Effects of quantum deformation on the spin-1/2 Aharonov-Bohm problem

TL;DR

This paper investigates how quantum deformation via the $k$-Poincaré-Hopf algebra affects the scattering and bound states of a spin-1/2 particle in the Aharonov-Bohm setup, revealing bounds on deformation and helicity conservation.

Contribution

It introduces a nonrelativistic limit of the $k$-deformed Dirac equation for the Aharonov-Bohm problem and applies self-adjoint extension methods to analyze scattering and bound states.

Findings

Bound states energies are derived from the pole structure of the S-matrix.

An upper bound on the deformation parameter is established.

Helicity conservation is analyzed within the deformed framework.

Abstract

In this letter we study the Aharonov-Bohm problem for a spin-1/2 particle in the quantum deformed framework generated by the $\kappa$-Poincar\'{e}-Hopf algebra. We consider the nonrelativistic limit of the $\kappa$-deformed Dirac equation and use the spin-dependent term to impose an upper bound on the magnitude of the deformation parameter $\varepsilon$. By using the self-adjoint extension approach, we examine the scattering and bound state scenarios. After obtaining the scattering phase shift and the $S$-matrix, the bound states energies are obtained by analyzing the pole structure of the latter. Using a recently developed general regularization prescription [Phys. Rev. D. \textbf{85}, 041701(R) (2012)], the self-adjoint extension parameter is determined in terms of the physics of the problem. For last, we analyze the problem of helicity conservation.