I would have added this as a comment, but I don't have enough reputation.

The Fourier transform is not a terribly useful thing to do on a generic GR background. In flat space the Fourier transform is useful because we have translation symmetry and momentum is conserved. But in a generic solution of Einstein's equations, the are no such symmetries (or Killing vectors).

As a consequence of this, most of the time when doing QFT on a curved space one works in position space, not in momentum space.

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