We explore the classical setting for the U(N ) framework for SU(2) intertwiners for loop quantum gravity (LQG) and describe the corresponding phase space in terms of spinors with the appropriate constraints. We show how its quantization leads back to the standard Hilbert space of intertwiner states defined as holomorphic functionals. We then explain how to glue these intertwiners states in order to construct spin network states as wave-functions on the spinor phase space. In particular, we translate the usual loop gravity holonomy observables to our classical framework. Finally, we propose how to derive our phase space structure from an action principle which induces non-trivial dynamics for the spin network states. We conclude by applying explicitly our framework to states living on the simple 2-vertex graph and discuss the properties of the resulting Hamiltonian.

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. ContentsIntroduction 2 I. The U(N )Framework VI. Conclusions and Outlook 25Acknowledgments 26 A. Diagonalizing the renormalized Hamiltonian 26 References 27Confidential: not for distribution. Submitted to IOP Publishing for peer review 2 October 2010 2 IntroductionLoop quantum gravity (LQG) presents a rigorous framework towards the quantization of general relativity. The main issues faced by this quantization scheme are the precise dynamics of the theory and the derivation of its semiclassical limit. There has been tremendous progress on these topics since the original formulation of the theory, but a punch line is still missing. Our main motivation for the present work is the study of the LQG dynamics. Since Thiemann's original proposal of a well-defined Hamiltonian constraint operator for LQG [1,2], there has been various more recent proposals among which we point out the algebraic quantum gravity framework [3] and the spinfoam approach e.g. [4,5]. Our main source of inspiration for our current approach is the recent model introduced by Rovelli and Vidotto [6]. The logic behind their model is to implement the LQG dynamics on the simplest non-trivial class of spin network states, thus constructing a first order truncation of the full theory. They considered spin network states based on a fixed graph with two vertices related by four edges, so that their model can be called "tetrahedron LQG model". Very interestingly, it was shown that this model can be understood as reproducing a cosmological setting in LQG and leads to a physical framework very similar to loop quantum cosmology [6,7]. It was also shown that the same procedure can be successfully applied to the current spinfoam models [8]. This "dipole quantum cosmology" is the starting point of our work.We consider the generalization of the Rovelli-Vidotto model to spin network states based on a graph with still 2 vertices but now with an arbitrary number N of edges. From their viewpoint, this should allow to introduce more anisotropy/inhomogeneity in their model. Here, we start anew with a thorough study of the algebraic structure of the Hilbert sp...

We give a complete and detailed description of the computation of black hole entropy in loop quantum gravity by employing the most recently introduced numbertheoretic and combinatorial methods. The use of these techniques allows us to perform a detailed analysis of the precise structure of the entropy spectrum for small black holes, showing some relevant features that were not discernible in previous computations. The ability to manipulate and understand the spectrum up to the level of detail that we describe in the paper is a crucial step towards obtaining the behavior of entropy in the asymptotic (large horizon area) regime.

We use the combinatorial and number-theoretical methods developed in previous work by the authors to study black hole entropy in the new proposal put forward by Engle, Noui and Pérez. Specifically we give the generating functions relevant for the computation of the entropy and use them to derive its asymptotic behavior including the value of the Immirzi parameter and the coefficient of the logarithmic correction. In this brief note we want to study some of the physical consequences that follow from the black hole entropy definition proposed, in the context of loop quantum gravity, by Engle, Noui and Pérez in [1]. The main reason to do this is to check wether this new definition satisfies the obvious physical requirement of reproducing the Bekenstein-Hawking formula for large black holes. Without going into the details of the theoretical foundations of this new proposal, this analysis can be seen as a straightforward consistency check. We also want to obtain corrections to this formula that can be eventually compared with equivalent results found in different approaches [2,3,4,5]. An additional reason to perform this study is to show the power of the combinatorial methods developed by the authors in [6,7,8,9].The problem of interest can be enunciated in the following way [1]. Given a value of the black hole area a H = 4πγℓ 2 P κ (where κ ∈ N is the level of the SU(2) Chern-Simons theory on the horizon, 1 ℓ P denotes the Planck length, and γ the Immirzi parameter), we have to determine the number of states labeled by spins j 1 , . . . , j n satisfying an inequality of the * Ivan.Agullo@uv.es † fbarbero@iem.cfmac.csic.es ‡ Enrique.Fernandez@uv.es § Jacobo.Diaz@uv.es ¶ ejsanche@math.uc3m.es 1 For an earlier treatment, based on different considerations, of the SU (2) Chern-Simons theory in this framework see [10].

Motivated by the analogy proposed by Witten between Chern-Simons and Conformal Field Theories, we explore an alternative way of computing the entropy of a black hole starting from the isolated horizon framework in Loop Quantum Gravity. The consistency of the result opens a window for the interplay between ConformalField Theory and the description of black holes in Loop Quantum Gravity.

Abstract. Loop Quantum Gravity provides a natural truncation of the infinite degrees of freedom of gravity, obtained by studying the theory on a given finite graph. We review this procedure and we present the construction of the canonical theory on a simple graph, formed by only two nodes. We review the U(N ) framework, which provides a powerful tool for the canonical study of this model, and a formulation of the system based on spinors. We consider also the covariant theory, which permits to derive the model from a more complex formulation, paying special attention to the cosmological interpretation of the theory.

Abstract. The implementation of the dynamics in Loop Quantum Gravity (LQG) is still an open problem. Here, we discuss a tentative dynamics for the simplest class of graphs in LQG: Two vertices linked with an arbitrary number of edges. We use the recently introduced U(N ) framework in order to construct SU(2) invariant operators and define a global U(N ) symmetry that will select the homogeneous/isotropic states. Finally, we propose a Hamiltonian operator invariant under area-preserving deformations of the boundary surface and we identify possible connections of this model with Loop Quantum Cosmology.

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