The physics of helical electron beam in a uniform magnetic field as a testing ground of gauge principle

According to Bliokh et al., allowing free propagation along the direction of a uniform magnetic field, the familiar Landau electron state can be regarded as a non-diffracting version of the helical electron beam propagating along the magnetic field. Based on this observation, they argued that, while propagating along the magnetic field, the Landau electrons receive characteristic rotation with three different angular velocities, depending on the eigen-value $m$ of the canonical OAM operator, which is generally gauge-variant, and this splitting was in fact experimentally confirmed. Through complete analyses of highly mysterious $m$-dependent rotational dynamics of the quantum Landau states, we try to make clear how and why their observation does not contradict the widely-believed gauge principle.