Black Hole S-Matrix

Question

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?

Andrew · Accepted Answer

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 $hbarrightarrow 0$. 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 & expleft[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 ufrac{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.

Black Hole S-Matrix

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