Quantum mechanics of the extended Snyder model

TL;DR

This paper explores a quantum harmonic oscillator within an extended Snyder model, revealing noncommutative geometry with associative star products and analyzing the spectrum and physical implications of added degrees of freedom.

Contribution

It introduces a novel extended Snyder model with tensorial degrees of freedom, leading to a noncommutative geometry with associative star products and a detailed spectral analysis.

Findings

The model exhibits a noncommutative geometry with associative star products.

The spectrum of the harmonic oscillator is characterized using creation and annihilation operators.

Additional degrees of freedom influence physical properties of the system.

Abstract

We investigate a quantum mechanical harmonic oscillator based on the extended Snyder model. This realization of the Snyder model is constructed as a quantum phase space generated by $D$ spatial coordinates and $D(D-1)/2$ tensorial degrees of freedom, together with their conjugate momenta. The \coo obey nontrivial \cor and generate a noncommutative geometry, which admits nicer properties than the usual realization of the model, in particular giving rise to an associative star product. The spectrum of the harmonic oscillator is studied through the introduction of creation and annihilation operators. Some physical consequences of the introduction of the additional degrees of freedom are discussed.