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.
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$\begingroup$ I don't know an explicit, definitive proof that it can't, but it seems unlikely. IMO this is exactly why QFT in curved spacetime is difficult; read about e.g. the Unruh effect (en.wikipedia.org/wiki/Unruh_effect). $\endgroup$– Martin C.Commented Dec 2, 2019 at 10:43
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$\begingroup$ @MartinC. Actually in Unruh effect there is no issue of curvature. The spacetime is flat if we negelect the back reaction which is assumed by default. Unruh effect happens because of 2 different set of complete function and therefore 2 different vacuua. $\endgroup$– aitfelCommented Dec 2, 2019 at 11:44
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$\begingroup$ I’m definitely not an expert in this field, so I’ll take your word for it. $\endgroup$– Martin C.Commented Dec 2, 2019 at 13:12
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$\begingroup$ The role of the vacuum in the formulation of QFT is to fix a representation of the CCR. Due to the infinite number of the degrees of freedom, the Stone-von-Neumann theorem doesn't apply to QFT and there are inequivalent representations on different Hilbert spaces. These representations all have Poincare-invariant states, but the one that contains the physical vacuum (the Fock representation for free theories) has its vacuum as the lowest-energy state. $\endgroup$– Prof. LegolasovCommented Dec 3, 2019 at 10:19
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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.
answered Dec 3, 2019 at 3:24