Position-dependent noncommutativity in quantum mechanics

TL;DR

This paper introduces a model for position-dependent noncommutativity in quantum mechanics, constructing a consistent algebra of operators that generalizes the Heisenberg algebra with variable noncommutativity.

Contribution

It proposes a new algebraic framework for position-dependent noncommutativity and a first-order Lagrangian that reproduces these relations upon quantization.

Findings

Constructed a complete algebra of position-dependent noncommutative operators.

Demonstrated the algebra satisfies the Jacobi identity.

Explored the localization of noncommutativity.

Abstract

The model of the position-dependent noncommutativety in quantum mechanics is proposed. We start with a given commutation relations between the operators of coordinates [x^{i},x^{j}]=\omega^{ij}(x), and construct the complete algebra of commutation relations, including the operators of momenta. The constructed algebra is a deformation of a standard Heisenberg algebra and obey the Jacobi identity. The key point of our construction is a proposed first-order Lagrangian, which after quantization reproduces the desired commutation relations. Also we study the possibility to localize the noncommutativety.