Note: the question in this post is basically originated from here. Please refer to it first if you don't get the sense.
I'd like to know how you compute the saddle point approximation in the path integral for an open quantum system.
What I here concern about is basically the quantum-to-classical transition, in which we have to consider an external environment so often. Many texts say when we take a limit $ \hbar \rightarrow 0 $ (more accurately, $ S/ \hbar \rightarrow \infty $, where $S$ stands for the action), we obtain the classical Lagrange equation. But this is not a perfect explanation for the quantum-to-classical transition, as there could be some cases where the system is still coherent after the approximation.
To clarify the process for the transition, the procedure called decoherence is representatively discussed. In this picture, the presence of an external environment is introduced, and the system we currently deal with is presupposed to be coupled with the environment. However, we know that in this case, the classical state we finally observe is determined by the external system and not by the system itself we concern with.
Hence, it seems that under the effect of the environment, it would be unable to determine the exact classical motion of the system only by considering the closed one. But I couldn't find any account for the saddle point approximation in the path integral formalism considering an open system, even though $ \hbar \rightarrow 0 $ may have an important role in the classical transition. (I guess there might be some cases we cannot obtain the classical Lagrange equation when taking the open system into.)
I think one should consider an open system for the first place to elaborate the transition process by integrating the explanation on the saddle point approximation and the decoherence. Is there any reference or paper on my question? I found a paper implanting the Lindblad equation in the path integral form, but it seems that there is no result computing the saddle point approximation in this kind of situation.