Path integral action of a particle in a magnetic field in the noncommutative plane and the Aharonov-Bohm effect

TL;DR

This paper derives the path integral action for a particle in a noncommutative plane with a magnetic field, explores the Aharonov-Bohm effect, and establishes equivalence with the noncommutative Schrödinger equation.

Contribution

It systematically formulates the path integral for noncommutative quantum mechanics, including the Aharonov-Bohm effect and dualities with other systems.

Findings

Explicit path integral action derived for noncommutative plane

Ground state energy and equations of motion obtained

Aharonov-Bohm phase and dualities demonstrated

Abstract

The formulation of noncommutative quantum mechanics as a quantum system represented in the space of Hilbert-Schmidt operators is used to systematically derive, using the standard time slicing procedure, the path integral action for a particle moving in the noncommutative plane and in the presence of a magnetic field and an arbitrary potential. Using this action, the equation of motion and the ground state energy for the partcle are obtained explicitly. The Aharonov-Bohm phase is derived using a variety of methods and several dualities between this system and other commutative and noncommutative systems are demonstrated. Finally, the equivalence of the path integral formulation with the noncommutative Schr\"{o}dinger equation is also established.