Geometric quantization of reduced contangent bundles

Abstract

The method of geometric quantization is applied to the symplectic space obtained by Marsden-Weinstein reduction of a cotangent bundle T ∗N . Specifically it is assumed that the symmetry [gauge] group acts freely on N so that N is a principal H-bundle over the assumed Riemannian manifold Q = N/ H. The reduced phase space obtained is the same of that for a charged particle moving on Q in an external Yang-Mills gauge field which is given by a connection on N → Q. An explicit map is found from a subalgebra of the classical observables to the corresponding quantum operators. These operators are found to be the generators of a representation of the semi-direct product group, Aut N ⋉ C ∞ (Q) . A generalized Aharanov-Bohm effect is shown to be a natural consequence of the quantization procedure. In particular, the role of the connection in the quantum mechanical system is made clear. The quantization of the Hamiltonian is also considered. Additionally, our approach allows the related quantization procedures proposed by Mackey and by Isham to be fully understood.