Physical symmetries and gauge choices in the Landau problem

TL;DR

This paper explores the gauge invariance and physical significance of conserved quantities in the Landau problem, emphasizing a gauge-independent formulation and its implications for gauge symmetry and angular momentum concepts.

Contribution

It clarifies the gauge choice in the Landau problem using a gauge-independent approach and discusses the physical meaning of gauge-invariant orbital angular momentum.

Findings

Eigen-functions are gauge-independent due to conserved quantities.

Gauge symmetry has a physically vacuous aspect in this context.

The gauge-invariant extension of orbital angular momentum is analyzed.

Abstract

Due to a special nature of the Landau problem, in which the magnetic field is uniformly spreading over the whole two-dimensional plane, there necessarily exist three conserved quantities, i.e. two conserved momenta and one conserved orbital angular momentum for the electron, independently of the choice of the gauge potential. Accordingly, the quantum eigen-functions of the Landau problem can be obtained by diagonalizing the Landau Hamiltonian together with one of the above three conserved operators with the result that the quantum mechanical eigen-functions of the Landau problem can be written down for arbitrary gauge potential. The purpose of the present paper is to clarify the meaning of gauge choice in the Landau problem based on this gauge-potential-independent formulation, with a particular intention of unraveling the physical significance of the concept of gauge-invariant-extension of the canonical orbital angular momentum advocated in recent literature on the nucleon spin decomposition problem. At the end, our analysis is shown to disclose a physically vacuous side face of the gauge symmetry.