Volume entropy

Abstract

© 2019 IOP Publishing Ltd Printed in the UK. 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 finite dimension, bounded by V log V. This implies that a notion of 'volume entropy' may be introduced on this state space, interpreted as the von Neumann entropy associated to the measurement of volume. However, it also becomes apparent that including the states with vanishing volume eigenvalues this entropy becomes divergent. We briefly discuss possible implications of this conundrum and difficulties arising for extending this analysis to higher valent nodes.