Entanglement entropy in Loop Quantum Gravity and geometrical area law

TL;DR

This paper develops an algebraic approach to define and analyze entanglement entropy in Loop Quantum Gravity, deriving an area law and exploring quantum corrections relevant to black hole entropy.

Contribution

It introduces a von Neumann algebra framework for entanglement entropy in LQG and derives an area law with quantum corrections for fixed-area states.

Findings

Entanglement entropy is proportional to the surface area.

Bulk entanglement renormalizes the area-law coefficient.

Logarithmic corrections to the entropy are identified.

Abstract

The non-factorizing nature of the Hilbert space in Loop Quantum Gravity (LQG) due to gauge invariance requires a generalized definition of entanglement entropy. This work employs the framework of von Neumann algebras to investigate the entanglement entropy in LQG. On a graph, the holonomy and flux operators within a region and on the boundary generate a non-factor type I von Neumann algebra, which is used to define the entanglement entropy for LQG states. This algebraic formalism is applied to ``fixed-area states''--superpositions of spin networks associated with a surface with a definite macroscopic area given by the LQG area spectrum. By maximizing the entropy, we derive a geometrical area law where the entanglement entropy is proportional to the area. In addition, we show that bulk entanglement can renormalize the area-law coefficient and produce logarithmic corrections. The results in this paper closely relate to LQG black hole entropy.