Noncommutative Quantum Mechanics with Path Integral

TL;DR

This paper explores noncommutative quantum mechanics with symmetric noncommutativity in phase space, providing a transformation to standard formalism and a method to compute path integrals, demonstrated on a charged particle in electromagnetic fields.

Contribution

It introduces a transformation to handle symmetric noncommutativity in phase space and develops a procedure for path integrals in noncommutative quantum mechanics with quadratic Lagrangians.

Findings

Explicit connection between quadratic Hamiltonians and Lagrangians in noncommutative regimes

Method to compute Feynman path integrals in noncommutative phase space

Application to charged particle in electromagnetic fields

Abstract

We consider classical and quantum mechanics related to an additional noncommutativity, symmetric in position and momentum coordinates. We show that such mechanical system can be transformed to the corresponding one which allows employment of the usual formalism. In particular, we found explicit connections between quadratic Hamiltonians and Lagrangians, in their commutative and noncommutative regimes. In the quantum case we give general procedure how to compute Feynman's path integral in this noncommutative phase space with quadratic Lagrangians (Hamiltonians). This approach is applied to a charged particle in the noncommutative plane exposed to constant homogeneous electric and magnetic fields.