3.3 The wave function of a black hole

Although the entanglement entropy can be defined for any co-dimension two surface, when the surface is a horizon particular care is required. In order to apply the general prescription outlined in Section 2.1, we first of all need to specify the corresponding wave function. Here we will follow the prescription proposed by Barvinsky, Frolov and Zelnikov [15Jump To The Next Citation Point]. This prescription is a natural generalization of the one in flat spacetime discussed in Section 3.8. On the other hand, it is similar to the “no-boundary” wave function of the universe introduced in [127]. We define the wave function of a black hole by the Euclidean path integral over field configurations on the half-period Euclidean instanton defined by the metric (50View Equation) with angular coordinate ϕ changing in the interval from 0 to π. This half-period instanton has Cauchy surface ℋ (on which we can choose coordinates x = (ρ, 𝜃)) as a boundary where we specify the boundary conditions in the path integral,
∫ Ψ [ψ − (x),ψ+ (x)] = 𝒟ψ e−W [ψ], (53 ) ψ (X )|ϕ=0 = ψ+(x) ψ (X )|ϕ= π = ψ− (x ),
where 1 ∫ ˆ W [ψ] = 2 ψ 𝒟 ψ is the action of the quantum field ψ. The functions ψ− (x ) and ψ+ (x ) are the boundary values defined on the part of the hypersurface ℋ, which is respectively inside (ℋ −) and outside (ℋ+) the horizon Σ. As was shown in [15Jump To The Next Citation Point], the wave function (53View Equation) corresponds to the Hartle–Hawking vacuum state [126Jump To The Next Citation Point].
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