# Semiclassical Behavior of Spinfoam Amplitude with Small Spins and Entanglement Entropy

## TL;DR

This paper explores the semiclassical behavior of small-spin spinfoam amplitudes in loop quantum gravity, revealing their relation to Regge calculus, and investigates their entanglement entropy, which obeys a thermodynamical-like law.

## Contribution

It demonstrates that small-spin generalized spinfoam amplitudes can define semiclassical states linked to Regge geometry and reveals their entanglement entropy satisfies a thermodynamical law.

## Key findings

- Small-spin spinfoam amplitudes relate to Regge calculus via coarse-graining.
- Semiclassical states are characterized by 4D Regge geometries.
- Entanglement entropy obeys a thermodynamical first law.

## Abstract

Spinfoam amplitudes with small spins can have interesting semiclassical behavior and relate to semiclassical gravity and geometry in 4 dimensions. We study the generalized spinfoam model (Spinfoams for all loop quantum gravity (LQG)) with small spins $j$ but a large number of spin degrees of freedom (DOFs), and find that it relates to the simplicial Engle-Pereira-Rovelli-Livine-Freidel-Krasnov (EPRL-FK) model with large spins and Regge calculus by coarse-graining spin DOFs. Small-$j$ generalized spinfoam amplitudes can be employed to define semiclassical states in the LQG kinematical Hilbert space. Each of these semiclassical states is determined by a 4-dimensional Regge geometry. We compute the entanglement R\'enyi entropies of these semiclassical states. The entanglement entropy interestingly coarse-grains spin DOFs in the generalized spinfoam model, and satisfies an analog of the thermodynamical first law. This result possibly relates to the quantum black hole thermodynamics in arXiv:1107.1320.