Quantum mechanics with coordinate dependent noncommutativity

TL;DR

This paper develops a mathematical framework for quantum mechanics on noncommutative spaces where the noncommutativity depends on coordinates, extending the theory to more general noncommutative geometries.

Contribution

It introduces a symplectic realization, constructs Darboux coordinates, and formulates a star product for coordinate-dependent noncommutativity in quantum mechanics.

Findings

Derived a formal series for noncommutative algebra of coordinates and momenta.

Applied framework to free particle and harmonic oscillator models.

Established a deformation of the Heisenberg algebra for coordinate-dependent noncommutativity.

Abstract

Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this paper we discuss the mathematical framework of such a theory. The noncommutativity is treated as an external antisymmetric field satisfying the Jacoby identity. First, we propose a symplectic realization of a given Poisson manifold and construct the Darboux coordinates on the obtained symplectic manifold. Then we define the star product on a Poisson manifold and obtain the expression for the trace functional. The above ingredients are used to formulate a nonrelativistic quantum mechanics on noncommutative spaces of general form. All considered constructions are obtained as a formal series in the parameter of noncommutativity. In particular, the complete algebra of commutation relations between coordinates and conjugated momenta is a deformation of the standard Heisenberg algebra. As examples we consider a free particle and an isotropic harmonic oscillator on the rotational invariant noncommutative space.