Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum, and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum $p_{\theta }$ breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as $p_i$ $(i=1,2,3)$ are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between $p_i$ generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.