Geometric Quantization of the Phase Space of a Particle in a Yang-Mills Field

Abstract

The method of geometric quantization is applied to a particle moving on an arbitrary Riemannian manifold $Q$ in an external gauge field, that is a connection on a principal $H$-bundle $N$ over $Q$. The phase space of the particle is a Marsden-Weinstein reduction of $T^*N$, hence this space can also be considered to be the reduced phase space of a particular type of constrained mechanical system. 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~$N\lx C^\infty_c(Q)$. A generalised Aharanov-Bohm effect is shown to be a natural consequence of the quantization procedure. In particular the rôle 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.\\