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

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 ${\cal O} (\beta)$.