The Weyl Algebra, Quantum Theory, and Normal Ordering 159
5.2.7.4 Geometric Quantization
A more geometric approach to quantization is geometric quantization.Hereonestarts
from phase space, that is, a manifold Ω with a symplectic structure ω (a particular kind
of two-form); see Remark 5.13. Then prequantization gives rise to a Hermitian line bundle
L over Ω, equipped with a U(1)-connection whose curvature equals iω. L is called the
prequantum line bundle. The Hilbert space H
0
= L
2
(Ω, L) of square integrable sections
is called the prequantum Hilbert space. This is not yet the Hilbert space of the quantized
theory – it is too big. But it is a step in the right direction. In particular, we can prequantize
classical observables: There is a map sending