Bloch Theory and Quantization of Magnetic Systems

TL;DR

This paper explores the relationship between geometric quantization of magnetic systems on manifolds and Bloch theory, providing new spectral insights beyond existing foundational results.

Contribution

It extends the connection between quantization and Bloch theory from the 2-torus to more general compact manifolds, offering structural spectral insights.

Findings

Established the relation between quantization and Bloch theory on general compact manifolds.

Provided spectral analysis insights into magnetic quantization.

Extended previous work from the 2-torus to broader classes of manifolds.

Abstract

Quantizing the motion of particles on a Riemannian manifold in the presence of a magnetic field poses the problems of existence and uniqueness of quantizations. Both of them are settled since the early days of geometric quantization but there is still some structural insight to gain from spectral theory. Following the work of Asch, Over & Seiler (1994) for the 2-torus we describe the relation between quantization on the manifold and Bloch theory on its covering space for more general compact manifolds.