Can QFT be constructed without using vacuum? So my question is simple when doing QFT in classical curved spacetime we found out that we have different sets of complete functions which leads to different vacuum states. And these different vacuum states actually leads to the fact that we observe particles at a different time or in a different frame depending on the situation.
Therefore I want to ask people who deal with axiomatic QFT that can we use some concept other than vacuum to deal with QFT in curved spacetime. But abandoning vacuua concept is same as not using Fock space because without vacuua we can't guarantee that our energy operator is bound from below.
 A: When spacetime is curved, the microlocal spectrum condition has been proposed as a substitute for the usual flat-spacetime spectrum condition. It doesn't select a unique vacuum state, but it does select a collection of states that can be used to build Hilbert-space representations in which we can still define particles. 
We can derive Hawking radiation without knowing what the microlocal spectrum is, because we can appeal to asymptotic conditions (such as the distant past, before a star collapses, when spacetime was approximately flat) in order to choose an appropriate "vacuum" state. The state isn't uniquely dictated by symmetry, like it is in the flat spacetime case, but we can still use it to build Hilbert-space representations with decent properties.
So yes, QFT can be constructed without having a unique vacuum state.

More formally, at least two different types of "axiom systems" for QFT have been proposed that are designed to work for generic spacetimes:


*

*One is... well, I don't know what it's called, but I listed several references about it in this question of mine.

*Another one is represented by the paper "The generally covariant locality principle — A new paradigm for local quantum physics" (https://arxiv.org/abs/math-ph/0112041).

One more thought: In a sense, gravity is inherently unstable. Think of the Jeans instability, or the fact that gravitationally-bound systems have negative specific heat. The usual spectrum condition in flat spacetime is a kind of stability condition, so maybe the fact that we lose a natural definition of "vacuum" when we consider curved spacetime can be regarded as forshadowing what we would encounter in a full quantum theory of gravity, which is still a work in progress.
