Gauge Covariance of the Aharonov-Bohm Phase in Noncommutative Quantum Mechanics

TL;DR

This paper investigates the gauge covariance of the Aharonov-Bohm phase in noncommutative quantum mechanics, revealing issues with traditional formulations and proposing a gauge covariant approach using noncommutative Wilson lines.

Contribution

It demonstrates that the standard path integral and coordinate shift methods lead to gauge non-covariance in NCQM and introduces a gauge covariant formulation via noncommutative Wilson lines.

Findings

Naive formulations are not gauge covariant in NCQM.

Noncommutative Wilson lines restore gauge covariance.

Potential for deriving Dirac quantization conditions in NCQM.

Abstract

The gauge covariance of the wave function phase factor in noncommutative quantum mechanics (NCQM) is discussed. We show that the naive path integral formulation and an approach where one shifts the coordinates of NCQM in the presence of a background vector potential leads to the gauge non-covariance of the phase factor. Due to this fact, the Aharonov-Bohm phase in NCQM which is evaluated through the path-integral or by shifting the coordinates is neither gauge invariant nor gauge covariant. We show that the gauge covariant Aharonov-Bohm effect should be described by using the noncommutative Wilson lines, what is consistent with the noncommutative Schr\"odinger equation. This approach can ultimately be used for deriving an analogue of the Dirac quantization condition for the magnetic monopole.