Dirac equation in a gauge-field background in the Moyal plane

TL;DR

This paper explores the Dirac equation in a noncommutative plane with a constant electromagnetic background, revealing how noncommutativity influences electromagnetic fields, the Hall effect, and atomic spectra, while maintaining core physical laws.

Contribution

It provides a gauge invariant formulation of the Dirac equation in a noncommutative setting and analyzes the effects on electromagnetic phenomena and atomic spectra.

Findings

Noncommutativity modifies effective magnetic and electric fields.

Hall conductivity remains unaffected to first order in NC parameter.

Hyperfine splitting of Hydrogen atom shows first order correction due to NC effects.

Abstract

Starting with the Dirac equation for an electron in a constant electromagnetic background on a noncommutative (NC) plane, we obtain a gauge invariant description of the system. Surprisingly, the dynamics of the system is dictated by the standard form of Lorentz force law, once the effective magnetic and electric fields $\left( B^{NC}, \, E^{NC} \right)$ correct up to leading order in the NC parameter are identified. The Hall effect is studied using the NC corrected fields in the non-relativistic (NR) limit. This shows that noncommutativity affects the cyclotron frequency, but leaves the Hall conductivity unaffected at least to first order in the NC parameter. Owing to the NC corrected magnetic field, the hyperfine splitting of Hydrogen atom spectrum also shows a first order correction which helps establish an upper bound on the spatial NC parameter.