### 7.1 Holographic proposal for entanglement entropy

Let be a -dimensional asymptotically anti-de Sitter spacetime. Its conformal boundary is
a -dimensional spacetime . On a slice of constant time in one picks a closed
-dimensional surface and defines the entropy of entanglement with respect to .
Now, on the constant slice of the -dimensional anti-de Sitter spacetime, consider a
-dimensional minimal surface such that its boundary in is the surface , .
According to [189, 188], the following quantity
where is the area of minimal surface and is Newton’s constant in the -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
() it correctly reproduced (10) and (11) for the entropy at finite size and at finite entropy
respectively. In higher dimensions ( the area of minimal surface diverges when it is
extended till . This is an important feature, typical of the AdS/CFT correspondence.
In fact, instead of a conformal boundary , one has to consider a regularized boundary
located at a small distance (measured in terms of some radial coordinate ). In the
AdS/CFT correspondence the divergence in has the interpretation of a UV divergence in
the boundary quantum field theory. Considering the regularized surface , which extends
to , one finds that its area, to leading order in , behaves as .
Taking this behavior of the area, one sees that the proposal (303) correctly reproduces the
proportionality of the entropy to the area of surface and its dependence on the UV cutoff
.
However, we note that in certain situations the choice of the minimal surface may not be unique. In
particular, if the dividing surface 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 . Different choices may correspond to different phases in the boundary
theory [157].