Does "QFT in curved spacetime" combine QFT and general relativity? From my understanding, QFT combines QM and special relativity. So doesn't QFT in curved spacetime combine QFT and general relativity? I realize that we need to quantize gravity to have the more accurate theory, but I want to know if GR and QFT are combined by QFT in curved spacetime. If it does, why do people say it's difficult to combine them?
 A: To echo what others have said: in QFT on curved space you allow the metric to be non-flat, but the metric itself (and thus spacetime) is not a quantum variable.
It is perhaps also worth emphasizing that although graviton interactions are non-renormalizable, you can still compute quantum corrections from gravitons at energies much lower than the Planck energy (see for instance ch. 22.4 of this textbook). It's once you reach energy scales on the order of $10^{19}$ GeV that the non-renormalizable nature of graviton interactions becomes a serious obstruction.
A: The problem arises due to the fact that if one wants to consider the quantum effects of gravity itself, then one has to compute graviton loop corrections. These are famously non-renormalizable, which means that one needs to include an infinite number of counter terms in the Lagrangian in order to be able to cure the divergences in the theory.
So yes, you can consider a quantum theory in the background of some curved space-time, you could derive quantum effects which are special to curved space time (e.g. Hawking radiation, Unruh effect etc) but you cannot truly call such a theory "quantized gravity", since by doing this, you are effectively only considering gravity at tree level. And that is, by definition, the classical approximation. So in a way studying fields in a curved background is a type of semi-classical approximation.
Gravity is classical but the rest is quantum.
A: QFT on curved spacetime describes the evolution of quantum fields defined on a curved spacetime, but this means abandoning some of the fundamental mathematical requirements of quantum theory, notably unitarity and with it the probability interpretation. In my view this is not just difficult, it is not even possible. I think a different approach altogether is necessary, one which recognises quantum mechanics itself does not require a fundamental spacetime. As Dirac put it,

“In the general case we cannot speak of an observable having a value for a particular
state, but we can … speak of the probability of its having a specified
value for the state, meaning the probability of this specified value being obtained
when one makes a measurement of the observable.”

In particular, since position does not exist in the general case in quantum mechanics, nor does spacetime. This was also encapsulated in von Neumann's treatment of quantum mechanics as a theory of measurement results. We should therefore consider spacetime as an emergent property from the mathematical structure of quantum mechanics, not as a fundamental on which quantum mechanics is based.
