Dynamics for a 2-vertex Quantum Gravity Model

We use the recently introduced U(N) framework for loop quantum gravity to study the dynamics of spin network states on the simplest class of graphs: two vertices linked with an arbitrary number N of edges. Such graphs represent two regions, in and out, separated by a boundary surface. We study the algebraic structure of the Hilbert space of spin networks from the U(N) perspective. In particular, we describe the algebra of operators acting on that space and discuss their relation to the standard holonomy operator of loop quantum gravity. Furthermore, we show that it is possible to make the restriction to the isotropic/homogeneous sector of the model by imposing the invariance under a global U(N) symmetry. We then propose a U(N) invariant Hamiltonian operator and study the induced dynamics. Finally, we explore the analogies between this model and loop quantum cosmology and sketch some possible generalizations of it.Comment: 28 pages, v2: typos correcte

Holomorphic Simplicity Constraints for 4d Riemannian Spinfoam Models

Starting from the reformulation of the classical phase space of Loop Quantum Gravity in terms of spinor variables and spinor networks, we build coherent spin network states and show how to use them to write the spinfoam path integral for topological BF theory in terms of Gaussian integrals in the spinors. Finally, we use this framework to revisit the simplicity constraints reducing topological BF theory to 4d Riemannian gravity. These holomorphic simplicity constraints lead us to a new spinfoam model for quantum gravity whose amplitudes are defined as the evaluation of the coherent spin networks.Comment: 4 pages. Proceedings of Loops'11, Madrid. To appear in Journal of Physics: Conference Series (JPCS

Many-nodes/many-links spinfoam: the homogeneous and isotropic case

I compute the Lorentzian EPRL/FK/KKL spinfoam vertex amplitude for regular graphs, with an arbitrary number of links and nodes, and coherent states peaked on a homogeneous and isotropic geometry. This form of the amplitude can be applied for example to a dipole with an arbitrary number of links or to the 4-simplex given by the compete graph on 5 nodes. All the resulting amplitudes have the same support, independently of the graph used, in the large j (large volume) limit. This implies that they all yield the Friedmann equation: I show this in the presence of the cosmological constant. This result indicates that in the semiclassical limit quantum corrections in spinfoam cosmology do not come from just refining the graph, but rather from relaxing the large j limit.Comment: 8 pages, 4 figure

Effect of the Influent COD Concentration on the Anaerobic Digestion of Winery Wastewaters from Grape-Red and Tropical Fruit (Guava) Wine Production in Fluidized Bed Reactors with Chilean Natural Zeolite for Biomass Immobilization

The effect of the influent COD concentration on the performance of anaerobic fluidized bed reactors treating winery wastewaters from grape-red wine (GRWW) and guava wine production (GWW) was studied at laboratory scale. Two reactors were used: one treating GRWW (AFB1) and the other processing GWW (AFB2). The behaviour of these reactors packed with Chilean zeolite as biomass immobilization support was compared at mesophilic temperature (35 °C). Influent COD varied from γ = 1–24 g L–1 and the HRT was maintained constant at 1 day throughout the experiment. During the experiment, influent and effluent pH, TVFA, COD and methane gas production were determined. COD removal efficiency increased with the influent COD up to a maximum of around γ = 19 g L–1 for GRWW and up to around 22 g L–1 for GWW due to the increase of the concentration of phenols. Process performance was slightly better with guava winery wastewater than with grape-red winery wastewater due its lower phenolic content. During the period of non-inhibition the methane yield was virtually constant

The picture of the Bianchi I model via gauge fixing in Loop Quantum Gravity

The implications of the SU(2) gauge fixing associated with the choice of invariant triads in Loop Quantum Cosmology are discussed for a Bianchi I model. In particular, via the analysis of Dirac brackets, it is outlined how the holonomy-flux algebra coincides with the one of Loop Quantum Gravity if paths are parallel to fiducial vectors only. This way the quantization procedure for the Bianchi I model is performed by applying the techniques developed in Loop Quantum Gravity but restricting the admissible paths. Furthermore, the local character retained by the reduced variables provides a relic diffeomorphisms constraint, whose imposition implies homogeneity on a quantum level. The resulting picture for the fundamental spatial manifold is that of a cubical knot with attached SU(2) irreducible representations. The discretization of geometric operators is outlined and a new perspective for the super-Hamiltonian regularization in Loop Quantum Cosmology is proposed.Comment: 6 page

Generating Functions for Coherent Intertwiners

We study generating functions for the scalar products of SU(2) coherent intertwiners, which can be interpreted as coherent spin network evaluations on a 2-vertex graph. We show that these generating functions are exactly summable for different choices of combinatorial weights. Moreover, we identify one choice of weight distinguished thanks to its geometric interpretation. As an example of dynamics, we consider the simple case of SU(2) flatness and describe the corresponding Hamiltonian constraint whose quantization on coherent intertwiners leads to partial differential equations that we solve. Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2) flatness on coherent spin networks for arbitrary graphs.Comment: 31 page

Revisiting the Simplicity Constraints and Coherent Intertwiners

In the context of loop quantum gravity and spinfoam models, the simplicity constraints are essential in that they allow to write general relativity as a constrained topological BF theory. In this work, we apply the recently developed U(N) framework for SU(2) intertwiners to the issue of imposing the simplicity constraints to spin network states. More particularly, we focus on solving them on individual intertwiners in the 4d Euclidean theory. We review the standard way of solving the simplicity constraints using coherent intertwiners and we explain how these fit within the U(N) framework. Then we show how these constraints can be written as a closed u(N) algebra and we propose a set of U(N) coherent states that solves all the simplicity constraints weakly for an arbitrary Immirzi parameter.Comment: 28 page

Classical Setting and Effective Dynamics for Spinfoam Cosmology

We explore how to extract effective dynamics from loop quantum gravity and spinfoams truncated to a finite fixed graph, with the hope of modeling symmetry-reduced gravitational systems. We particularize our study to the 2-vertex graph with N links. We describe the canonical data using the recent formulation of the phase space in terms of spinors, and implement a symmetry-reduction to the homogeneous and isotropic sector. From the canonical point of view, we construct a consistent Hamiltonian for the model and discuss its relation with Friedmann-Robertson-Walker cosmologies. Then, we analyze the dynamics from the spinfoam approach. We compute exactly the transition amplitude between initial and final coherent spin networks states with support on the 2-vertex graph, for the choice of the simplest two-complex (with a single space-time vertex). The transition amplitude verifies an exact differential equation that agrees with the Hamiltonian constructed previously. Thus, in our simple setting we clarify the link between the canonical and the covariant formalisms.Comment: 38 pages, v2: Link with discretized loop quantum gravity made explicit and emphasize

Black hole entropy in loop quantum gravity

We discuss the recent progress on black hole entropy in loop quantum gravity, focusing in particular on the recently discovered discretization effect for microscopic black holes. Powerful analytical techniques have been developed to perform the exact computation of entropy. A statistical analysis of the structures responsible for this effect shows its progressive damping and eventual disappearance as one increases the considered horizon area. © Published under licence by IOP Publishing Ltd