Volume Entropy

TL;DR

This paper introduces the concept of volume entropy in Loop Quantum Gravity by analyzing the finite-dimensional space of quadrivalent, diffeomorphism-invariant states with bounded volume, revealing a logarithmic bound.

Contribution

It demonstrates that the space of certain quantum states with bounded volume has finite dimension, enabling the definition of volume entropy in this context.

Findings

Finite dimension of state space bounded by V log V

Introduction of volume entropy as von Neumann entropy

Bounded state space for regions with no zero-volume nodes

Abstract

Building on a technical result by Brunnemann and Rideout on the spectrum of the Volume operator in Loop Quantum Gravity, we show that the dimension of the space of the quadrivalent, diffeomorphism invariant states with no zero-volume nodes describing a region with total volume smaller than $V$, has \emph{finite} dimension, bounded by $V \log V$. This allows us to introduce the notion of "volume entropy" for this phase space: the von Neumann entropy associated to the measurement of volume.