  Classical mechanics has a natural mathematical setting in symplectic geometry
and it may be asked if the same is true for quantum mechanics. More precisely,
is it possible to capture certain quantum idiosyncrasies within the symplectic
framework of classical mechanics? Beginning with Dirac, the idea of quantizing
a classical system involved associating the phase space variables with
Hermitian operators which act on some Hilbert space, as well as associating the
Poisson bracket on the phase space with the commutator for the operators.
Mathematically the phase space is associated with some symplectic manifold $M$
and the non-degenerate 2-form $\omega$, which comes with $M$. Geometric
prequantization is a process by which one does this in a mathematically
"rigorous" manner and we shall attempt to just introduce the methods here. We
do this by exploring this contruction for ($\mathbb{R}^{2n}, \sum_{i=1}^n dp_i
\wedge dq_i$) which avoids using complex line bundles in any non-trivial way.
One should note however that the Hilbert space one obtains is in fact too
"big", in the sense that it has too many functions in order to correspond with
actual physically significant Hilbert spaces. Geometric quantization remedies
this situation but it should be noted that not all manifolds are
prequantizable. We shall not discuss either of these issues however.
