We give an efficient method, combining number-theoretic and combinatorial ideas, to exactly compute black hole entropy in the framework of loop quantum gravity. Along the way we provide a complete characterization of the relevant sector of the spectrum of the area operator, including degeneracies, and explicitly determine the number of solutions to the projection constraint. We use a computer implementation of the proposed algorithm to confirm and extend previous results on the detailed structure of the black hole degeneracy spectrum.

Ever since the pioneering works of Bekenstein and Hawking, black hole entropy has been known to have a quantum origin. Furthermore, it has long been argued by Bekenstein that entropy should be quantized in discrete (equidistant) steps given its identification with horizon area in (semi-)classical general relativity and the properties of area as an adiabatic invariant. This lead to the suggestion that the black hole area should also be quantized in equidistant steps to account for the discrete black hole entropy. Here we shall show that loop quantum gravity, in which area is not quantized in equidistant steps, can nevertheless be consistent with Bekenstein's equidistant entropy proposal in a subtle way. For that we perform a detailed analysis of the number of microstates compatible with a given area and show consistency with the Bekenstein framework when an oscillatory behavior in the entropy-area relation is properly interpreted. Black hole entropy is one of the most intriguing constructs of modern theoretical physics. On the one hand, it has a correspondence with the black hole horizon area through the laws of (classical) black hole mechanics. On the other hand, it is assumed to have a quantum statistical origin given that the proper identification between entropy and area S A=4' 2 p came only after an analysis of quantum fields on a fixed background [1].Furthermore, it has long been argued by Bekenstein that the proportionality between entropy and area, for large, classical black holes, can be justified from the adiabatic invariance properties of horizon area when subject to different scenarios (see [2,3] for a review). Further heuristic quantization arguments lead to the suggestion that area, when quantized, should have a discrete, equidistant spectrum in the large horizon limitwith a parameter and n integer. The relation between area and entropy that one expects to encounter in the large horizon radius is then extrapolated to the full spectrum. This would imply that entropy too would have a discrete spectrum, a property that might also be expected if entropy is to be associated with (the logarithm of) the number of microstates compatible with a given macrostate. When this condition is imposed, then the area is expected to have an spectrum of the formwith k and n integers [4]. Even when appealing and physically well motivated, these arguments remain somewhat heuristic and have no detailed microscopic quantum gravity formalism to support them. A quantum canonical description of black holes that has offered a detailed description of the quantum horizon degrees of freedom is given by loop quantum gravity (LQG) [5]. This formalism allows the inclusion of several matter couplings (including nonminimal couplings) and black holes far from extremality, in four dimensions. There is no restriction in the values of the matter charges. The approach uses as a starting point isolated horizon (IH) boundary conditions at the classical level, where the interior of the black hole is excluded from the region under consideration...

Quantum black holes within the loop quantum gravity (LQG) framework are considered. The number of microscopic states that are consistent with a black hole of a given horizon area A 0 are counted and the statistical entropy, as a function of the area, is obtained for A 0 up to 550 ℓ 2 Pl . The results are consistent with an asymptotic linear relation and a logarithmic correction with a coefficient equal to −1/2. The Barbero-Immirzi parameter that yields the asymptotic linear relation compatible with the Bekenstein-Hawking entropy is shown to coincide with a value close to γ = 0.274, which has been previously obtained analytically. However, a new and oscillatory functional form for the entropy is found for small, Planck size, black holes that calls for a physical interpretation.

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].

Abstract. We review the black hole entropy calculation in the framework of Loop Quantum Gravity based on the quasi-local definition of a black hole encoded in the isolated horizon formalism. We show, by means of the covariant phase space framework, the appearance in the conserved symplectic structure of a boundary term corresponding to a Chern-Simons theory on the horizon and present its quantization both in the U (1) gauge fixed version and in the fully SU (2) invariant one. We then describe the boundary degrees of freedom counting techniques developed for an infinite value of the Chern-Simons level case and, less rigorously, for the case of a finite value. This allows us to perform a comparison between the U (1) and SU (2) approaches and provide a state of the art analysis of their common features and different implications for the entropy calculations. In particular, we comment on different points of view regarding the nature of the horizon degrees of freedom and the role played by the Barbero-Immirzi parameter. We conclude by presenting some of the most recent results concerning possible observational tests for theory.

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