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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?

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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.

This definition doesn't work out in general due to the fact that both no state may be invariant under Poincaré transformation (ie any spacetime that isn't maximally symmetric) and that states of lowest energy may fail to be unique (for instance the Higgs vacua). If we allow for arbitrary coordinate transformations, the notion of a state of lowest energy may even fail to be invariant under such.

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.

For convenience (and because it is called a vacuum after all), it's typically a state which obeys some properties. It's often going to be a state of lowest energy in a given coordinate system (maybe for some observer), with some arbitrary choice if there is more than one such state (for instance pick a Higgs vacua with a certain phase). Different observers will see different vacua as is known from the Unruh and Hawking radiation, so we just require conditions for specific observers.

For instance you will have three commonly used vacuum in Schwarzschild spacetime. The Boulware vacuum is roughly equivalent to a vacuum state from the perspective of an observer at infinity, the Unruh vacuum is one for a free-falling observer, while the Hartle-Hawking vacuum has the benefit of being well-behaved on both sides of the horizon (this isn't so much the case for Boulware or Unruh) but isn't really a vacuum state as it is never empty of radiation.

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  • $\begingroup$ If we allow general coordinate transformations, the notion of energy itself doesn’t make sense anymore. Energy is the Noether charge associated with time translations. GCTs contain time translations as a gauge redundancy rather than a symmetry. In background independent setting, there simply is no global time. $\endgroup$ – Prof. Legolasov Sep 20 '19 at 12:56

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