Algebraic Quantization, Good Operators and Fractional Quantum Numbers

TL;DR

This paper investigates algebraic quantization of systems with periodic boundary conditions, revealing how 'good' operators and fractional quantum numbers emerge, explaining phenomena like flux quantization and the fractional quantum Hall effect.

Contribution

It introduces a framework for understanding fractional quantum numbers through algebraic quantization, extending traditional geometric quantization methods.

Findings

Identification of 'good' and 'bad' operators in algebraic quantization.

Demonstration of fractional quantum numbers in specific physical systems.

Explanation of flux quantization and fractional quantum Hall effect phenomena.

Abstract

The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the ``failure" of the Ehrenfest theorem is clarified in terms of the already defined notion of {\it good} (and {\it bad}) operators. The analysis of ``constrained" Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric Quantization. This study is illustrated with the examples of the free particle on the circumference and the charged particle in a homogeneous magnetic field on the torus, both examples featuring ``anomalous" operators, non-equivalent quantization and the latter, fractional quantum numbers. These provide the rationale behind flux quantization in superconducting rings and Fractional Quantum Hall Effect, respectively.