Why is it impossible to formulate unitary QFT in a dynamical background? I cannot recall the exact argument but I remember my professor saying something like unitary time evolution in a dynamical background "kicks" a state out of the Hilbert space constructed on curved Cauchy hypersurfaces, such as in the interior of Schwarzschild black holes. (It is best to model matter as open quantum systems in such backgrounds). On the other hand, unitary quantum field theory is pretty well-defined in static curved spacetimes.
Can somebody provide some mathematical structure to the above argument, if valid at all? Any reference to literature which studies the difficulty of formulating unitary quantum field theory in dynamical backgrounds is highly appreciated as well. Thank you.
 A: I have a few comments

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*QFT is well-defined also on dynamical backgrounds, and there are various traditional and more modern approaches to constructing free and perturbative QFTs on fixed gravitational backgrounds.


*Just from the title I would interpret "unitary QFT" meaning a QFT with unitary S-matrix. This is also true if the dynamical background is ''sufficiently nice'', as has been shown by Wald (see https://www.sciencedirect.com/science/article/pii/0003491679901350).
To your main concern, which is the unitarity of the time-evolution:

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*In the non-static case there is no time-translation symmetry of the system, i.e. the construction of an ordinary Schrödinger picture must fail physically because the system looks different at different times. It seems that a good discussion of these problems including technical aspects and many references can be found in Sec. 6 of https://sites.google.com/a/umich.edu/ruetsche-laura/ourweyl.pdf?attredirects=0.
A similar problem already appears when you have couplings to external fields on Minkowski space-time. In the case of external electromagnetic fields there seem to be some more recent attempts, e.g. involving 'time-varying Fock-spaces' https://arxiv.org/abs/1510.03890. I must admit that to me they appear quite mathematically involved, but perhaps one can do something similar for gravitational fields.


*Such questions were an important motivation for the development of the 'algebraic approach' to QFT. There can talk about `time-evolution' as automorphisms of the algebras of observables at different times. In this language the question of the unitary time evolution then becomes the mathematical question of the 'unitary implementability' of the automorphism group of time translations.
