Path integral action of a particle in the non commutative plane

TL;DR

This paper develops a path integral formulation for a particle in a non-commutative plane, deriving a non-local action that can be made local with auxiliary fields, and computes propagators for free and harmonic oscillators.

Contribution

It introduces a coherent state path integral approach for non-commutative quantum mechanics and derives a new action that captures particle dynamics in this setting.

Findings

Derived a non-local action for particles in non-commutative space.

Showed how to localize the action using auxiliary fields.

Explicitly computed propagators for free particle and harmonic oscillator.

Abstract

Non commutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on non commutative configuration space. Taking this as departure point, we formulate a coherent state approach to the path integral representation of the transition amplitude. From this we derive an action for a particle moving in the non commutative plane and in the presence of an arbitrary potential. We find that this action is non local in time. However, this non locality can be removed by introducing an auxilary field, which leads to a second class constrained system that yields the non commutative Heisenberg algebra upon quantization. Using this action the propagator of the free particle and harmonic oscillator are computed explicitly. For the harmonic oscillator this action yields the frequencies already obtained by other means in the literature.