  Geometric quantization often produces not one Hilbert space to represent the
quantum states of a classical system but a whole family $H_s$ of Hilbert
spaces, and the question arises if the spaces $H_s$ are canonically isomorphic.
[ADW] and [Hi] suggest to view $H_s$ as fibers of a Hilbert bundle $H$,
introduce a connection on $H$, and use parallel transport to identify different
fibers. Here we explore to what extent this can be done. First we introduce the
notion of smooth and analytic fields of Hilbert spaces, and prove that if an
analytic field over a simply connected base is flat, then it corresponds to a
Hermitian Hilbert bundle with a flat connection and path independent parallel
transport. Second we address a general direct image problem in complex
geometry: pushing forward a Hermitian holomorphic vector bundle $E-->Y$ along a
non-proper map $Y-->S$. We give criteria for the direct image to be a smooth
field of Hilbert spaces. Third we consider quantizing an analytic Riemannian
manifold $M$ by endowing $TM$ with the family of adapted K\"ahler structures
from arxiv:1004.4069 [LSz]. This leads to a direct image problem. When $M$ is
homogeneous, we prove the direct image is an analytic field of Hilbert spaces.
For certain such $M$---but not all---the direct image is even flat; which means
that in those cases quantization is unique.
