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

Due to the absence of well-defined concepts of time and energy in background independent systems, formulating statistical equilibrium in such settings remains an open issue. Even more so in the full quantum gravity context, not based on any of the usual spacetime notions but on non-spatiotemporal degrees of freedom. In this paper, after having clarified different general notions of statistical equilibrium, on which different construction procedures 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 operatorial formulation of group field theory to define its statistical mechanical framework, and, based on this, 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 flow 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