# Role of guiding center in Landau level system and mechanical and pseudo orbital angular momenta

## TL;DR

This paper explores the roles and differences of pseudo orbital angular momentum and mechanical OAM in Landau level systems, emphasizing the guiding center's role and their physical interpretations.

## Contribution

It clarifies the physical meanings of pseudo and mechanical OAMs and their dependence on coordinate origin, highlighting the guiding center's significance in Landau systems.

## Key findings

- Pseudo OAM is gauge-invariant and extends canonical OAM.
- Mechanical OAM correlates with classical cyclotron motion.
- Both OAMs depend on the coordinate origin choice.

## Abstract

There is an interesting but not so popular quantity called pseudo orbital angular momentum (OAM) in the Landau-level system, besides the well-known canonical and mechanical OAMs. The pseudo OAM can be regarded as a gauge-invariant extension of the canonical OAM, which is formally gauge invariant and reduces to the canonical OAM in a certain gauge. Since both of the pseudo OAM and the mechanical OAM are gauge invariant, it is impossible to judge which of those is superior to the other solely from the gauge principle. However, these two OAMs have totally different physical meanings. The mechanical OAM shows manifest observability and clear correspondence with the classical OAM of the cyclotron motion. On the other hand, we demonstrate that the standard canonical OAM as well as the pseudo OAM in the Landau problem are the concepts which crucially depend on the choice of the origin of the coordinate system. We try to reveal the relation between the pseudo OAM and the mechanical OAM as well as their observability by paying special attention to the role of guiding-center operator in the Landau problem.