Geometric Quantization

Geometric quantization is a mathematical framework for constructing quantum Hilbert spaces from classical symplectic manifolds. The moment map plays a central role through the "quantization commutes with reduction" principle.

Prequantization

Definition5.7Prequantum Line Bundle

A prequantization of a symplectic manifold $(M, \omega)$ is a Hermitian line bundle $(L, \nabla, h)$ with connection $\nabla$ whose curvature satisfies $\mathrm{curv}(\nabla) = -2\pi i \omega$ . A prequantization exists if and only if $[\omega/2\pi]$ is an integral cohomology class: $[\omega] \in H^2(M; \mathbb{Z})$ .

Definition5.8Polarization

A polarization of $(M, \omega)$ is an integrable Lagrangian distribution $\mathcal{P} \subseteq T_\mathbb{C}M$ (a subbundle of the complexified tangent bundle that is Lagrangian, involutive, and of complex dimension $n$ ). A Kähler polarization is one where $\mathcal{P} \cap \overline{\mathcal{P}} = 0$ , corresponding to a compatible complex structure.

ExampleQuantization of $S^2$

For $S^2$ with area form $\omega = k \cdot \omega_{\text{std}}$ ( $k \in \mathbb{Z}_{>0}$ ), the prequantum line bundle is $\mathcal{O}(k) \to \mathbb{CP}^1$ . The quantum Hilbert space (with respect to the Kähler polarization) is $H^0(\mathbb{CP}^1, \mathcal{O}(k)) \cong \mathbb{C}^{k+1}$ , a $(k+1)$ -dimensional space of degree- $k$ homogeneous polynomials.

Guillemin-Sternberg Conjecture

RemarkQuantization Commutes with Reduction

The Guillemin-Sternberg conjecture (proved by Meinrenken, Tian-Zhang, and others) states: for a compact Hamiltonian $G$ -space with prequantum line bundle, the quantization of the reduced space $M_0 = \mu^{-1}(0)/G$ equals the $G$ -invariant part of the quantization of $M$ : $Q(M_0) \cong Q(M)^G.$ This is the mathematical formulation of the physical principle that "quantization commutes with reduction."