Prequantization ($[\omega /(2\pi \hslash )]\in H^2(P,\mathbb{Z})$)

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.