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.

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.

The purpose of this paper is to analyze in detail the Hamiltonian formulation for the compact Gowdy models coupled to massless scalar fields as a necessary first step toward their quantization. We will pay special attention to the coupling of matter and those features that arise for the S 1 × S 2 and S 3 topologies that are not present in the well-studied T 3 case-in particular the polar constraints that come from the regularity conditions on the metric. As a byproduct of our analysis we will get an alternative understanding, within the Hamiltonian framework, of the appearance of initial and final singularities for these models.

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].

We introduce an iterative method to univocally determine the adiabatic expansion of the modes of Dirac fields in spatially homogeneous external backgrounds. We overcome the ambiguities found in previous studies and use this new procedure to improve the adiabatic regularization/renormalization scheme. We provide details on the application of the method for Dirac fields living in a four-dimensional Friedmann-Lemaître-Robertson-Walker spacetime with a Yukawa coupling to an external scalar field. We check the consistency of our proposal by working out the conformal anomaly. We also analyze a two-dimensional Dirac field in Minkowski space coupled to a homogeneous electric field and reproduce the known results on the axial anomaly. The adiabatic expansion of the modes given here can be used to properly characterize the allowed physical states of the Dirac fields in the above external backgrounds.

We analyze in full-detail the geometric structure of the covariant phase space (CPS) of any local field theory defined over a space-time with boundary. To this end, we introduce a new frame: the "relative bicomplex framework". It is the result of merging an extended version of the "relative framework" (initially developed in the context of algebraic topology by R. Bott and L.W. Tu in the 1980s to deal with boundaries) and the variational bicomplex framework (the differential geometric arena for the variational calculus). The relative bicomplex framework is the natural one to deal with field theories with boundary contributions, including corner contributions. In fact, we prove a formal equivalence between the relative version of a theory with boundary and the non-relative version of the same theory with no boundary. With these tools at hand, we endow the space of solutions of the theory with a (pre)symplectic structure canonically associated with the action and which, in general, has boundary contributions. We also study the symmetries of the theory and construct, for a large group of them, their Noether currents and charges. Moreover, we completely characterize the arbitrariness (or lack thereof for fiber bundles with contractible fibers) of these constructions. This clarifies many misconceptions about the role of the boundary terms in the CPS description of a field theory. Finally, we provide what we call the CPS-algorithm to construct the aforementioned (pre)symplectic structure and apply it to some relevant examples.

We introduce, in a systematic way, a set of generating functions that solve all the different combinatorial problems that crop up in the study of black hole entropy in loop quantum gravity. Specifically we give generating functions for the following: the different sources of degeneracy related to the spectrum of the area operator, the solutions to the projection constraint, and the black hole degeneracy spectrum. Our methods are capable of handling the different countings proposed and discussed in the literature. The generating functions presented here provide the appropriate starting point to extend the results already obtained for microscopic black holes to the macroscopic regime-in particular those concerning the area law and the appearance of an effectively equidistant area spectrum.

We study several issues related to the different choices of time available for the classical and quantum treatment of linearly polarized cylindrical gravitational waves. We pay special attention to the time evolution of creation and annihilation operators and the definition of Fock spaces for the different choices of time involved. We also discuss the issue of microcausality and the use of field commutators to extract information about the causal properties of quantum spacetime.