevant feature, we find that in Galilean Snyder spacetimes the time coordinate does not commute with space coordinates, in contrast with previous proposals for non-relativistic Snyder models, which assume that time and space decouple in the non-relativistic limit $c\to \infty$. This remnant mixing between space and time in the non-relativistic limit is a quite general Planck-scale effect found in several quantum spacetime models.