Here is a little puzzle I just came up with when in today's hep-th serving I found

arXiv:2304.14351 [pdf, other]

But to explain it, I should probably say one or two things about thermal states in the algebraic QFT language: There (as we teach for example in our "Mathematical Statistical Physics" class) you take take to distinguish (quasi-local) observables which form a C*-algebra and representations of these on a Hilbert space. In particular, like for example for Lie algebras, there can be inequivalent representations that is different Hilbert spaces where the observables act as operators but there are no (quasi-local) operators that you can use to act on a vector state in one Hilbert space that brings you to the other Hilbert space. The different Hilbert space representations are different super-selection sectors of the theory.

A typical example are states of different density in infinite volume: The difference in particle number is infinite but any finite product of creation and annihilation operators cannot change the particle number by an infinite amount. Or said differently: In Fock space, there are only states with arbitrary but finite particle number, trying to change that you run into IR divergent operators.

Similarly, assuming that the (weak closure) of the representation on one Hilbert space if a type III factor as it should be for a good QFT, states of different temperatures (KMS states in that language) are disjoint, meaning they live in different Hilbert spaces and you cannot go from one to the other by acting with a quasi-local operator. This is proven as Theorem 5.3.35 in volume 2 of the Bratelli/Robinson textbook.

Now to the AdS black holes: Start with empty AdS space also encoded by the vacuum in the holographic boundary theory. Now, at t=0 you act with two boundary operators (obviously quasi-local) to create two strong gravitational wave packets heading towards each other with very small impact parameter. Assuming the hoop conjecture, they will create a black hole when they collide (probably plus some outgoing gravitational radiation).

Then we wait long enough for things to settle (but not so long as the black hole starts to evaporate in a significant amount). We should be left with some AdS-Kerr black hole. From the boundary perspective, this should now be a thermal state (of the black hole temperature) according to the usual dictionary.

So, from the point of the boundary, we started from the vacuum, acted with local operators and ended up in a thermal state. But this is exactly what the abstract reasoning above says is impossible.

How can this be? Comments are open!

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