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.