Bloch Theory and Quantization Classes

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. Introduction In geometric quantization for symplectic manifolds one is faced with questions of existence and uniqueness (see e.g. Woodhouse, 1980) which do not arise for the common phase space T M (with standard symplectic structure) of Hamiltonian mechanics. But, when incorporating magnetic fields (closed 2-forms b ∈ Ω(M)) into the picture one is forced either to choose magnetic potentials (a ∈ Ω(M) with da = b) or to “charge” the standard symplectic structure by the magnetic field (see remark 1 below). In either case, the questions of existence and uniqueness come up now even for the phase space T M . Indeed, these questions arise for prequantizations, whereas — given a prequantization — the choice of a quantization is canonical when the phace space is T M with a charged symplectic structure. On the other hand, the cohomological obstructions and degrees of freedom for geometric quantization vanish on the covering space X := M̃ . Since the