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I know how we define a vacuum in flat space QFT and also in a curved space QFT. But, can somebody tell me how do the choice of vacuum state in (say) the CFT side of AdS/CFT changes the choice of vacuum state in gravity side? Let me ask the other way. I mean if we pick a vacuum (say in bulk side, because it may not be unique), how does it reflect on the CFT vacuum (and vice versa)? So, my question is how does this choice reflect on both sides and how do we generally make the identification?


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The vacuum is an eigenstate of the energy operator corresponding to the lowest allowed eigenvalue. When there are two equivalent descriptions of a physical model, the vacua of both descriptions are obviously identified with each other because they satisfy the same condition I just described. If you're asking about something else than the trivialities I just said, then it could help if you were a little bit more comprehensible. –  Luboš Motl Oct 5 '11 at 17:08
Sorry for my misunderstandings. So, I don't have any problem with CFT side. But, I was thinking that because in the bulk side, we sometimes use field theory and its not a flat space, so whether it will be possible to choose different vacua by Bogoliubov transformations as is done in QFT in curved space subject. If that is so, then how will those different choices of vacua be identified with vacuum in CFT side. I mean in that case, what I am talking about are different mode expansions of a field in bulk side. So, will that similarly correspond to different modes in CFT side? –  user1349 Oct 5 '11 at 20:58
So, I have this two theories, one at bulk and other at boundary. And being equivalent their vacuua must be identified as you said. Now, suppose I make a Bogoliubov transformation to pick another set of creation/annihilation operator and hence a new vacuua. How did then the CFT side vacuum changed? E.g. in dS space I have this series of alpha vacuua. So, if a CFT description of dS QG exists, then how will going from one alpha vacuua to other, change the CFT vacuua? –  user1349 Oct 5 '11 at 21:05
Dear Disinterested, a random Bogoliubov transformation that renames combinations of creation and annihilation operators in the bulk as annihilation operators isn't a particularly natural operation in general and you shouldn't call the state annihilated by such new "annihilation" operators a vacuum. A vacuum is the lowest-energy eigenstate of an energy operator. In AdS/CFT, it's the global $H$ energy generating translations in time. –  Luboš Motl Oct 6 '11 at 5:11
On both sides, one may also consider the spectrum of other generators of the superconformal groups, and if there are ground states, they match as well. If you want to play with individual creation/annihilation operators in the bulk of the AdS, you won't find an easy dictionary in the CFT. The locality itself fails to be manifest on the CFT side even though it's manifest in the AdS bulk description. ... dS/CFT is a completely different issue and according to everything I can say, it's just wrong: we had long discussions with Andy Strominger about it and he would surely disagree. –  Luboš Motl Oct 6 '11 at 5:13

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I only know of this problem being discussed for scalars. In AdS, there's a unique SO(d-1,2) invariant vacuum, so your question doesn't apply. In de Sitter space, on the other hand, you have a one-parameter family of dS invariant vacua, labeled by a complex parameter alpha. Switching between these vacua can be accomplished by what's called a Mottola-Allen transform, and this corresponds to perturbing the CFT by some marginal deformation (at least in three-dimensional dS). See Bousso, Maloney, and Strominger for details.

I'm not really sure if these alpha vacua are so physical though. Taking the standard Euclidean vacuum, which is the analytic continuation of the vacuum from the sphere, corresponds to demanding that the fields start as plane waves, which sounds pretty reasonable. Also, Harlow and Stanford show that analytically continuing the infrared wavefunction from AdS gives the dS wavefunction with Euclidean initial conditions, so the alpha vacua are in some sense not as "preferred".

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