I have been learning QFT in curved spaces from various sources (Birrell/Davies, Tom/Parker, some papers), and one thing that confuses me the most is the choice of vacua in various spacetimes, and the definition used in various contexts. They seem to be everywhere and I am somewhat disoriented.
For convenience, I will break the question into three parts. If I have misunderstood something fundamental, I would appreciate the correction. I am not sure if it is too long as a single post, but they look connected enough to be on the same post.
UPDATE: it has been suggested that this question is too long (especially Part 1), but I think what I need is coherence so multiple answers that can be read together into coherent explanations will work. I am looking for basically something similar to an explanation about differences between correlators in this StackExchange post here, which helps because most sources do not put them side by side or use the same language.
(1) What are the definitions of the following vacua for spacetimes where you can define them?
- Bunch-Davies vacuum
- Instantaneous vacuum
- Adiabatic vacuum
- Conformal vacuum$^\dagger$
- Unruh vacuum
- Hartle-Hawking vacuum
- Boulware vacuum
- Static vacuum*
To start with, I do know some basic ideas about how these states are defined; usually, you need to pick how to split the Hilbert space into positive and negative frequency solutions, and they are different based on boundary conditions that the eigenmodes have to satisfy (positive frequency means the Lie derivative with respect to the appropriate Killing time is proportional to the frequency); furthermore, for some states (e.g. Unruh/HH/Boulware), often the narrative is based on where the non-zero stress energy is, though strictly speaking it's a matter of what Killing time you pick. What I am more interested in is the definitions, and perhaps more importantly the distinguishing features between these states (if at all).
Only for $^\dagger$ do I know how to define it precisely, since it requires the use of conformal coupling and conformal invariance of some sort to define it, therefore it is also independent of the precise spacetime you use it on so long as the symmetry is there. For some of these, I know roughly how they work in specific spacetimes, but I am still somewhat confused, and I suspect that for some of those, they coincide. For example, Unruh/HH/Boulware is used in BH spacetimes, but if I were to switch to cosmological spacetimes with horizons, would these names be applicable? In the de Sitter case for the massless conformally coupled scalar field, some sources say adiabatic/conformal/Bunch-Davies are the same thing, though not clear how so.
Furthermore, for some vacua in say cosmological contexts, they were defined with respect to some length scale (super/subhorizon scales) which I don't understand. For static vacuum, it is a name I give it myself because I cannot find it fitting anywhere; it is the vacuum for static coordinates in de Sitter spacetimes. Does it coincide with some of these (or say if I have instead Schwarzschild-de Sitter black holes)? What about say the AdS spacetime, which is often studied in many other contexts, yet I only really heard of conformal vacuum being used in most cases (so is that Poincare coordinates?), though presumably in presence of black holes these would go back to Unruh/HH/Boulware?
(2) Is it necessary that a good vacuum state be "Hadamard", i.e. have a particular short-distance behaviour similar to Minkowski? It looks to me that it has to be, but I have heard that some seemingly good/well-motivated choice of states violate this condition (e.g. Sorkin-Johnston state, or some vacua in the massless minimally coupled scalar field in de Sitter --- I don't know which vacua, hence the question in Part 1)? This is one reason I removed SJ state from part 1 (for now).
(3) There have been quantization methods involving null hypersurfaces (this is done e.g. in Unruh effect derivation in Wald's text, or Frolov's null surface quantization paper). I know that strictly speaking, the choice of vacua is not about the choice of coordinates but rather the boundary conditions we put on the field modes, and some of these are very natural in some coordinates but not others. However, null slicing introduces null coordinates as natural time, and it seems to me that it would make some choice of vacua hard to define since you might be using $\mathscr{I}^\pm$ to define the boundary conditions. Even worse in the case of rotating spacetimes, where the Cauchy surface would need to include the horizons, as I think how the Frolov-Thorne vacuum (?) is defined. Is the choice of null slicing essential for defining certain vacua, and if so for what "textbook" states are these?
