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

TL;DR

This paper investigates whether the Weyl ordering prescription yields correct energy levels for a quantum particle on a D-dimensional sphere by comparing two different Hamiltonian construction methods.

Contribution

It compares the energy eigenvalues obtained via Weyl ordering in second class and first class constrained systems, revealing discrepancies.

Findings

Weyl ordering produces consistent energy levels for D=2 in the second class system.

Conversion to a first class system alters the energy eigenvalues.

Different Hamiltonian formulations lead to different energy spectra.

Abstract

The energy eigenvalues of the quantum particle constrained in a surface of the sphere of D dimensions embedded in a $R^{D+1}$ space are obtained by using two different procedures: in the first, we derive the Hamiltonian operator by 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 the Hermitian operators. When D=2 we verify that there is no constant parameter in the expression of the eigenvalues energy, a result that is in agreement with the fact that an extra term would change the level spacings in the hydrogen atom; in the second procedure it is adopted the non-abelian BFFT formalism to convert the second class constraints into first class ones. The non-abelian first class Hamiltonian operator is symmetrized by also using the Weyl ordering rule. We observe that their energy eigenvalues differ from 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.