The vacuum of quantum field theories in curved spacetime The vacuum of a quantum field theory in flat spacetime is postulated to be unique for all inertial observers. If $\hat{P}_\mu$ denote the generators for spacetime translations, and $\hat{J}_{\mu\nu}$ denotes the generators of homogeneous Lorentz transformations, this postulate means $$\hat{P}_\mu|0\rangle=0, ~\hat{J}_{\mu\nu}|0\rangle=0.$$
Why can't we postulate that the vacuum is unique for all observers also in curved spacetime by demanding that the vacuum is invariant under general coordinate transformations?
 A: Here is an explanation I have: even if axiomatic QFT (in flat Minkowski spacetime) places the vacuum on a pedestal by including it in one of the axioms, in "real life QFT" it is not a primordial notion, but a derived one. Namely, as soon as you find a so-called "mode expansion" of the unquantized free field, by imposing the standard quantization rule, you end up with the vacuum defined using annihilation and creation operators.
But how did you get there in the first place? The possibility of a mode expansion by "transfourier-ing" the linear classical field equation rests on one important thing:
- the existence of a set of four rectangular coordinates which enables one to account for the needed "wave vector" with which the so-called "positive frequency modes" can be defined.
This in turn has a neat geometrical explanation. It stems from what is known as the isometry group of the space time manifold, which for a flat Minkowski spacetime is the Poincaré group. Symmetries and globally conserved quantities are encoded in the so-called Killing vectors of the manifold.
The problem, of course, is that an arbitrary curved 4 D manifold has no global Killing vectors, thus any mode expansion (hence annihilation operators) is only local. So you cannot have a unique vacuum, even if you wanted to. You have a distinct vacuum at each point for each set of local coordinates. Different vacua means different annihilation operators which you can link through the a so-called  Bogoliubov transformation.
This is all in some detail explained in a QFT book such as Birrell & Davies which builds all at an easy pace (compared to a more involved approach by Wald, for which his GR book is a prerequsite).
