Black Hole S-Matrix I am reading arXiv:2006.03606 where through Eq. (1.1) they say that the transition amplitude for collapse of matter from initial state $\Psi_{i}$ into a black hole and eventually evaporation of black hole into final state $\Psi_{f}$ is given as :-
$$
\mathcal{A}_{fi} = \langle \Psi_{f} | \mathcal{\hat S} | \Psi_{i} \rangle = \int \mathcal{D}\Phi\ e^{iS^{\prime}[\Phi]}\ \Psi_{f}^{*}[\Phi]\ \Psi_{i}[\Phi] \longrightarrow \mathrm{Saddle\ Point\ Approximation} \longrightarrow e^{iS^{\prime}[\Phi_{cl}]}\ \Psi_{f}^{*}[\Phi_{cl}]\ \Psi_{i}[\Phi_{cl}]
$$
where the semiclassical configuration $\Phi_{cl}$ extremizes the above integrand.
I have two doubts here:

*

*Are we choosing the $\Phi_{cl}$ so that it extremizes the integrand or is it a given for any $\Phi_{cl}$ because there are no histories to be summed over and then all the $\Phi_{cl}$ are equivalent upto a non-consequential phase (disregarding any Aharonov-Bohm type experiments for the moment)?


*Without the saddle-point approximation it appears that we have $\hbar = 1$ in the integrand but when we take the approximation by considering $\Phi \to \Phi_{cl}$ then the action is classical (right?) so we have that $\hbar \to 0$. This appears to be quite contradictory. How are they maintaining that $\hbar = 1$?
Can someone please help me with the above issues?
 A: As far as I can tell, the context for BH scattering is actually not relevant, so I will answer within the context of a scalar field theory.
Question 2 is actually fairly easy to answer. You should think of $\hbar$ in this context as a formal expansion parameter, and not a constant of nature. You may recall, that in ordinary perturbation theory, a common trick is to introduce a formal parameter $\epsilon$ which is only meant to count the order of perturbation theory, and which is taken to 1 at the end of the calculation after it is no longer needed. The role of $\hbar$ is exactly the same here.
For question 1...
Let's write the path integral with the $\hbar$ dependence intact...
\begin{equation}
Z = \int \mathcal{D}\phi e^{i S[\phi]/\hbar}F[\phi]
\end{equation}
where $F[\phi]=\Psi_i^\star[\phi]\Psi_f[\phi]$.
To use the saddle point approximation, we formally take the limit $\hbar\rightarrow 0$. Since the exponent is oscillating strongly in this limit, we  expand around a saddle point, which is to say a field configuration $\phi_{cl}$ which satisfies
\begin{equation}
\frac{\delta S}{\delta \phi}\Big|_{\phi=\phi_{cl}}=0
\end{equation}
We've suggestively called the saddle point as $\phi_{\rm cl}$ since it obeys the classical equations of motion.
We then do a field redefinition in the path integral, $\phi=\phi_{\rm cl}+\sqrt{\hbar}\chi$. (We won't actually use the $\sqrt{\hbar}$ here, but it is useful if you go to higher orders).
Then...
\begin{align}
Z = F[\phi_{\rm cl}] e^{i S[\phi_{cl}]/\hbar} \int \mathcal{D} \chi & \exp\left[i \int {\rm d}^4 y {\rm d}^4 y' \frac{\delta^2 S}{\delta \phi(y) \delta \phi(y') }\Big|_{\phi=\phi_{cl}}\chi(y) \chi(y')\right] \\
& \times  \left(1+\sqrt{\hbar} \int {\rm d}^4 u\frac{\delta \log F}{\delta \phi(u)}\Big|_{\phi=\chi_{cl}} \chi(u) + O(\chi^2)\right)
\end{align}
Now, we can actually express this whole thing as $Z=F[\phi_{cl}]e^{iS[\phi_{cl}]/\hbar}(1+O(\sqrt{\hbar}))$, and therefore to leading order in $\hbar$ (leading order in the WKB approximation) we recover what you have written. We can now set $\hbar=1$, since we don't need it anymore.
To summarize, we can recover your expression to leading order in the WKB approximation, if we expand around a saddle point $\phi_{cl}$ which obeys the classical equations of motion.
This formalism also allows us to go to higher orders, if desired... we "simply" have to do the path integral over the fluctuations $\chi$, order by order.
