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

TL;DR

This paper develops a quantum mechanics framework on Riemannian manifolds using Schwinger's approach, highlighting the role of group actions and the necessity of additional assumptions for complete description.

Contribution

It extends Schwinger's quantization to Riemannian manifolds with intransitive isometry groups, clarifying the conditions for unique quantum descriptions and addressing gauge ambiguities.

Findings

Quantum mechanics is well-defined only on submanifolds with simply transitive group actions.

Additional assumptions akin to canonical quantization are needed for full degrees of freedom.

An undetermined gauge field of order extit{ extbar}hbar extbar{} exists, vanishing as extit{ extbar}hbar extbar{} approaches zero.

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 $\hbar\to 0$.