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Entanglement entropy in Loop Quantum Gravity and geometrical area law

Muxin Han

TL;DR

This work develops an algebraic framework for entanglement entropy in Loop Quantum Gravity, where gauge invariance prevents a simple factorization of the Hilbert space. By formulating entropy with non-factor von Neumann algebras and a renormalized trace, it derives a geometrical area law S ∝ Ar(𝒮) from fixed-area states and shows bulk entanglement can renormalize the area-law coefficient and produce logarithmic corrections. The analysis connects LQG microstate counting to black hole entropy and extends to infinite-dimensional settings, where edge modes and hidden sectors systematically account for boundary gauge freedoms. Overall, the paper provides a rigorous, gauge-consistent pathway from quantum geometry to horizon thermodynamics, clarifying how boundary and bulk degrees of freedom shape entanglement in quantum gravity.

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.

Entanglement entropy in Loop Quantum Gravity and geometrical area law

TL;DR

This work develops an algebraic framework for entanglement entropy in Loop Quantum Gravity, where gauge invariance prevents a simple factorization of the Hilbert space. By formulating entropy with non-factor von Neumann algebras and a renormalized trace, it derives a geometrical area law S ∝ Ar(𝒮) from fixed-area states and shows bulk entanglement can renormalize the area-law coefficient and produce logarithmic corrections. The analysis connects LQG microstate counting to black hole entropy and extends to infinite-dimensional settings, where edge modes and hidden sectors systematically account for boundary gauge freedoms. Overall, the paper provides a rigorous, gauge-consistent pathway from quantum geometry to horizon thermodynamics, clarifying how boundary and bulk degrees of freedom shape entanglement in quantum gravity.

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.

Paper Structure

This paper contains 16 sections, 4 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Hilbert space
  3. Von Neumann algebra and entanglement entropy: Finite-dimensional Hilbert space
  4. Fixed-area state and geometrical area law
  5. Fixed-area states
  6. Maximizing the entanglement entropy and microstate-counting
  7. A general scheme of state-sum
  8. Bulk entropy
  9. Random states
  10. Renormalization of area-law
  11. Logarithmic correction
  12. Von Neumann algebra and entanglement entropy: Infinite-dimensional Hilbert space
  13. Von Neumann algebra of holonomies and fluxes
  14. Renormalized trace, density operator, and entanglement entropy
  15. Hidden sector Hilbert space
  16. ...and 1 more sections

Key Result

Lemma 5.1

Given two sequence $\{\alpha_n\}_{n=1}^\infty$ and $\{\beta_n\}_{n=1} ^\infty$ where $\beta_n\in\mathbb{C}$ and $\alpha_n>0$, if $\sum_{n=1}^\infty\left|\beta_{n}\right|e^{-\alpha_{n}\mathrm{Re}(s)}<\infty$ for some $\mathrm{Re}(s)>0$, we have for the cut-off $a$ that does not coincide with any $\alpha_n$. The parameter $T>0$ is greater than the real part of all singularities given by the integra

Figures (2)

  • Figure 1: Graphic notations for $|\psi_{A,{\bar{A}}}\rangle$, $\bm{\rho}$, and \ref{['averagerhorho']}.
  • Figure 2: The graphic computations of the averages $\langle Z_1\rangle$ and $\langle Z_0\rangle$.

Theorems & Definitions (4)

  • Lemma 5.1
  • Lemma 7.1
  • Lemma 7.2
  • Theorem 7.3