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Lorentzian spinfoam gravity path integral and geometrical area-law entanglement entropy

Muxin Han

TL;DR

This work presents a Lorentzian spinfoam path-integral framework to compute entanglement entropy in 3+1D LQG by summing over a family of 2-complexes (spinfoam stacks). Using a replica-trick formalism adapted to the spinfoam setting and a Laplace-transform-based state-sum, the leading entanglement entropy scales with the boundary area, $S \simeq \beta a$, where $a$ is fixed by the LQG area spectrum and the coefficient $\beta$ is independent of the chosen 2-complex; for $0<\gamma\lesssim 1/2$ one can reproduce the Bekenstein-Hawking result $S=\mathrm{Ar}(\mathfrak{S})/(4\ell_P^2)$ by tuning the coupling constants to the Barbero-Immirzi parameter. The analysis avoids Euclidean-contour ambiguities and demonstrates the robustness of the area law under variations in coarse-graining, initial states, and alternative area-fixing prescriptions, while providing a Lorentzian, gauge-consistent derivation of gravitational entropy from the quantum geometry of spinfoams.

Abstract

This paper investigates entanglement entropy in 3+1 dimensional Lorentzian covariant Loop Quantum Gravity (LQG). We compute the entanglement entropy for a spatial region from states dynamically generated by a spinfoam path integral that sums over a family of 2-complexes. The resulting entropy exhibits a geometric area law, $S \simeq βa$, where the area $a$ of the entangling surface is determined by the LQG area spectrum and the leading coefficient $β>0$ is independent of the underlying 2-complexes. By relating the coupling constant of the sum over 2-complexes to the Barbero-Immirzi parameter $γ$, we reproduce the Bekenstein-Hawking formula for the range $0 < γ\lesssim 1/2$. This work provides a Lorentzian path integral approach to gravitational entropy without the need for contour prescriptions.

Lorentzian spinfoam gravity path integral and geometrical area-law entanglement entropy

TL;DR

This work presents a Lorentzian spinfoam path-integral framework to compute entanglement entropy in 3+1D LQG by summing over a family of 2-complexes (spinfoam stacks). Using a replica-trick formalism adapted to the spinfoam setting and a Laplace-transform-based state-sum, the leading entanglement entropy scales with the boundary area, , where is fixed by the LQG area spectrum and the coefficient is independent of the chosen 2-complex; for one can reproduce the Bekenstein-Hawking result by tuning the coupling constants to the Barbero-Immirzi parameter. The analysis avoids Euclidean-contour ambiguities and demonstrates the robustness of the area law under variations in coarse-graining, initial states, and alternative area-fixing prescriptions, while providing a Lorentzian, gauge-consistent derivation of gravitational entropy from the quantum geometry of spinfoams.

Abstract

This paper investigates entanglement entropy in 3+1 dimensional Lorentzian covariant Loop Quantum Gravity (LQG). We compute the entanglement entropy for a spatial region from states dynamically generated by a spinfoam path integral that sums over a family of 2-complexes. The resulting entropy exhibits a geometric area law, , where the area of the entangling surface is determined by the LQG area spectrum and the leading coefficient is independent of the underlying 2-complexes. By relating the coupling constant of the sum over 2-complexes to the Barbero-Immirzi parameter , we reproduce the Bekenstein-Hawking formula for the range . This work provides a Lorentzian path integral approach to gravitational entropy without the need for contour prescriptions.

Paper Structure

This paper contains 14 sections, 3 theorems, 84 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Stacking spin-networks
  3. Stacking spinfoams
  4. Laplace transform and state sum
  5. Laplace transform
  6. Stationary phase approximation
  7. Modular entanglement entropy
  8. Entanglement entropy of spinfoam state
  9. The spinfoam state
  10. Replica trick
  11. Computing the replica partition function
  12. Spinfoam state with fix areas
  13. An analog of black hole entropy
  14. Solutions of $\sum_{k=1}^\infty (\pm 1)^k d_k^{2}e^{- s_\pm \sqrt{k(k+2)}} = \lambda^{-1}$

Key Result

Lemma 4.1

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 (9)

  • Figure 1: The spin-network stack.
  • Figure 2: Coarse-graining $p$ stacked links with spins $k_1/2,\cdots,k_p/2$ to the macroscopic area $A_\mathfrak{l}$ associated to the surface $\mathfrak{S}_\mathfrak{l}$.
  • Figure 3: The spin-network stack evolves to the spinfoam stack: The spin-network link $\mathfrak{l}$ evolves to the spinfoam face $f$. the spin-network nodes $\mathfrak{n}_1,\mathfrak{n}_2$ evolve to the spinfoam edges $e_1,e_2$. The faces evolves from the dashed links are not shown on this figure. The leftmost complex is the root complex. The power of coupling constant $\lambda_f$ counts the number of stacked faces.
  • Figure 4: A spinfoam state $\psi$ with the initial state $\psi_0$.
  • Figure 5: The replica partition function $\mathcal{Z}_{\mu,n}$ for $n=2$: The gluing with the blue arrows is due to $\mathrm{Tr}_{\overline{R},\mu}$ defining the reduced density operator $\bm\rho_{R,\mu}$. The gluing with the red arrows corresponds to the product of $\bm\rho_{R,\mu}$ and the trace $\mathrm{Tr}_{R,\mu}$. The gluing gives a manifold with branch surface $\mathfrak{S}=R\cap\overline{R}$.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3