Algebraic Structure of Dirac Hamiltonians in Non-Commutative Phase Space

TL;DR

This paper explores the algebraic structure of two-dimensional Dirac Hamiltonians in non-commutative phase space using graded Lie algebras, providing insights into their spectra and applications to models like Landau levels and cylindrical wells.

Contribution

It introduces an algebraic framework based on $rak{sl}(2|1)$ for analyzing Dirac Hamiltonians in non-commutative space, connecting algebraic structures to physical spectra.

Findings

Constructed representation spaces of $rak{sl}(2|1)igoplus rak{so}(2)$ for spectral analysis.

Applied the framework to models like Landau levels and finite cylindrical wells.

Provided a method to analyze energy spectra algebraically in non-commutative settings.

Abstract

In this article we study two-dimensional Dirac Hamiltonians with non-commutativity both in coordinates and momenta from an algebraic perspective. In order to do so, we consider the graded Lie algebra $\mathfrak{sl}(2|1)$ generated by Hermitian bilinear forms in the non-commutative dynamical variables and the Dirac matrices in $2+1$ dimensions. By further defining a total angular momentum operator, we are able to express a class of Dirac Hamiltonians completely in terms of these operators. In this way, we analyze the energy spectrum of some simple models by constructing and studying the representation spaces of the unitary irreducible representations of the graded Lie algebra $\mathfrak{sl}(2|1)\oplus \mathfrak{so}(2)$. As application of our results, we consider the Landau model and a fermion in a finite cylindrical well.