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In Dirac's canonical quantization theory on systems with second-classconstraints, the commutators between the position, momentum and Hamiltonianform a set of algebraic relations that are fundamental in construction of boththe quantum momentum and the Hamiltonian. For a free particle on atwo-dimensional sphere or a spherical top, results show that the well-knowncanonical momentum p_{{\theta}} breaks one of the relations, while threecomponents of the momentum expressed in the three-dimensional Cartesian systemof axes as p_{i} (i=1,2,3) are satisfactory all around. This momentum is notonly geometrically invariant but also self-adjoint, and we call it geometricmomentum. The nontrivial commutators between p_{i} generate three components ofthe orbital angular momentum; thus the geometric momentum is fundamental to theangular one. We note that there are five different forms of the geometricmomentum proposed in the current literature, but only one of them turns out tobe meaningful.

Geometric Momentum: the Proper Momentum for a Free Particle on a Two-dimensional Sphere