Statistical Equilibrium in Quantum Gravity: Gibbs states in Group Field Theory

Gibbs states are known to play a crucial role in the statistical description of a system with a large number of degrees of freedom. They are expected to be vital also in a quantum gravitational system with many underlying fundamental discrete degrees of freedom. However, due to the absence of well-defined concepts of time and energy in background independent settings, formulating statistical equilibrium in such cases is an open issue. This is even more so in a quantum gravity context that is not based on any of the usual spacetime structures, but on non-spatiotemporal degrees of freedom. In this paper, after having clarified general notions of statistical equilibrium, on which two different construction procedures for Gibbs states can be based, we focus on the group field theory formalism for quantum gravity, whose technical features prove advantageous to the task. We use the operator formulation of group field theory to define its statistical mechanical framework, based on which we construct three concrete examples of Gibbs states. The first is a Gibbs state with respect to a geometric volume operator, which is shown to support condensation to a low-spin phase. This state is not based on a pre-defined symmetry of the system and its construction is via Jaynes' entropy maximisation principle. The second are Gibbs states encoding structural equilibrium with respect to internal translations on the GFT base manifold, and defined via the KMS condition. The third are Gibbs states encoding relational equilibrium with respect to a clock Hamiltonian, obtained by deparametrization with respect to coupled scalar matter fields.