Loop quantum gravity and Planck-size black hole entropy

The Loop Quantum Gravity (LQG) program is briefly reviewed and one of its main applications, namely the counting of black hole entropy within the framework is considered. In particular, recent results for Planck size black holes are reviewed. These results are consistent with an asymptotic linear relation (that fixes uniquely a free parameter of the theory) and a logarithmic correction with a coefficient equal to -1/2. The account is tailored as an introduction to the subject for non-experts.Comment: 21 pages, 5 figures. Contribution to the Proceedings of the NEB XII International Conferenc

Note on Self-Duality and the Kodama State

An interesting interplay between self-duality, the Kodama (Chern-Simons) state and knot invariants is shown to emerge in the quantum theory of an Abelian gauge theory. More precisely, when a self-dual representation of the CCR is chosen, the corresponding vacuum in the Schroedinger representation is precisely given by the Kodama state. Several consequences of this construction are explored.Comment: 4 pages, no figures. References and discussion added. Final version to appear in PR

Effective constrained polymeric theories and their continuum limit

The classical limit of polymer quantum theories yields a one parameter family of `effective' theories labeled by \lambda. Here we consider such families for constrained theories and pose the problem of taking the `continuum limit', \lambda -> 0. We put forward criteria for such question to be well posed, and propose a concrete strategy based in the definition of appropriately constructed Dirac observables. We analyze two models in detail, namely a constrained oscillator and a cosmological model arising from loop quantum cosmology. For both these models we show that the program can indeed be completed, provided one makes a particular choice of \lambda-dependent internal time with respect to which the dynamics is described and compared. We show that the limiting theories exist and discuss the corresponding limit. These results might shed some light in the problem of defining a renormalization group approach, and its associated continuum limit, for quantum constrained systems.Comment: 20 pages, 5 figures. Typos corrected, discussion expanded, references added. Version to be published in PR

Loop quantization of the Schwarzschild interior revisited

The loop quantization of the Schwarzschild interior region, as described by a homogeneous anisotropic Kantowski-Sachs model, is re-examined. As several studies of different -inequivalent- loop quantizations have shown, to date there exists no fully satisfactory quantum theory for this model. This fact poses challenges to the validity of some scenarios to address the black hole information problem. Here we put forward a novel viewpoint to construct the quantum theory that builds from some of the models available in the literature. The final picture is a quantum theory that is both independent of any auxiliary structure and possesses a correct low curvature limit. It represents a subtle but non-trivial modification of the original prescription given by Ashtekar and Bojowald. It is shown that the quantum gravitational constraint is well defined past the singularity and that its effective dynamics possesses a bounce into an expanding regime. The classical singularity is avoided, and a semiclassical spacetime satisfying vacuum Einstein's equations is recovered on the "other side" of the bounce. We argue that such metric represents the interior region of a white-hole spacetime, but for which the corresponding "white-hole mass" differs from the original black hole mass. Furthermore, we find that the value of the white-hole mass is proportional to the third power of the starting black hole mass.Comment: Revised version. Comparison with Ashtekar-Bojowald quantization expanded. A figure showing dependence of the white hole mass on the fiducial cell in Ashtekar-Bojowald quantization added. To appear in CQ

Quantum Superposition Principle and Geometry

If one takes seriously the postulate of quantum mechanics in which physical states are rays in the standard Hilbert space of the theory, one is naturally lead to a geometric formulation of the theory. Within this formulation of quantum mechanics, the resulting description is very elegant from the geometrical viewpoint, since it allows to cast the main postulates of the theory in terms of two geometric structures, namely a symplectic structure and a Riemannian metric. However, the usual superposition principle of quantum mechanics is not naturally incorporated, since the quantum state space is non-linear. In this note we offer some steps to incorporate the superposition principle within the geometric description. In this respect, we argue that it is necessary to make the distinction between a 'projective superposition principle' and a 'decomposition principle' that extend the standard superposition principle. We illustrate our proposal with two very well known examples, namely the spin 1/2 system and the two slit experiment, where the distinction is clear from the physical perspective. We show that the two principles have also a different mathematical origin within the geometrical formulation of the theory.Comment: 10 pages, no figures. References added. V3 discussion expanded and new results added, 14 pages. Dedicated to Michael P. Ryan on the occasion of his sixtieth bithda

Loop quantum cosmology of Bianchi IX: Effective dynamics

We study numerically the solutions to the effective equations of Bianchi IX spacetimes within Loop Quantum Cosmology. We consider Bianchi IX models with and without inverse triad corrections whose matter content is a scalar field without mass. The solutions are classified using the classical observables. We show that both effective theories --with lapse N=V and N=1-- solve the big bang singularity and reproduce the classical dynamics far from the bounce. Moreover, due to the spatial compactness, there is an infinity number of bounces and recollapses. We study the limit of large volume and show that both effective theories reproduce the same dynamics, thus recovering general relativity. We implement a procedure to identify amongst the Bianchi IX solutions, those that behave like k=0,1 FLRW as well as Bianchi I, II, and VII_0 models. The effective solutions exhibit Bianchi I phases with Bianchi II transitions and also Bianchi VII_0 phases, which had not been studied before, at the quantum nor effective level. We comment on the possible implications of these results for a quantum modification to the classical BKL behaviour.Comment: This version to be published in the CQG special issu