Geometric quantization

Geometric quantization associates quantum states and observables with a symplectic manifold. A first step uses a complex line bundle with a connection on a vector bundle whose curvature is proportional to the symplectic form divided by $\hslash$; its existence imposes an integrality condition. A real polarization is an integrable half-dimensional distribution with Lagrangian submanifolds as leaves. Requiring sections to be covariantly constant along the leaves reduces the variables to a configuration-space description. Restricting the curvature to a Lagrangian leaf gives zero, permitting local flat sections; global sections can require trivial holonomy. Exact Lagrangian graphs also describe local semiclassical phases through their generating functions.

A polarization is an involutive complex rank-half distribution on which the symplectic form vanishes. Polarized prequantum sections satisfy ${\nabla }_{X}s=0$ for $X\in \mathcal{F}$. A real polarization is tangent to a Lagrangian foliation; a complex Kähler polarization uses the antiholomorphic tangent spaces. Only observables whose Hamiltonian flows preserve the chosen distribution act directly on the polarized space. Singular leaves, holonomy and changes of polarization are genuine quantization difficulties.

The pairing compares state spaces belonging to two polarizations using their prequantum sections and half-forms. When it defines a suitable identification or projection, a classical Hamiltonian flow can transport a state to its transported polarization and the pairing can return it to the original state space. Differentiating this construction supplies operators for some observables that do not directly preserve the original polarization. Existence, domains and unitarity require additional hypotheses; it is not a universal canonical equivalence of all polarizations.

Tensor the prequantum bundle with a square root of the canonical determinant bundle of a polarization, when such a square root exists. The half-form contributes to the state pairing and to the observable action. It improves coordinate invariance and accounts for familiar Maslov and oscillator zero-point shifts. Its existence is additional global data, not automatic from prequantum integrality.

Prequantization realizes a symplectic manifold using a Hermitian line bundle with connection whose curvature is $i\omega /\hslash$ in the convention ${\iota }_{X_f}\omega =df$. Its existence requires integral symplectic periods. On sections, the Kostant-Souriau prequantum operator represents every classical observable and its Poisson bracket, but the resulting states depend on the full phase space. A polarization in geometric quantization removes this excess of variables.

For ${\iota }_{X_f}\omega =df$ and $F_{\nabla }=i\omega /\hslash$, the curvature commutator and $[X_f,X_g]=-X_{\{f,g\}}$ give $[\hat{f},\hat{g}]=i\hslash \widehat{\{f,g\}}$. In cotangent coordinates, $\theta =pdq$ gives $\hat{f}=-i\hslash X_f+f-\theta (X_f)$. Restriction to a vertical polarization yields $\hat{q}=q$ and $\hat{p}=-i\hslash {\partial }_q$.

A prequantum line bundle is a Hermitian line bundle over a symplectic manifold with unitary connection on a vector bundle of the displayed curvature. With $\omega =-d\theta$, a local connection is $\nabla =d-i\theta /\hslash$. Reversing the Hamiltonian-vector-field convention or passing to the dual line bundle reverses the corresponding curvature convention. Integrality of $\omega /(2\pi \hslash )$ on closed two-cycles is the topological existence condition.