Is it possible to define Feynman diagrams in curved space-time? I have a very simple question:
"Is it possible to talk about Amplitudes and Feynman diagrams assuming a different background than the usual Minkowski one?
Let's assume for example that the background is the Schwarzschild one.
Is it possible to define an S-matrix for a theory of scalar particles interacting through gravitons?"
 A: Yes, but it's only done in special cases of spacetimes.
The only thing that changes about the Feynman rules in a different spacetime is the propagator. For a scalar, that comes from solving the equation
$$
\nabla^\mu\nabla_\mu G(x) =  \delta^4(x)
$$
(with some boundary conditions). This can be done perfectly well in a curved background. One you get the propagator, you can expand correlation functions like $\langle \phi \phi \phi \phi\rangle$ in terms of Feynman diagrams.
In fact, this is done in the context of AdS/CFT, and the Feynman diagrams are called "Witten Diagrams."
Now, scattering amplitudes are a little trickier than correlation functions. For Minkowski, spatial correlation functions need to essentially be Fourier transformed to give scattering amplitudes. On a curved background, there won't necessarily be an appropriate notion of such a Fourier transform. In AdS, for example, which has the topology of a 'soup can,' you don't really talk about S-matrices for this reason. However, as long as the space is asymptotically Minkowski, you should be able to do it.
As for the case of Schwarzschild, I imagine you might run into a bit of trouble due to the horizon and singularity, but I don't know anything about this case in particular.
