Path-integral action of a particle in the noncommutative phase-space

TL;DR

This paper develops a path integral formulation for quantum particles in noncommutative phase-space, deriving the action, energy spectra, and revealing a connection to particles in magnetic fields.

Contribution

It introduces a novel path integral approach for noncommutative phase-space and derives the associated action and spectra, linking to magnetic field analogies.

Findings

Action resembles that of a particle in a magnetic field

Energy spectra for free particle and harmonic oscillator obtained

Nonlocal action leads to noncommutative Heisenberg algebra

Abstract

In this paper we construct a path integral formulation of quantum mechanics on noncommutative phase-space. We first map the system to an equivalent system on the noncommutative plane. Then by applying the formalism of representing a quantum system in the space of Hilbert-Schmidt operators acting on noncommutative configuration space, the path integral action of a particle is derived. It is observed that the action has a similar form to that of a particle in a magnetic field in the noncommutative plane. From this action the energy spectrum is obtained for the free particle and the harmonic oscillator potential. We also show that the nonlocal nature (in time) of the action yields a second class constrained system from which the noncommutative Heisenberg algebra can be recovered.