Entanglement Entropy in Quantum Gravity and the Plateau Problem

In a quantum gravity theory the entropy of entanglement $S$ between the fundamental degrees of freedom spatially divided by a surface is discussed. The classical gravity is considered as an emergent phenomenon and arguments are presented that: 1) $S$ is a macroscopical quantity which can be determined without knowing a real microscopical content of the fundamental theory; 2) $S$ is given by the Bekenstein-Hawking formula in terms of the area of a co-dimension 2 hypesurface $\cal B$; 3) in static space-times $\cal B$ can be defined as a minimal hypersurface of a least volume separating the system in a constant time slice. It is shown that properties of $S$ are in agreement with basic properties of the von Neumann entropy. Explicit variational formulae for $S$ in different physical examples are considered.