The spectral distance in the Moyal plane

JOURNAL OF GEOMETRY AND PHYSICS

卷:61

Article

Cagnache, Eric²  D'Andrea, Francesco³  Martinetti, Pierre^1,4  Wallet, Jean-Christophe² 

[1] Univ Gottingen, Inst Theoret Phys, D-37077 Gottingen, Germany

[2] Univ Paris 11, Theoret Phys Lab, F-91405 Orsay, France

[3] Catholic Univ Louvain, Dept Math, B-1348 Louvain, Belgium

[4] Univ Gottingen, Courant Ctr Higher Order Struct Math, D-37073 Gottingen, Germany

关键词: Noncommutative geometry;    Compact quantum metric space;    Spectral triple;    Moyal space;   

DOI  :  10.1016/j.geomphys.2011.04.021

来源: Elsevier

【 摘 要 】

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 (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 M-n (C) with arbitrary n is an element of N, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n = 2. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

Free