Statistical equilibrium in quantum gravity: Gibbs states in group field theory

Abstract

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 welldefined 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 (GFT) formalism for quantum gravity, whose technical features prove advantageous to the task. We use the operator formulation of GFT 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. specific model), and in terms of which quantum spacetime is indeed (tentatively) described as a quantum manybody system, albeit of a very exotic nature.One of the foundational concepts in statistical physics is that of equilibrium. Equilibrium configurations are those that are invariant under time evolution (in turn identified, in flat space, with time translations), generated by the Hamiltonian of the system. But how does one define statistical equilibrium when there is no preferred time and Hamiltonian? This is the case in classical constrained systems such as general relativity. This is also the case in quantum gravitational contexts, especially in formalisms that are not based on continuum spacetime structures, like differentiable spacetime manifolds etc. This is the open problem of defining statistical equilibrium in a (non-spatiotemporal) background independent system. Still, since equilibrium states hold a special place in statistical physics, this is where we start, for developing a statistical mechanical formulation of quantum gravity within a GFT formalism.In this work we aim to construct Gibbs equilibrium states for a GFT system. Whether these states provide a truly comprehensive characterisation of statistical equilibrium in quantum gravity in general is a different (and challenging) issue that is not considered here. We also do not analyse here the physical consequences of our results for a descrip...

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