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 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 (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.
Highlights
INDEPENDENCE, STATISTICAL EQUILIBRIUM AND GIBBS STATESWhat characterises statistical equilibrium? In a non-relativistic system, the answer is unambiguous
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
In this work we have tackled the issue of defining statistical equilibrium in full quantum gravity, i.e. in a complete background independent context and dealing with the fundamental microscopic degrees of freedom of a quantum spacetime
Summary
What characterises statistical equilibrium? In a non-relativistic system, the answer is unambiguous. KMS states take the explicit form of Gibbs states, whose density operators have the standard form proportional to e−βH This characterisation of equilibrium is unambiguous because of the special role played by time and its conjugate energy in non-relativistic mechanics, where time is absolute, modelled as the unique, external parameter encoding the dynamics of the system. The following discussion is not restricted to the group field theory formalism, or even to classical or quantum sectors Rather it attempts to present in a coherent way the perspectives and strategies employed in past studies ([6,7,8,9, 13, 14] and related works) for defining Gibbs states and statistical equilibrium in a background independent system. For infinite systems, it is wellknown that equilibrium states are not of the exponential Gibbs form, and are algebraic states that are more generally characterised by the KMS condition (and entropy maximisation for global equilibrium configurations)
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