It is possible to extend a quantum field theory to a curved spacetime. But does this lead to predictions that can be tested and measured? Had it been confirmed?
The underlying reason I am asking this is: curved spacetime means emergence of gravity and therefore General Relativity regime. And we know that GR and QFT are incompatible. I realise that in order to include gravity, one should put its Lagrangian in from the very beginning and this, I guess, does not work. But does the current mathematical framework for extending the known field theories to a curved spacetime work?
Answer
The biggest prediction of QFT on curved (not dynamical!) space-time is the Hawking radiation . This radiation can in principle be measured experimentally, even though it's an effect so small that with current technology there's probably no hope of a measurement. It's still possible that with some clever way of maximizing the experimental signal we could achieve such goal (for instance with something like the measurement of the proton life time, in which there's no hope to follow a single proton for $10^{33} yr$, but it's "easy" to do that with $10^{33}$ protons.)
Moreover, in the solar system gravity is weak, the space is only slightly curved. Therefore, while QFT on flat space-time can be routinely tested in a laboratory on earth, for a significant curved space-time effect you have to usually look at astrophysical or cosmological experiments, with all the related uncertainties.
The math: Is known that a theory with interacting gravitons (spin 2 massless particles) is not renormalizable. QFT with gravitons as an effective field theory could work, in the sense that you can make predictions like in the Fermi's theory of weak interactions. For instance see the beautiful treatment of Schwartz (Quantum Field Theory and The Standard Model, p.404) in which he finds the quantum gravity predictions to the Mercury's perihelion shift.
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