A generalization of the Dirac's canonical quantization theory for a system with second-class constraints is proposed as the fundamental commutation relations that are constituted by all commutators between positions, momenta and Hamiltonian so they are simultaneously quantized in a self-consistent manner, rather than by those between merely positions and momenta so the theory either contains redundant freedoms or conflicts with experiments. The application of the generalized theory to quantum motion on a torus leads to two remarkable results: i) The theory formulated purely on the torus, i.e., based on the so-called the purely intrinsic geometry, conflicts with itself. So it provides an explanation why an intrinsic examination of quantum motion on torus within the Schrödinger formalism is improper. ii) An extrinsic examination of the torus as a submanifold in three dimensional flat space turns out to be self-consistent and the resultant momenta and Hamiltonian are satisfactory all around.

A two dimensional surface can be considered as three dimensional shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of two dimensional sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = i δij − xixj/r 2 rather than [xi, pj] = i δij that does not hold true any more. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

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With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx + iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyond the intrinsic geometry of the surfaces. PACS numbers: 03.65.-w Quantum mechanics; 02.40.-k Differential geometry; 92.60.hc Mesosphere; 73.20.Fz Quantum localization on surfaces and interfacesThe coherent states on two-dimensional sphere, generated by the annihilation operators, were discovered around the turn of present century, independently by Hall [1][2][3][4][5][6] in the Bargmann representation, and by in the position representation, respectively. Once each group of them got to know the work of the other, both soon realized that their coherent states are essentially the same [6,9], and the equivalence was also noted by other group [11]. However, it is puzzling that in the annihilation operators they introduced, there is a fundamental quantity that is represented by a non-hermitian operator and has the same dimension of linear momentum, but it does not bear a transparent physical nor geometric meaning. This article points out that the physical and geometric interpretation of the fundamental quantity is easily available, based on the geometric momentum that is recently introduced to offer a proper description of momentum on sphere [12], whose general form for an arbitrary twodimensional curved surface with M denoting the mean curvature, is given by [12][13][14], *

It is pointed out that the current form of the extrinsic equation of motion for a particle constrained to remain on a hypersurface is in fact a half-finished version; for it is established without regard to the fact that the particle can never depart from the geodesics on the surface. Once this fact is taken into consideration, the equation takes the same form as that for the centripetal force law, provided that the symbols are re-interpreted so that the law is applicable for higher dimensions. The controversial issue of constructing operator forms of these equations is addressed, and our studies show the quantization of constrained system based on the extrinsic equation of motion is preferable.