for the moment the conent that would go here is rather at quantum observable
geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
In the contect of geometric quantization a prequantum operator is a linear operator that presents an observable in quantum mechanics/quantum field theory once a polarization is chosen.
More in detail, the quantomorphism group $\mathbf{Aut}(\mathbf{c}_{conn})$ naturally acts on the space of sections $\mathbf{\Gamma}_X(E)$ of the prequantum line bundle.
For $O \in \mathbf{Aut}(\mathbf{c}_{conn})$ a given Hamiltonian symplectomorphism with Hamiltonian, the corresponding map
is the prequantum operator that quantizes $O$.