In Geometric Quantization the pre-quantum operator of a phase-space function $f$ is defined by $$ \mathbb P_f = -i\hbar X_f - \theta(X_f) +f $$ with Poisson bracket $X_f = \{\, \cdot \, , f\}$ and the 1-form $\theta = p \cdot dq$. The commutators of such pre-quantum operators have the nice property $$ [\mathbb P_f, \mathbb P_g] = i \hbar \mathbb P_{\{f,g\}}\,. $$ I can easily check that $\mathbb P_q = q $ and $ \mathbb P_p = - i \hbar \frac{\partial}{\partial q}$, but the Hamiltonian pre-quantum operator has wrong sign w.r.t. the canonical canonical quantum mechanics $$ \mathbb P_H = - i\hbar \frac{\partial}{\partial t} \,\,\, ? $$ Am I missing something?
My guess is that it has something to do with the extended phase-space.
Please help me!