In QFT we usually deal with scattering problems. On the asymptotic past we have free particles, they interact, and we compute the overlap with distinct free particle states on the asymptotic future to find the amplitude for various processes.

The $S$-matrix is then defined as follows. Let $|\Phi_i\rangle$ be the initial state with the said initial free particle content. Let $|\Phi(t)\rangle$ be its evolution. Then

$$S_{if}=\langle \Phi_f|S|\Phi_i\rangle=\lim_{t\to \infty}\langle \Phi_f|\Phi(t)\rangle.$$

So the $S$-matrix encodes the amplitudes of transitions between some initial state and some final state in a scattering problem.

Now the problem is: I've heard people talking about $S$ matrices in the process of black hole evaporation by means of Hawking radiation emission.

In particular this is briefly mentioned in this paper when the author states that to fully understand the unitarity issues of black hole evaporation an $S$-matrix is required.

Now, how can black hole evaporation be scattering problem in the end?

I'm failing to see how. In the usual derivation one starts with the vacuum $|0\rangle_{\text{in}}$ in the asymptotic past $\mathscr{I}^-$ and one just computes what $|0\rangle_{\text{in}}$ is as seen by an observer at $\mathscr{I}^+$.

So it seems we are just taking one state $|0\rangle_{\text{in}}$ and expanding in terms of one convenient basis. It actualy is weird that it seems there is no evolution at all.

So what is the precise relation between Hawking radiation and scattering? How this is a scattering problem if we are just starting with a vacuum and expanding in another basis?

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    $\begingroup$ You don't start with a vacuum. You start with a whole bunch of free particles that come together to form a black hole. $\endgroup$ – Peter Shor Oct 13 '18 at 15:56
  • $\begingroup$ But this is qualitative, is there a way to make it quantitative and precise? First a black hole is formed by gravitational collapse of a highly massive body. Such a body I believe would be macroscopic, formed by one extremely high number of elementary particles bound together. Finally, these particles wouldn't be free since they are the constituents of the body (they are held together to form atoms, molecules, etc). These bond states seem to not be known how to describe with QFT. So this seems quite complicated actually. $\endgroup$ – user1620696 Oct 13 '18 at 16:11
  • $\begingroup$ There's no reason you couldn't make a black hole by starting with a whole bunch of free particles, aimed so as to all collide near some point. And if you can't get an $S$-matrix for this case, getting an $S$-matrix for the gravitational collapse of a massive body is completely hopeless. $\endgroup$ – Peter Shor Oct 13 '18 at 16:19
  • $\begingroup$ Is there some full description of that (the initial free state that could give rise to a black hole), or it is currently unknown? Perhaps the Vaidya geometry with $M(v)=M_0\theta(v-v_0)$ which is one of the simplest examples of gravitational collapse could be the resulting geometry of some simple setting like that? $\endgroup$ – user1620696 Oct 13 '18 at 17:00
  • $\begingroup$ There’s a strange premature consensus in the community that S-matrix is a valid description in the strong field case. $\endgroup$ – Prof. Legolasov Oct 14 '18 at 13:52

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