Kostant-Souriau prequantum operator
= Kostant-Souriau prequantum operator
{c}
{title2=$\widehat f=-i\hbar\nabla_{X_f}+f$}
= Prequantum operator
{synonym}
For $\iota_{X_f}\omega=df$ and $F_\nabla=i\omega/\hbar$, the curvature <commutator> and $[X_f,X_g]=-X_{\{f,g\}}$ give $[\widehat f,\widehat g]=i\hbar\widehat{\{f,g\}}$. In cotangent coordinates, $\theta=p\,dq$ gives $\widehat f=-i\hbar X_f+f-\theta(X_f)$. Restriction to a vertical polarization yields $\widehat q=q$ and $\widehat p=-i\hbar\partial_q$.