Bloch Theory and Quantization of Magnetic Systems

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.