# What is an vacuum in QFT in curved spacetime?

Sometimes in some public lectures about General Relativity (GR) and Quantum Mechanics, in college, the professors dealt with the vacuum concept, precisely in the context of Quantum Field Theory (QFT), like: Minkowski Vacuum, Rindler Vacuum.

I'm understanding this concept as the background spacetime, like for instance, in Schwarschild geometry the trajectory of light-like geodesics and time-like geodesics are profoundly different from flat geometries like Minkowski spacetime. From this point of view we have then a Schwarschild Vacuum, but I'm not sure if this is the right way to see this QFT picture.

So what is a Vacuum in GR and QFT?

A vacuum isn't a terribly well-defined concept in general, but in Minkowski space for free field theories bounded from below, we have the good luck of all the definitions lining up : there's a unique state $$\Omega \in \mathcal{H}$$ that is invariant under Poincaré transformations and such that $$\langle \hat{H} \rangle_\Omega$$ minimizes the energy.
In the broadest sense, a vacuum state is basically any state we wish to pick up to build the theory from. Given some operator algebra $$\mathscr{A}$$, you pick a cyclic state $$\psi \in \mathcal{H}$$ (so that applying $$\mathscr{A}$$ on $$\psi$$ is dense in $$\mathcal{H}$$). As there are no prefered vacuum in general theory, all we can really ask is that it relates to every other state.