The spectral distance in the Moyal plane

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 AA of the algebra of Schwartz functions on R2R2, we explicitly compute Connes’ spectral distance between the pure states of AA 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 (Gayral et al. (2004) [17]) is not a spectral metric space in the sense of Bellissard et al. (2010) [19]. This motivates the study of truncations of the spectral triple, based on Mn(C)Mn(C) with arbitrary n∈Nn∈N, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n=2n=2.