Suppose I have some quantum theory described by path integral with non-quadratic action. I may take a semi-classical approximation version of a given theory. I guess these two theories have the same classical limit i.e. $\hbar\rightarrow 0$ limit, because the difference (in some sense) of two theories is proportional to $\hbar$. Then, can I think of these two theories as a 'different quantization', like choosing some specific ordering for the $xp$ (ordering problem) or something like that?
This question originated from the relation between the Hamiltonian path integral and the Lagrangian path integral. In the Hamiltonian path integral, if the Hamiltonian is quadratic in $p$, we can get an exactly same path integral when substitute $\dot{q}=\partial H/\partial p$, and this is Lagrangian path integral. However, if the Hamiltonian is not quadratic, integrating out $p$ generally gives a Lagrangian path integral with a most general Lagrangian compatible with symmetries. On the other hand, I can just $ignore$ the Hamiltonian side and treat the Lagrangian path integral with the original Lagrangian as a quantization of my theory (this is essentially a semi-classical or saddle point approximation of the previous one). It seems these two things share the same classical limit.
