Does the Weyl ordering prescription lead to the correct energy levels for the quantum particle on the D-dimensional sphere?

The energy eigenvalues of the quantum particle constrained in a surface of the sphere of D dimensions embedded in a RD+1 space are obtained by using t wo different procedures: in the first, we derive the Hamiltonian operator b y squaring the expression of the momentum, written in Cartesian components, which satisfies the Dirac brackets between the canonical operators of this second-class system. We use the Weyl ordering prescription to construct th e Hermitian operators. When D = 2 we verify that there is no constant param eter in the expression of the eigenvalues energy, a result that is in agree ment with the fact that an extra term would change the level spacings in th e hydrogen atom; in the second procedure it is adopted the non-Abelian BFFT formalism to convect the second-class constraints into first-class ones. T he non-Abelian first-class Hamiltonian operator is symmetrized by also usin g the Weyl ordering rule. We observe that their energy eigenvalues differ f rom a constant parameter when we compare with the second-class system. Thus , a conversion of the D-dimensional sphere second-class system for a first- class one does not reproduce the same values.