The spectral distance on the Moyal plane

TL;DR

This paper investigates the spectral distance in the noncommutative Moyal plane, computing explicit distances for certain states and exploring the properties of truncated spectral triples as compact quantum metric spaces.

Contribution

It provides explicit calculations of spectral distances on the Moyal plane and introduces truncated spectral triples that form compact quantum metric spaces.

Findings

Spectral distance between eigenstates of the harmonic oscillator is explicitly computed.

For some pure states, the spectral distance is infinite, indicating the spectral triple is not a spectral metric space.

Truncated spectral triples based on M_n(C) are shown to be compact quantum metric spaces.

Abstract

We study the noncommutative geometry of the Moyal plane from a metric point of view. Starting from a non compact spectral triple based on the Moyal deformation A of the algebra of Schwartz functions on R^2, we explicitly compute Connes' spectral distance between the pure states of A corresponding to eigenfunctions of the quantum harmonic oscillator. For other pure states, we provide a lower bound to the spectral distance, and show that the latest is not always finite. As a consequence, we show that the spectral triple [20] is not a spectral metric space in the sense of [5]. This motivates the study of truncations of the spectral triple, based on M_n(C) with arbitrary integer n, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n=2.