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 β. Even though the minimal length exists, the cross-section is invariant under the simultaneous change ϕ →−ϕ, α′→−α′, where ϕ and α′ are azimuthal angle and magnetic flux parameter. However, unlike the usual Aharonv–Bohm scattering, the cross-section exhibits discontinuous behavior at every integer α′. The symmetries, which the cross-section has in the absence of GUP, are shown to be explicitly broken at the level of 𝒪(β).

中文翻译：

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广义不确定性原理校正的量子力学中的 Aharonov-Bohm 样散射

我们在存在最小长度的情况下讨论经典电动力学和 Aharonov-Bohm 效应。在前者中，我们推导出经典的运动方程和相应的拉格朗日。在后者中，我们采用广义不确定性原理（GUP）并计算散射截面，直至 GUP 参数的一阶β. 即使存在最小长度，横截面在同时变化下是不变的φ →-φ,α'→-α'， 在哪里φ和α'是方位角和磁通量参数。然而，与通常的 Aharonv-Bohm 散射不同，横截面在每个整数处都表现出不连续的行为α'. 在没有 GUP 的情况下横截面的对称性在𝒪(β).