I'm reading a bit about geometric quantization and, among the axioms of this construction, is one requiring that the operator $\hat f = -\textrm i \hbar \nabla _{X_f} + f$ associated to the classical observable $f$ be self-adjoint whenever $f$ takes real values (where $X_f$ is the Hamiltonian field of $f$ and the second term acts by multiplication). It is not difficult to see that $\hat f$ is symmetric, but why is it also self-adjoint? ($\hat f$ acts on the completion in the $L^2$ norm of the space of sections in some complex Hermitian line bundle $L$ over $M$.) Frustratingly, none of the texts that I have been reading bothers showing this, they just state it (at best).

As a side-note, does anyone know of a serious, solid, trustworthy text on the subject? The ones that I have been reading skip a lot of details and rely on plenty of hand-waving (including Woodhouse's book from '91), what a disappointment).