Can the S-Matrix formalism (or an extension of it) describe the transition of matter falling into a black hole into Hawking radiation?
What I mean is start with an initial state |i>=|ψi(−∞)> that describes the infalling matter far away from the black hole and assume a final state |f>=|ψf(∞)> where the black hole is evaporated and only Hawking radiation is left.
The transition probability would as usually be wi→f=|Mif|2 where Mif=⟨f|ˆS|i⟩ defines an element of the S-Matrix
ˆS=ˆTexp(−iℏ∞∫−∞dtˆWIP(t))
ˆT is the time ordering operator and ˆWIP describes perturbation caused by the black hole in the interaction picture.
If the S-Matrix formalism can be applied to this situation, how would one in particular construct ˆWIP?