Typicality in spin network states of quantum geometry

In this work, we extend the so-called typicality approach, originally formulated in statistical mechanics contexts, to $SU(2)$-invariant spin-network states. Our results do not depend on the physical interpretation of the spin network; however, they are mainly motivated by the fact that spin-network states can describe states of quantum geometry, providing a gauge-invariant basis for the kinematical Hilbert space of several background-independent approaches to quantum gravity. The first result is, by itself, the existence of a regime in which we show the emergence of a typical state. We interpret this as the proof that in that regime there are certain (local) properties of quantum geometry which are "universal". Such a set of properties is heralded by the typical state, of which we give the explicit form. This is our second result. In the end, we study some interesting properties of the typical state, proving that the area law for the entropy of a surface must be satisfied at the local level, up to logarithmic corrections which we are able to bound.Comment: Typos and mistakes fixe

Statistical mechanics of covariant systems with multi-fingered time

Recently, in [Class. Quantum Grav. 33 (2016) 045005], the authors proposed a new approach extending the framework of statistical mechanics to reparametrization-invariant systems with no additional gauges. In this work, the approach is generalized to systems defined by more than one Hamiltonian constraints (multi-fingered time). We show how well known features as the Ehrenfest- Tolman effect and the J\"uttner distribution for the relativistic gas can be consistently recovered from a covariant approach in the multi-fingered framework. Eventually, the crucial role played by the interaction in the definition of a global notion of equilibrium is discussed.Comment: 5 pages, 2 figure

On the fate of the Hoop Conjecture in quantum gravity

We consider a closed region $R$ of 3d quantum space modeled by $SU(2)$ spin-networks. Using the concentration of measure phenomenon we prove that, whenever the ratio between the boundary $\partial R$ and the bulk edges of the graph overcomes a finite threshold, the state of the boundary is always thermal, with an entropy proportional to its area. The emergence of a thermal state of the boundary can be traced back to a large amount of entanglement between boundary and bulk degrees of freedom. Using the dual geometric interpretation provided by loop quantum gravity, we interprete such phenomenon as a pre-geometric analogue of Thorne's "Hoop conjecture", at the core of the formation of a horizon in General Relativity.Comment: 7 pages, 2 figures, minor improvement

Thermodynamic aspects of gravity

In this thesis we consider a scenario where gravitational dynamics emerges from the holographic hydrodynamics of some microscopic, quantum system living in a local Rindler wedge. We start by considering the area scaling properties of the entanglement entropy of a local Rindler horizon as a conceptually basic realization of the holographic principle. From the generalized second law and the Bekenstein bound we derive the gravitational dynamics via the entropy balance approach developed in [Jacobson 1995]. We show how this setting can account for the equilibrium and the nonequilibrium features associated with the gravitational dynamics and extend the thermodynamical derivation from General Relativity to generalized Brans-Dicke theories. We then concentrate on the possibility to define a version of fluid/gravity correspondence within the local Rindler wedge setting. We show how the hydrodynamical description of the horizon can be directly associated to a hydrodynamical description of the thermal fields. Because of the holographic behavior, the properties of the Rindler wedge thermal gauge theory are effectively encoded in a codimension one system living close to the Rindler horizon. In a large scale analysis, this system can be thought of as a fluid living on a codimension one stretched horizon membrane. This sets an apparent duality between the horizon local geometry and the fluid limit of the thermal gauge theory. Beyond the connection between the classical Navier-Stokes equations and a classical geometry, we discuss the possibility to realize such a duality at any point in spacetime by means of the equivalence principle. Given the shared local Rindler geometric setting, we eventually deal with the intriguing possibility to link the fluid/Rindler correspondence to the derivation of the gravitational field equations from a local non-equilibrium spacetime thermodynamics

Group Field theory and Tensor Networks: towards a Ryu-Takayanagi formula in full quantum gravity

We establish a dictionary between group field theory (thus, spin networks and random tensors) states and generalized random tensor networks. Then, we use this dictionary to compute the R\'{e}nyi entropy of such states and recover the Ryu-Takayanagi formula, in two different cases corresponding to two different truncations/approximations, suggested by the established correspondence.Comment: 54 pages, 10 figures; v2: replace figure 1 with a new version. Matches submitted version. v3: remove Renyi entropy computation on the random tensor network, focusing on GFT computation and interpretatio

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 non-local, combinatorial gluing constraints that are modelled as multi-particle interactions. We focus on Gibbs equilibrium states, constructed using Jaynes' principle of constrained maximisation of entropy, which has been shown recently to play an important role in characterising 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.Comment: v3 published version; v2 18 pages, 4 figures, improved text in sections IIIC & IVB, minor changes elsewher

Coupling and thermal equilibrium in general-covariant systems

A fully general-covariant formulation of statistical mechanics is still lacking. We take a step toward this theory by studying the meaning of statistical equilibrium for coupled, parametrized systems. We discuss how to couple parametrized systems. We express the thermalization hypothesis in a general-covariant context. This takes the form of vanishing of information flux. An interesting relation emerges between thermal equilibrium and gauge.Comment: 8 pages, 3 figure