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

Understand

Passerini and Severini have recently shown that the Braunstein-Ghosh-Severini (BGS) entropy S(Gamma) = -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 non-relativistic 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.