More on the gauge principle and nonobservability of some quantum numbers characterizing the Landau eigen-states

The eigen-states of the Landau Hamiltonian in the symmetric gauge are characterized by two integers $n$ and $m$. Here, $n$ denotes the familiar Landau quantum number, while $m$ represents the eigen-value of the canonical orbital angular momentum (OAM) operator $\hat{L}^{can}_z$. On the other hand, the eigen-states in the 1st Landau gauge are characterized by two integers $n$ and $k_x$, here $n$ is the Landau quantum number, while $k_x$ is the eigen-value of the canonical momentum operator $\hat{p}^{can}_x$. Since the canonical momentum and the canonical OAM are both gauge-variant quantities, their eigenvalues $k_x$ and $m$ are standardly believed not to correspond to observables. However, this wide-spread view was suspected in a recent paper based on the logical development of the gauge-potential-independent formulation of the Landau problem, which predicts the existence of two conserved momenta $\hat{p}^{cons}_x$ and $\hat{p}^{cons}_y$ and one conserved OAM $\hat{L}^{cons}_z$. They are regarded as Noether charges of the Landau Hamiltonian, the conservation of which is guaranteed {\it independently} of the choice of the {\it auge potential}. In particular, on the basis of novel covariant gauge transformation properties of these conserved operators, the eigen-values of which are characterized by the quantum numbers $k_x$, $k_y$, and $m$, it was claimed that these quantum numbers correspond to observables at least in principle. The purpose of the present paper is to show that this claim is not justified, regardless of the differences in the two theoretical formulations of the Landau problem, i.e. the traditional formulation and the gauge-potential-independent formulation.