Improved Dirac quantization of a free particle

In the framework of Dirac quantization with second-class constraints, a fre e particle moving on the surface of a (d - 1)-dimensional sphere has an amb iguity in the energy spectrum due to the arbitrary shift of canonical momen ta. We explicitly show that this spectrum obtained by the Dirac method is c onsistent with the result of the Batalin-Fradkin-Tyutin formalism, which is an improved Dirac method, at the level of the first-class constraint by fi xing the ambiguity, and discuss its physical consequences.