Quantum Mechanics in Curved Configurational Space

Abstract

Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasiclassical, and path integration formalisms are considered for quantization of geodesic motion on the Riemannian configuration spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for noninvariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between operators of coordinates and momenta of a particle. A cosmological consequence of the deformation is presented in short.