Noncommutative Quantum Mechanics and Seiberg-Witten Map

TL;DR

This paper reformulates noncommutative quantum mechanics using the Seiberg-Witten map to address ambiguities, and finds that key physical effects like the Aharonov-Bohm phase and Hall conductivity are unaffected at linear order in the noncommutative parameter.

Contribution

It introduces a reformulation of noncommutative quantum mechanics with the Seiberg-Witten map, showing that certain physical effects remain unchanged at linear order in the noncommutative parameter.

Findings

Aharonov-Bohm phase remains unchanged at linear order in 

Hall conductivity formula is unaffected by noncommutativity at linear order

The reformulation resolves ambiguity issues in noncommutative theory

Abstract

In order to overcome ambiguity problem on identification of mathematical objects in noncommutative theory with physical observables, quantum mechanical system coupled to the NC U(1) gauge field in the noncommutative space is reformulated by making use of the unitarized Seiberg-Witten map, and applied to the Aharonov-Bohm and Hall effects of the NC U(1) gauge field. Retaining terms only up to linear order in the NC parameter \theta, we find that the AB topological phase and the Hall conductivity have both the same formulas as those of the ordinary commutative space with no \theta-dependence.