### 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 scalars and Dirac fermions. The induced gravitational
action in this model, to lowest order in curvature, is
where the induced Newton’s constant is
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
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 (319). 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 (319) is set by the UV cutoff . 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 [95, 97, 87]. One considers a multiplet of scalar
fields of mass and non-minimal coupling and a set of massive Dirac fields with mass . 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
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 , 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
where the quantity is determined by the expectation value of the non–minimally-coupled scalar fields
on the horizon
This quantity is UV divergent. For a single field it is similar to the quantity (142). 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.