nLab
quantum operator (in geometric quantization)

for the moment the conent that would go here is rather at quantum observable

Idea

In the contect of geometric quantization a prequantum operator is an operator that presents an observable in quantum mechanics/quantum field theory once a polarization is chosen.

More in detail, the quantomorphism group $\mathrm{Aut}(c_{\mathrm{conn}})$ naturally acts on the space of sections ${\Gamma }_X(E)$ of the prequantum line bundle.

$$\widehat{(-)}:{\Gamma }_X(E)\times \mathrm{Aut}(c_{\mathrm{conn}})\to {\Gamma }_X(E).$$\widehat {(-)} : \mathbf{\Gamma}_X(E) \times \mathbf{Aut}(\mathbf{c}_{conn}) \to \mathbf{\Gamma}_X(E) \,.

For $O\in \mathrm{Aut}(c_{\mathrm{conn}})$ a given Hamiltonian symplectomorphism with Hamiltonian, the corresponding map

$$\hat{O}:{\Gamma }_X(E)\to {\Gamma }_X(E)$$\widehat{O} : \mathbf{\Gamma}_X(E) \to \mathbf{\Gamma}_X(E)

is the prequantum operator that quantizes $O$.