Polarization in geometric quantization ($\mathcal{F}\subset T_{\mathbb{C}}P$)

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.