Entanglement Entropy in Quantum Gravity and the Plateau Problem

TL;DR

This paper explores how entanglement entropy in quantum gravity relates to classical concepts like the Bekenstein-Hawking formula and minimal hypersurfaces, suggesting entropy is a macroscopic, emergent quantity consistent with von Neumann entropy properties.

Contribution

It demonstrates that entanglement entropy in quantum gravity can be described by classical geometric quantities without detailed microscopic knowledge, linking it to minimal hypersurfaces and the Bekenstein-Hawking formula.

Findings

Entanglement entropy is a macroscopic quantity determined by surface area.

In static spacetimes, the entropy corresponds to minimal hypersurfaces.

The properties of entanglement entropy align with von Neumann entropy characteristics.

Abstract

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.