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

TL;DR

This paper explores the connection between Braunstein-Ghosh-Severini entropy of a graph and the physical geometry in Loop Quantum Gravity, proposing that the entropy may represent a physical measure of geometric information.

Contribution

It demonstrates that the BGS entropy relates to the Hamiltonian of a quantum particle in Loop Quantum Gravity, linking graph entropy to physical geometry.

Findings

The matrix rho(Gamma) corresponds to the Hamiltonian operator in Loop Quantum Gravity.

BGS entropy can be interpreted as a measure of geometric information.

Discussion on the physical interpretation and challenges of entropy in quantum geometry.

Abstract

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. 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.