7.1 Holographic proposal for entanglement entropy

Let M be a (d + 1)-dimensional asymptotically anti-de Sitter spacetime. Its conformal boundary is a d-dimensional spacetime ∂M. On a slice of constant time t in ∂M one picks a closed (d − 2)-dimensional surface Σ and defines the entropy of entanglement with respect to Σ. Now, on the constant t slice of the (d + 1)-dimensional anti-de Sitter spacetime, consider a (d − 1)-dimensional minimal surface γ such that its boundary in ∂M is the surface Σ, ∂γ = Σ. According to [189, 188], the following quantity
A (γ) S = ------, (303 ) 4Gd+1
where A(γ) is the area of minimal surface γ and Gd+1 is Newton’s constant in the (d + 1)-dimensional gravitational theory, is equal to the entanglement entropy one has calculated in the boundary conformal field theory. This holographic proposal has passed many tests and never failed. It has correctly reproduced the entropy in all cases where it is explicitly known. In particular, in two spacetime dimensions (d = 2) it correctly reproduced (10View Equation) and (11View Equation) for the entropy at finite size and at finite entropy respectively. In higher dimensions (d > 2) the area of minimal surface γ diverges when it is extended till ∂M. This is an important feature, typical of the AdS/CFT correspondence. In fact, instead of a conformal boundary ∂M, one has to consider a regularized boundary ∂M 𝜖 located at a small distance 𝜖 (measured in terms of some radial coordinate ρ). In the AdS/CFT correspondence the divergence in 𝜖 has the interpretation of a UV divergence in the boundary quantum field theory. Considering the regularized surface γ, which extends to ∂M 𝜖, one finds that its area, to leading order in 𝜖, behaves as A (γ) ∼ A (Σ )∕𝜖d−2. Taking this behavior of the area, one sees that the proposal (303View Equation) correctly reproduces the proportionality of the entropy to the area of surface Σ and its dependence on the UV cutoff 𝜖.

However, we note that in certain situations the choice of the minimal surface Σ may not be unique. In particular, if the dividing surface Σ has several components or if the quantum field resides inside a cavity instead of being defined on an infinite space, there is more than one natural choice of the minimal surface γ. Different choices may correspond to different phases in the boundary theory [157].

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