Single particle in quantum gravity and Braunstein-Ghosh-Severini entropy of a spin network

Passerini and Severini have recently shown that the Braunstein-Ghosh-Severini (BGS) entropy $S_{\Gamma }=-\mathrm{Tr}[{\rho }_{\Gamma }\log {\rho }_{\Gamma }]$ of a certain density matrix ${\rho }_{\Gamma }$ naturally associated to a graph $\Gamma$, is maximized, among all graphs with a fixed number of links and nodes, by regular graphs. We ask if this result can play a role in quantum gravity, and be related to the apparent regularity of the physical geometry of space. We show that in loop quantum gravity the matrix ${\rho }_{\Gamma }$ is precisely the Hamiltonian operator (suitably normalized) of a nonrelativistic quantum particle interacting with the quantum gravitational field, if we restrict elementary area and volume eigenvalues to a fixed value. This operator provides a spectral characterization of the physical geometry, and can be interpreted as a state describing the spectral information about the geometry available when geometry is measured by its physical interaction with matter. It is then tempting to interpret its BGS entropy $S_{\Gamma }$ as a genuine physical entropy: we discuss the appeal and the difficulties of this interpretation.