Inequivalent Quantizations and Holonomy Factor from the Path-Integral Approach

TL;DR

This paper introduces a path-integral quantization method for homogeneous spaces G/H, unifying algebraic and Dirac approaches, and clarifies the origin of inequivalent quantizations and holonomy factors.

Contribution

It presents a simple, unified path-integral procedure for quantizing on G/H, revealing the connection between different quantization approaches and their physical equivalence.

Findings

Derived inequivalent quantizations and holonomy factors using the path-integral approach.

Proved the physical equivalence of matrix-valued and scalar-valued path-integrals.

Unified various quantization methods within a common framework.

Abstract

A path-integral quantization on a homogeneous space G/H is proposed based on the guiding principle `first lift to G and then project to G/H'. It is then shown that this principle gives a simple procedure to obtain the inequivalent quantizations (superselection sectors) along with the holonomy factor (induced gauge field) found earlier by algebraic approaches. We also prove that the resulting matrix-valued path-integral is physically equivalent to the scalar-valued path-integral derived in the Dirac approach, and thereby present a unified viewpoint to discuss the basic features of quantizing on $G/H$ obtained in various approaches so far.