Quantum mechanics on a curved Snyder space

TL;DR

This paper explores the mathematical structure of a quantum model with two fundamental scales, revealing discrete spectra for position and momentum operators within a curved Snyder space framework.

Contribution

It introduces a new representation of the Snyder-de Sitter algebra using geometric properties of phase space, highlighting its invariance under Born reciprocity.

Findings

Position and momentum operators have discrete spectra.

The algebra admits representations based on Grassmannian manifold geometry.

The model incorporates two fundamental scales: Planck mass and cosmological constant.

Abstract

We study the representations of the three-dimensional Euclidean Snyder-de Sitter algebra. This algebra generates the symmetries of a model admitting two fundamental scales (Planck mass and cosmological constant) and is invariant under the Born reciprocity for exchange of positions and momenta. Its representations can be obtained starting from those of the Snyder algebra, and exploiting the geometrical properties of the phase space, that can be identified with a Grassmannian manifold. Both the position and momentum operators turn out to have a discrete spectrum.