Introduction

Abstract

It is well known that canonical quantization of a free point particle in curved space is a long standing and controversial problem in quantum mechanics [1–13]. Indeed, for such system, correspondence of classical mechanics and quantum one does not uniquely define Hamiltonian operator and this ambiguity affects an energy spectrum of the physical system. In order to quantize rigorously physical systems subjected to constraints, Dirac Hamiltonian scheme [14] has been widely used in theoretical physics: it has appeared from string theory to produce Virasoro conditions, to nuclear phenomenology. However, the resulting Dirac brackets may be field dependent and nonlocal, and thus pose serious ordering problems for the quantization of the theory. In order to avoid this problem, the improved Dirac Hamiltonian scheme converting the second class constraints into first class ones has been developed [15–22] and it restricts quantum mechanical Hilbert space instead of configuration space. The operators representing the first class constraints are then generators of gauge transformations, and the physical states are all found by going into gauge invariant subspace of the Hilbert space.