Path integral and geometric quantization I was wondering how one obtains geometric quantization from a path integral.  It's often assumed that something like this is possible, for example, when working with Chern-Simons theory, but rarely explained in detail.  One problem I run into is that, when trying to repeat the usual derivation of the path integral, we want to insert a complete set of states, but typically here the Hilbert space is finite dimensional, so I don't see how to interpret this as the integral over some manifold, as in the usual case.  The simplest case I can think of is $S^2$ with $j$ times its usual symplectic form, which gives the spin $j$ representation of $SU(2)$.  Is there a way to recover this from an integral over paths on $S^2$?
 A: In the special case of geometric quantization with respect to a Kähler
polarization, (which covers the path integral for spin over $S^2$ mentioned in the question),  there exits a rigorous way to define a path integral, i.e., with respect to a well defined measure on the space of paths. Please see for example the following
article by Laurent Charles. This type of path integral was proposed by F.A. Berezin in his famous article on covariant and contravariant operator symbols.
(There is an on-line version in Russian in which a discretized version is
given in the last page (before the references)).
Actually, Witten used this path integral in his seminal work Quantum field theory and the Jones polynomial, but did not show it explicitly. The form used by Witten , which will be described here, is a path integral representation of a Wilson loop. Here I'll write this special case of the path integral in a more comprehensible form and try to explain the physical intuition behind it.
$\left < tr_{\mathcal{H}}T\{exp(i\int_0^T B^{a}(t) \sigma_a)\}\right > = \mathrm{lim}_{m\rightarrow \infty}\int exp\big (i\int _0^T \alpha^{\mathcal{H}}_i\dot{z}^i - \bar{\alpha}^{\mathcal{H}}_i \dot{\bar{z}}^i + \frac{m}{2}g_{i\bar{j}}\dot{z}^i \dot{\bar{z}}^j+B^{a} (t)\Sigma^{\mathcal{H}}_a(z, \bar{z})\big) \mathcal{D}z\mathcal{D}\bar{z}$
The left hand side is the trace of a time ordered product of a spin (or
more generally an element of  a Lie algebra $\sigma_a)$) coupled to an external
magnetic field $B^{a}$ in the representation $\mathcal{H}$. As explained
by Witten according to the Borel Weil Bott theorem, to every
representation there corresponds a coadjoint orbit with a given Kähler
form  $ \omega^{\mathcal{H}}$ depending on the representation $\mathcal{H}$. The path integral is performed on this manifold. It is a Riemannian path integral whose action contains 3 types of terms:
The first terms depending on the symplectic potentials $
\alpha^{\mathcal{H}}$ satisfying: $ \omega^{\mathcal{H}}
=d\alpha^{\mathcal{H}}$ . This term is of the form of an interaction with
a magnetic monopole.
The second term is the Riemannian kinetic energy term. The third term is
proportional to the Hamiltonian functions $\Sigma^{\mathcal{H}}_a(z,
\bar{z})$ whose Poisson brackets satisfy the Lie algebra on the coadjoint
orbit. This term is of the type of interaction with an external time
varying magnetic field.
Thus in summary the path integral is a nonrelativitic path integral over a Riemannian
manifold in the presence of magnetic fields. In other words the
corresponding Hamiltonian is a magnetic Schrödinger operator. As very
well known the solutions to these types of problems are Landau levels a
and the lowest landau level is degenerate in general.
The crucial
observation is that when the particle mass $m$ tends to infinity, the
excited Landau level energies become very high and decouple, thus we
remain with the lowest Landau level which happens to be exactly the
representation $\mathcal{H}$ we started from.
It is worthwhile to mention that this form of the path integral has other
applications and was used in works on Heisenberg ferromagnets and quark
confinement.
A: I have to admit, that I'm not fully understand your question - but I try to answer it :)
After performing the geometric quantization procedure (prequantization and polarization) you get a well-defined hilbert space (which in general is infinite-dimensional). Furthermore, under suitable conditions you can quantize the classical observables to obtain quantum operators. Assume you get a Hamilton-operator $H$. Then you are able to reformulate this "quantum" theory in the path-integral formalism. 
A discussion of path integral in the framework of geometric quantization can be found in the standard book: Woodhouse: Geometric quantization!
Side remark: Path integrals are defined for finite dimensional Hilbert spaces, too. They are even simpler as the "integral over all paths" reduces to a "sum over all possible intermediate states". Google should help you with path integrals in two/many state systems (eg http://theory.tifr.res.in/~sgupta/courses/qm2008/lec22.pdf).
