Statistical equilibrium of tetrahedra from maximum entropy principle

Discrete formulations of (quantum) gravity in four spacetime dimensions build space out of tetrahedra. We investigate a statistical mechanical system of tetrahedra from a many-body point of view based on nonlocal, combinatorial gluing constraints that are modeled as multiparticle interactions. We focus on Gibbs equilibrium states, constructed using Jaynes's principle of constrained maximization of entropy, which has been shown recently to play an important role in characterizing equilibrium in background-independent systems. We apply this principle first to classical systems of many tetrahedra using different examples of geometrically motivated constraints. Then for a system of quantum tetrahedra, we show that the quantum statistical partition function of a Gibbs state with respect to some constraint operator can be reinterpreted as a partition function for a quantum field theory of tetrahedra, taking the form of a group field theory.