Quantum mechanics on Riemannian Manifold in Schwinger's Quantization Approach II

TL;DR

This paper extends Schwinger's quantization to quantum mechanics on homogeneous Riemannian manifolds with group actions, revealing a gauge structure linked to the manifold's fiber bundle geometry.

Contribution

It introduces a method to construct quantum mechanics on G/H manifolds using Schwinger's approach, highlighting the gauge connection as a principal fiber bundle connection.

Findings

Quantum mechanics on G/H has an inherent gauge structure.

The coordinate representation and physical state selection are developed.

The gauge potential is identified as the connection 1-form of a principal fiber bundle.

Abstract

Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold $M$ is a homogeneous Riemannian space with the given action of isometry transformation group. Using the identification of $M$ with the quotient space $G/H$, where $H$ is the isotropy group of an arbitrary fixed point of $M$, we show that quantum mechanics on $G/H$ possesses a gauge structure, described by the gauge potential that is the connection 1-form of the principal fiber bundle $G(G/H, H)$. The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of states are developed.