Equivalence of curvature and noncommutativity in a physical space: Harmonic oscillator on sphere

Abstract

We study the two-dimensional harmonic oscillator on a noncommutative plane. We show that by introducing appropriate Bopp shifts, one can obtain the Hamiltonian of a two-dimensional harmonic oscillator on a sphere according to the Higgs model. By calculating the commutation relations, we show that this noncommutativity is strictly dependent on the curvature of the background space. In other words, we introduce a kind of duality between noncommutativity and curvature by introducing noncommutativity parameters as functions of curvature. Also, it is shown that the physical realization of such model is a charged harmonic oscillator in the presence of electromagnetic field.