Aharonov-Bohm-Like Scattering in the Generalized Uncertainty Principle-corrected Quantum Mechanics

TL;DR

This paper explores how the generalized uncertainty principle (GUP) affects Aharonov-Bohm scattering, revealing symmetry breaking and discontinuities in the scattering cross section due to minimal length effects.

Contribution

It introduces the first-order GUP correction to Aharonov-Bohm scattering and analyzes the resulting symmetry breaking and discontinuous behavior in the cross section.

Findings

Cross section invariant under combined transformations of flux and angle.

Discontinuous behavior of cross section at integer flux values.

Explicit symmetry breaking at first-order GUP correction.

Abstract

We discuss classical electrodynamics and the Aharonov-Bohm effect in the presence of the minimal length. In the former we derive the classical equation of motion and the corresponding Lagrangian. In the latter we adopt the generalized uncertainty principle (GUP) and compute the scattering cross section up to the first-order of the GUP parameter $\beta$. Even though the minimal length exists, the cross section is invariant under the simultaneous change $\phi \rightarrow -\phi$, $\alpha' \rightarrow -\alpha'$, where $\phi$ and $\alpha'$ are azimuthal angle and magnetic flux parameter. However, unlike the usual Aharonv-Bohm scattering the cross section exhibits discontinuous behavior at every integer $\alpha'$. The symmetries, which the cross section has in the absence of GUP, are shown to be explicitly broken at the level of ${\cal O} (\beta)$.