8.2 Entanglement entropy in induced gravity

One, possibly very natural, way, originally proposed by Jacobson [141], to attack these problems is to consider gravity as an induced phenomenon, in the spirit of Sakharov’s ideas [190] (for a review on a modern touch on these ideas see [220]). In this approach the gravitational field is not fundamental but arises as a mean field approximation of the underlying quantum field theory of fundamental particles (the constituents). This is based on the fact that, even if there is no gravitational action at tree level, it will appear at one-loop. The details of this mechanism will, of course, depend on the concrete model.

Model with minimally-coupled fields.
To start with, let us consider a simple model in which the constituents are minimally-coupled fields: we consider N 0 scalars and N 1∕2 Dirac fermions. The induced gravitational action in this model, to lowest order in curvature, is

∫ W = − ----1--- R√gd4x , (318 ) ind 16 πGind E
where the induced Newton’s constant is
--1-- --N--- G = 12π 𝜖2 , N = N0 + 2N1 ∕2, (319 ) ind
N being the number of field species in this model. The renormalization statement, which is valid for the minimally-coupled fields, guarantees that there is a precise balance between the induced Newton’s constant and the entanglement entropy, so that
N 1 Sent = -----2A(Σ ) = ------A(Σ ) = SBH , (320 ) 48π 𝜖 4Gind
i.e., the entanglement entropy of the constituents is precisely equal to the Bekenstein–Hawking entropy, expressed in terms of the induced Newton’s constant (319View Equation). Thus, if at a fundamental level the constituents in nature were only minimal fields, the Bekenstein–Hawking entropy, as this example shows, would be explained as the entropy of entanglement. Of course, this example ignores the fact that there are elementary particles, namely the gauge bosons, which are non-minimally coupled.

Models with non-minimally coupled fields.
In the model with minimal fields the induced Newton’s constant (319View Equation) is set by the UV cutoff 𝜖. If one wants to deal with the UV finite quantities, one has to add fields, which contribute negatively to Newton’s constant. Excluding non-physical fields with wrong statistics, the only possibility is to include non–minimally-coupled fields, scalars or vectors. Models of this type have been considered by Frolov, Fursaev and Zelnikov [95Jump To The Next Citation Point, 97Jump To The Next Citation Point, 87Jump To The Next Citation Point]. One considers a multiplet of scalar fields of mass ms and non-minimal coupling ξs and a set of massive Dirac fields with mass md. The number of fields and their parameters are fine tuned so that the ultra-violet divergences in the cosmological constant and in Newton’s constant are canceled. The induced Newton’s constant then

(∑ ∑ ) -1---= --1- (1 − 6ξs)m2 ln m2 + 2 m2 ln m2 (321 ) Gind 12 π s s s d d
is dominated by the mass of the heaviest constituents. However, as soon as we include the non–minimally-coupled fields the precise balance between Newton’s constant and the entanglement entropy is violated, so that the Bekenstein–Hawking entropy SBH = A (Σ )∕4Gind, defined with respect to the induced Newton’s constant, is no longer equal to the entanglement entropy. In the model considered in [95, 97, 87] (various models of a similar nature are considered in [88, 96, 93, 91, 92, 89, 106, 105]), the exact relation between two entropies is
SBH = Sent − Q , (322 )
where the quantity Q is determined by the expectation value of the non–minimally-coupled scalar fields on the horizon Σ
∫ ∑ 2 Q = 2π ξs Σ < ϕs > . (323 ) s
This quantity is UV divergent. For a single field it is similar to the quantity (142View Equation). Thus, the sharp difference between the entanglement entropy and the Bekenstein–Hawking entropy in this model can be summarized as follows: even though the induced Newton’s constant is made UV finite, the entanglement entropy still (and, in fact, always) remains UV divergent. Thus, we conclude that, in the model of Frolov, Fursaev and Zelnikov, the entanglement entropy is clearly different from the Bekenstein–Hawking entropy.
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