Volume Entropy

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.