A procedure that assigns to a symplectic manifold a Hilbert space together with operators representing classical observables, implementing a correspondence between Poisson brackets and operator commutators to produce a quantum model from a classical phase space.

Principle

Choose a prequantum line bundle with connection whose curvature is the symplectic form, select a polarization to reduce states, and promote classical observables to operators so that the Poisson algebra is represented (up to factors of iħ) in the resulting Hilbert space.

Demonstration

For the cotangent bundle T*Q with its canonical symplectic form, one constructs the prequantum line bundle and, with the vertical (position) polarization, recovers square-integrable wavefunctions on Q and the usual position and momentum operator representation for simple systems like a particle on R^n.

Misapplication

Applying the procedure without checking the integrality condition for the symplectic form or ignoring the need to choose an appropriate polarization can yield an inconsistent or trivial Hilbert space, or produce operators that do not respect the required commutation relations.

Consequence

When successfully carried out, the process yields a concrete Hilbert space and a representation of a chosen subalgebra of classical observables; different choices (line bundle, connection, polarization) can lead to inequivalent quantum theories.

Reversal

Deformation quantization inverts the viewpoint by deforming the algebra of classical observables (introducing a noncommutative star product) while leaving the classical phase space manifold intact instead of producing a Hilbert space from it.

Boundary

Applies to smooth symplectic manifolds (or orbifolds) satisfying the necessary topological integrality conditions; it is not directly defined for singular phase spaces, general Poisson manifolds without integrality, or most infinite-dimensional field-theory phase spaces without extra structure.

Semantic Tension

Tension exists between the desire for a functorial, canonical quantization and the manifest choice-dependence (line bundle, connection, polarization) that yields many inequivalent constructions; this contrasts with algebraic or path-integral quantization approaches.

Synthesis

Geometric quantization is a structured method that converts a symplectic phase space into a quantum Hilbert space by imposing topological and geometric data (prequantum bundle, connection, polarization) and promoting classical observables to operators, with the outcome sensitive to the chosen auxiliary structures.