An exact representation isotropic and anisotropic noncommutative phase spaces, and their relations

Abstract

Noncommutative phase space of an arbitrary dimension is considered. The both of operators coordinates and momenta in noncommutative phase space may be noncommutative. In this paper, we introduce momentum-momentum noncommutativity in addition to coordinate-coordinate noncommutativity. We find an exact form for the linear coordinate transformation which relates a noncommutative phase space to the corresponding ordinary one. As an example, the Hamiltonian of a three-dimensional harmonic oscillator is examined.