Geometric quantization of reduced contangent bundles

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 ⋉ C ∞ 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.

Quantization of a particle in a background Yang–Mills field

“…[7], [8]). If we are to quantize an observable that is a pullback of f on the phase space of the particle T * Q, the connection α plays an important role.…”

“…Moreover, suppose the co-adjoint orbit O µ is integral, so that the quantization of this coadjoint orbit gives a irreducible representation space H µ of G [5] [6]. A theorem of Guillemin-Sternberg [7] (see also [8]) then suggests that the quantization ofAnd this result holds independent of whether there is a Yang-Mills field present in the backgroud. Thus when some technical assumptions are made so that the procedure of geometric quantization can be carried out smoothly, the Yang-Mills field plays no role in the quantization of the phase space…”

“…Moreover, suppose the co-adjoint orbit O µ is integral, so that the quantization of this coadjoint orbit gives a irreducible representation space H µ of G [5] [6]. A theorem of Guillemin-Sternberg [7] (see also [8]) then suggests that the quantization of…”

“…In section 3 we introduce local coordinates to facilitate our calculation, and state some results concerning the Hamiltonian vector fields for our observables. We follow closely the treatment of [8] on the polarization chosen for the phase space in section 4. Our results when f is polarization preserving are given in section 5, and the quantization of …”

“…Moreover, suppose the co-adjoint orbit O µ is integral, so that the quantization of this coadjoint orbit gives a irreducible representation space H µ of G [5] [6]. A theorem of Guillemin-Sternberg [7] (see also [8]) then suggests that the quantization of J −1 (O µ )/G is given by Hom G (H µ , L 2 (N )), the space of G−equivariant linear maps from H µ to L 2 (N ). And this result holds independent of whether there is a Yang-Mills field present in the backgroud.…”