Geometric Quantization
(May 2011)Abstract
Pre-Quantization. Given a symplectic manifold M, construct a Lie algebra
of operators on a Hilbert space associated to M together with a map
of Lie algebras from smooth functions on M with the Poisson bracket
to the Lie algebra of operators. The Hilbert space is a space of
sections of a line bundle over M. The line bundle is built with a
connection whose curvature is the original symplectic form. The problem
with pre-quantization is that the Hilbert space you get is too big.
It is not a mathematical problem, it is a physics problem. The (sometimes)
solution to this problem is called polarization.