On the geometric quantization of symplectic Lie group actions

A general method for the geometric quantization of connected and simply connected symplectic manifolds and the lifting of symplectic Lie group actions is developed. In particular, a geometric construction of multipliers for a Lie group based on the action of the group on a potential of the symplectic form on the manifold is given. These methods are then employed to quantize the 'massive' symplectic homogeneous spaces of the Galilei group and the group action, thereby emphazising the affine structure of the group and deriving a novel form of phase space representations. In the case of nonzero spin we quantize the action of the covering group of the Galilei group. We derive the spin bundles needed from frame bundles over spheres equipped with their natural Levi Civita connection. Furthermore we give a new geometric description of the 'massless' symplectic homogeneous spaces (the coadjoint orbits) of the Galilei group including a description of the group actions and the symplectic forms. We then describe their geometric quantization as well as the lifting of the group action.