Editing Canonical quantization (section)

===Geometric quantization===
{{main|Geometric quantization}}
In contrast to the theory of deformation quantization described above, geometric quantization seeks to construct an actual Hilbert space and operators on it. Starting with a symplectic manifold <math>M</math>, one first constructs a prequantum Hilbert space consisting of the space of square-integrable sections of an appropriate line bundle over <math>M</math>. On this space, one can map ''all'' classical observables to operators on the prequantum Hilbert space, with the commutator corresponding exactly to the Poisson bracket. The prequantum Hilbert space, however, is clearly too big to describe the quantization of <math>M</math>.

One then proceeds by choosing a polarization, that is (roughly), a choice of <math>n</math> variables on the <math>2n</math>-dimensional phase space. The ''quantum'' Hilbert space is then the space of sections that depend only on the <math>n</math> chosen variables, in the sense that they are covariantly constant in the other <math>n</math> directions. If the chosen variables are real, we get something like the traditional Schrödinger Hilbert space. If the chosen variables are complex, we get something like the [[Segal–Bargmann space]].