Classical and quantum mechanics of the nonrelativistic Snyder model

TL;DR

This paper explores the classical and quantum mechanics of the nonrelativistic Snyder model, revealing how the sign of a coupling constant affects properties like bounded momenta and quantized positions, with exact solutions for the harmonic oscillator.

Contribution

It provides a detailed analysis of the nonrelativistic Snyder model's mechanics, highlighting the dependence on the coupling constant and solving the harmonic oscillator exactly.

Findings

Negative coupling constant leads to bounded momenta.

Positive coupling constant results in quantized positions and areas.

Harmonic oscillator frequency depends on energy.

Abstract

The Snyder model is an example of noncommutative spacetime admitting a fundamental length scale $\beta$ and invariant under Lorentz transformations, that can be interpreted as a realization of the doubly special relativity axioms. Here, we consider its nonrelativistic counterpart, i.e. the Snyder model restricted to three-dimensional Euclidean space. We discuss the classical and the quantum mechanics of a free particle in this framework, and show that they strongly depend on the sign of a coupling constant $\lambda$, appearing in the fundamental commutators and proportional to $\beta^2$. For example, if $\lambda$ is negative, momenta are bounded. On the contrary, for positive $\lambda$, positions and areas are quantized. We also give the exact solution of the harmonic oscillator equations both in the classical and the quantum case, and show that its frequency is energy dependent.