Noncommutative \(n\)-torus in the magnetic field: volume, scalar curvature, and quantum stochastic equation

Abstract

Motivated by the works published in 2003 by Chakraborty et al. [J. Operator Theory, 49 (2003), 185–201], and by Sakamoto and Tanimura [J. Math. Phys., 44 (2003), 5042–5069], we investigate the noncommutative \(n\)-torus in a magnetic field. We study the invariance of volume, integrated scalar curvature, and volume form using the method of perturbation by the inner derivation of the magnetic Laplacian in this geometric framework. Moreover, we derive the magnetic stochastic process describing the motion of a particle in a uniform magnetic field in this torus and deduce the properties of solutions of the corresponding magnetic quantum stochastic differential equation.