One intriguing issue in the nucleon spin decomposition problem is the existence of two types of decompositions, which are representably characterized by two different orbital angular momenta (OAMs) of quarks. The one is the manifestly gauge-invariant mechanical OAM, while the other is the so-called gauge-invariant canonical (g.i.c.) OAM, the concept of which was introduced by Chen et al. To get a deep insight into the difference of these two decompositions, it is therefore vitally important to understand the the physical meanings of the above two OAMs correctly. Also to be clarified is the implication of the gauge symmetry that is immanent in the concept of g.i.c. OAM. We find that the famous Landau problem provides us with an ideal tool to answer these questions owing to its analytically solvable nature. After deriving a complete relation between the standard eigen-functions of the Landau Hamiltonian in the Landau gauge and in the symmetric gauge, we try to unravel the physics of the the canonical OAM and the mechanical OAM, by paying special attention to their gauge-dependence. We also argue that, different from the mechanical OAM of the electron, the canonical OAM or its gauge-invariant version would not correspond to any direct observables at least in the Landau problem. Also briefly discussed is the uniqueness or non-uniqueness problem of the nucleon spin decomposition, which arises from the arbitrariness in the definition of the so-called physical component of the gauge field.
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