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1. Restating the question

You are asked to list and explain the most common “vacuum states’’ that appear in quantum field theory on curved (or non‑trivial) backgrounds.
For each of the following names you must give

  • a precise definition (how the mode functions are chosen, what symmetry or boundary condition is imposed, etc.);
  • the physical or mathematical property that makes the state special (e.g. regularity at a horizon, invariance under a Killing field, minimisation of particle production, etc.); and
  • a short comment on how it differs from the other vacua in the same space‑time.

The list to be covered is

Vacuum name Space‑times where it is usually defined
Bunch‑Davies (BD) de Sitter (global or planar)
Instantaneous (or “instant‑time’’) any globally hyperbolic spacetime, used as a bookkeeping device
Adiabatic slowly‑varying (e.g. FLRW) backgrounds
Conformal conformally flat spacetimes, massless conformally coupled fields
Unruh black‑hole exterior (or Rindler) with a future horizon
Hartle‑Hawking (HH) eternal black hole (Kruskal) – thermal equilibrium
Boulware static black‑hole exterior – “empty’’ at infinity
Static any static spacetime – the state defined by positive‑frequency w.r.t. the static Killing time

Below we give a step‑by‑step construction of each vacuum, then summarise the key distinguishing features, and finally list the common pitfalls.

2. Detailed constructions

2.1 General set‑up

Consider a free real scalar field (\phi) (mass (m) and curvature coupling (\xi)) on a globally hyperbolic spacetime ((\mathcal M,g_{\mu\nu})).
The field equation is

[ \bigl(\Box_g + m^2 + \xi R\bigr)\phi = 0 . ]

Because the equation is linear, we may expand (\phi) in a complete set of solutions ({u_{\bf k}(x)}),

[ \phi(x)=\sum_{\bf k}\bigl(a_{\bf k}u_{\bf k}(x)+a_{\bf k}^{\dagger}u_{\bf k}^*(x)\bigr) . ]

A vacuum state is completely specified once we decide which linear combinations of solutions are to be called positive‑frequency modes.
The annihilation operators (a_{\bf k}) associated with those modes then annihilate the vacuum:

[ a_{\bf k}\,|0\rangle =0\quad\forall{\bf k}. ]

Thus the definition of a vacuum reduces to a choice of mode basis (or, equivalently, a choice of complex structure on the space of classical solutions).
All the vacua below are distinguished by how this choice is made.

2.2 Bunch–Davies (BD) vacuum

Spacetime: (spatially flat) de Sitter space, usually written in conformal coordinates

[ ds^2 = \frac{1}{(H\eta)^2}\bigl(-d\eta^2 + d\mathbf{x}^2\bigr),\qquad \eta\in(-\infty,0). ]

Mode construction:

  1. Write the field equation in Fourier space: (\phi_{\bf k}(\eta) = \int d^3x\,e^{-i{\bf k}\cdot\mathbf{x}}\,\phi(\eta,\mathbf{x})).

  2. The mode functions satisfy

    [ \phi_{\bf k}’’ +\frac{2}{\eta}\phi_{\bf k}’ +\Bigl(k^2+\frac{m^2}{H^2\eta^2}\Bigr)\phi_{\bf k}=0 . ]

  3. The two independent solutions are Hankel functions (H^{(1)}{\nu}(-k\eta)) and (H^{(2)}{\nu}(-k\eta)) with (\nu =\sqrt{\frac{9}{4}-\frac{m^2}{H^2}}).

  4. BD prescription: pick the positive‑frequency solution that behaves like a Minkowski plane wave for (\eta\to -\infty) (i.e. early‑time, sub‑horizon limit).

    [ u^{\rm BD}{\bf k}(\eta,\mathbf{x}) = \frac{\sqrt{\pi}}{2} H^{(1)}{\nu}(-k\eta)\frac{e^{i{\bf k}\cdot\mathbf{x}}}{(2\pi)^{3/2}} . ]

    The asymptotic form (\sim e^{-ik\eta}) guarantees that an inertial observer in the far past sees no particles.

Key property:

  • It is de Sitter‑invariant (the two‑point function depends only on the invariant distance).
  • It is the unique Hadamard state that is regular at the past conformal boundary (\mathscr I^{-}).

2.3 Instantaneous (or “instant‑time’’) vacuum

Spacetime: Any globally hyperbolic manifold with a chosen Cauchy surface (\Sigma_{t_0}).

Idea: At a fixed time (t_0) we diagonalise the instantaneous Hamiltonian (the generator of evolution with respect to the chosen time function).

Construction steps

  1. Choose a foliation (\Sigma_t) with lapse (N) and shift (N^i).
  2. On the slice (\Sigma_{t_0}) define the canonical variables (\phi(\mathbf{x})) and its momentum (\pi(\mathbf{x})).

  3. Expand them in eigenfunctions of the spatial Laplacian (-\Delta_{\Sigma}):

    [ -\Delta_{\Sigma} Y_{n}(\mathbf{x}) = \omega_n^2(t_0) Y_n(\mathbf{x}) . ]

  4. The instantaneous positive‑frequency mode is

    [ u_n^{\rm inst}(t,\mathbf{x}) = \frac{1}{\sqrt{2\omega_n(t_0)}}\, e^{-i\omega_n(t_0)(t-t_0)}\,Y_n(\mathbf{x}), ]

    evaluated only at the chosen instant (t=t_0).

  5. Define the vacuum ( 0_{t_0}\rangle) by (a_n^{\rm inst} 0_{t_0}\rangle=0).

Key property:

  • It is time‑dependent: the definition changes if you pick a different Cauchy surface.
  • It is useful as a reference when discussing particle production (the number of particles measured at a later time with respect to the instantaneous vacuum at the earlier time).

2.4 Adiabatic vacuum

Spacetime: Typically an FLRW universe with scale factor (a(t)) that varies slowly compared with the field frequency.

Motivation: In a slowly varying background one can construct mode functions that minimise the amount of particle creation order‑by‑order in a WKB/adiabatic expansion.

Construction (to (n^{\rm th}) adiabatic order)

  1. Write the mode equation for a spatial Fourier mode (k):

    [ \ddot{\chi}_k + \omega_k^2(t)\,\chi_k = 0,\qquad \omega_k^2(t)=\frac{k^2}{a^2}+m^2+\Bigl(\xi-\tfrac{1}{6}\Bigr)R . ]

  2. Seek a WKB ansatz

    [ \chi_k^{(n)}(t)=\frac{1}{\sqrt{2W_k^{(n)}(t)}}\exp!\Bigl(-i\int^t!W_k^{(n)}(t’)dt’\Bigr), ]

    where (W_k^{(n)}) is expanded recursively:

    [ \bigl(W_k^{(n)}\bigr)^2 = \omega_k^2 - \frac{1}{2}\frac{\ddot W_k^{(n-1)}}{W_k^{(n-1)}}+\frac{3}{4}\Bigl(\frac{\dot W_k^{(n-1)}}{W_k^{(n-1)}}\Bigr)^2, ]

    with the zeroth‑order choice (W_k^{(0)}=\omega_k).

  3. The adiabatic vacuum of order (n) is defined by taking the positive‑frequency modes (\chi_k^{(n)}) as the basis.

  4. In the limit (n\to\infty) (if the series converges) one obtains a preferred Hadamard state; in practice one stops at the lowest order that renders the renormalised stress tensor finite.

Key property:

  • It is locally defined (depends only on the metric and its derivatives at the chosen time) and is Hadamard to the adiabatic order used.
  • It reduces to the BD vacuum in de Sitter when the expansion is exactly exponential (the adiabatic series can be summed).

2.5 Conformal vacuum

Spacetime: Any conformally flat background, i.e. (g_{\mu\nu}= \Omega^2(x)\,\eta_{\mu\nu}).

Field: A massless scalar with conformal coupling (\xi = 1/6).

Construction:

  1. Use the conformal map (\phi = \Omega^{-1}\tilde\phi). The action for (\tilde\phi) is just that of a free massless field in flat Minkowski space.

  2. Choose the Minkowski vacuum for (\tilde\phi): positive‑frequency plane waves (\tilde u_{\bf k}\propto e^{-i \mathbf{k} \tilde t + i\mathbf{k}\cdot\mathbf{x}}).

  3. Pull back to the original spacetime:

    [ u_{\bf k}^{\rm conf}(x)=\Omega^{-1}(x)\,\tilde u_{\bf k}\bigl(\tilde x(x)\bigr). ]

  4. The state annihilated by the corresponding (a_{\bf k}) is the conformal vacuum.

Key property:

  • The two‑point function is simply (\langle 0_{\rm conf} \phi(x)\phi(x’) 0{\rm conf}\rangle = \Omega^{-1}(x)\Omega^{-1}(x’)\,\langle 0{\rm M} \tilde\phi(\tilde x)\tilde\phi(\tilde x’) 0_{\rm M}\rangle).
  • It is invariant under the full conformal group (when the background admits that symmetry).

2.6 Unruh vacuum

Spacetime: An eternal black‑hole geometry (e.g. Schwarzschild) or flat space in Rindler coordinates.
We will give the Schwarzschild version; the Rindler case is analogous.

Geometric picture: The maximally extended Schwarzschild spacetime contains two asymptotically flat regions (right R and left L) and a future horizon (\mathcal H^{+}).

Mode basis

Region Positive‑frequency definition
Right exterior (R) Modes that are positive‑frequency w.r.t. the Killing time (t) (∂/∂t) that is static in region R.
Future horizon (\mathcal H^{+}) Modes that are positive‑frequency w.r.t. the affine parameter (U) (null Kruskal coordinate) on (\mathcal H^{+}).
Left exterior (L) No condition – the state is empty there (no particles coming from L).

Construction steps

  1. Expand the field in “in‑going’’ modes that originate from past null infinity (\mathscr I^{-}) and propagate into the right exterior. These have the usual (e^{-i\omega t}) dependence at large (r).

  2. On the future horizon use Kruskal coordinates ((U,V)) where (U = -e^{-\kappa (t-r_*)}) (with surface gravity (\kappa)). Positive‑frequency there means dependence (e^{-i\Omega \ln(-U)}

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