Quantum mechanics on Riemannian Manifold in Schwinger's Quantization approach III

Abstract

Using extended Schwinger's quantization approach quantum mechanics on a Riemannian manifold $M$ with a given action of an intransitive group of isometries is developed. It was shown that quantum mechanics can be determined unequivocally only on submanifolds of $M$ where $G$ acts simply transitively (orbits of $G$-action). The remaining part of degrees of freedom can be described unequivocally after introducing some additional assumptions. Being logically unmotivated, these assumptions are similar to canonical quantization postulates. Besides this ambiguity that has a geometrical nature there is undetermined gauge field of $\hbar$ (or higher) order, vanishing in the classical limit $\hbar\to 0$.