[Show abstract][Hide abstract] ABSTRACT: We discuss, within the simplified context provided by the polymeric harmonic
oscillator, a construction leading to a separable Hilbert space that preserves
some of the most important features of the spectrum of the Hamiltonian
operator. This construction can be generalized to loop quantum cosmology and is
helpful to sidestep some of the issues that appear in that context. In
particular those related to superselection and the definition of suitable
ensembles for the statistical mechanics of these types of systems.
[Show abstract][Hide abstract] ABSTRACT: In a previous paper, we classified and obtained the exponential generating
functions for the large class of bivariate linear recurrences proposed by
Graham, Knuth, and Patashnik. In this paper, we obtain general Rodrigues-like
formulas for the corresponding univariate row generating polynomials. We make
special emphasis on two-parameter generalizations of the Eulerian numbers <n,k>
and <<n,k>>. In these cases, starting from the exponential generating function,
we achieve a complete solution to the problem: i.e., closed formulas for these
numbers. Finally, we briefly discuss some other applications of combinatorial
[Show abstract][Hide abstract] ABSTRACT: We consider the Problem 6.94 posed in the book Concrete Mathematics by
Graham, Knuth, and Patashkin, and solve it using bivariate exponential
generating functions. This problem includes many particular cases of great
combinatorial interest. We find a complete classification in four types of the
solution for this problem, and for each type, we obtain the corresponding
exponential generating function. For some families the generating function
requires the introduction of a generalization of the tree function. We provide
many applications of our general results to particularly interesting cases.
Journal of Combinatorial Theory Series A 07/2013; · 0.77 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The purpose of this paper is to study in detail the constraint structure of
the Hamiltonian and symplectic-Lagrangian descriptions for the scalar and
electromagnetic fields in the presence of spatial boundaries. We carefully
discuss the implementation of the geometric constraint algorithm of Gotay,
Nester and Hinds with special emphasis on the relevant functional analytic
aspects of the problem. This is an important step towards the rigorous
understanding of general field theories in the presence of boundaries, very
especially when these fail to be regular. The geometric approach developed in
the paper is also useful with regard to the interpretation of the physical
degrees of freedom and the nature of the constraints when both gauge symmetries
and boundaries are present.
[Show abstract][Hide abstract] ABSTRACT: We discuss the detailed structure of the spectrum of the Hamiltonian for the
polymerized harmonic oscillator and compare it with the spectrum in the
standard quantization. As we will see the non-separability of the Hilbert space
implies that the point spectrum consists of bands similar to the ones appearing
in the treatment of periodic potentials. This feature of the spectrum of the
polymeric harmonic oscillator may be relevant for the discussion of the polymer
quantization of the scalar field and may have interesting consequences for the
statistical mechanics of these models.
Classical and Quantum Gravity 05/2013; 30(16). · 3.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
Journal of Physics Conference Series 05/2012; 360(1):2035-.
[Show abstract][Hide abstract] ABSTRACT: We give a short introduction to the approaches currently used to describe
black holes in loop quantum gravity. We will concentrate on the classical
issues related to the modeling of black holes as isolated horizons, give a
short discussion of their canonical quantization by using loop quantum gravity
techniques, and a description of the combinatorial methods necessary to solve
the counting problems involved in the computation of the entropy.
[Show abstract][Hide abstract] ABSTRACT: The international conference LOOPS'11 took place in Madrid from the 23–28 May 2011. It was hosted by the Instituto de Estructura de la Materia (IEM), which belongs to the Consejo Superior de Investigaciones Cientĺficas (CSIC). Like previous editions of the LOOPS meetings, it dealt with a wealth of state-of-the-art topics on Quantum Gravity, with special emphasis on non-perturbative background-independent approaches to spacetime quantization. The main topics addressed at the conference ranged from the foundations of Quantum Gravity to its phenomenological aspects. They encompassed different approaches to Loop Quantum Gravity and Cosmology, Polymer Quantization, Quantum Field Theory, Black Holes, and discrete approaches such as Dynamical Triangulations, amongst others. In addition, this edition celebrated the 25th anniversary of the introduction of the now well-known Ashtekar variables and the Wednesday morning session was devoted to this silver jubilee. The structure of the conference was designed to reflect the current state and future prospects of research on the different topics mentioned above. Plenary lectures that provided general background and the ‘big picture’ took place during the mornings, and the more specialised talks were distributed in parallel sessions during the evenings. To be more specific, Monday evening was devoted to Shape Dynamics and Phenomenology Derived from Quantum Gravity in Parallel Session A, and to Covariant Loop Quantum Gravity and Spin foams in Parallel Session B. Tuesday's three Parallel Sessions dealt with Black Hole Physics and Dynamical Triangulations (Session A), the continuation of Monday's session on Covariant Loop Quantum Gravity and Spin foams (Session B) and Foundations of Quantum Gravity (Session C). Finally, Thursday and Friday evenings were devoted to Loop Quantum Cosmology (Session A) and to Hamiltonian Loop Quantum Gravity (Session B). The result of the conference was very satisfactory and enlightening. Not only was it a showroom for the research currently being carried out by many groups throughout the world, but there was also a permanent look towards the future. During these days, the CSIC Campus witnessed many scientific conversations triggered by the interaction amongst the people and groups that participated in LOOPS'11 Madrid and which, in many cases, will crystallise into new results and advances in the field. The conference would not have been possible without the generous help of a number of national and international institutions. The organizers would like to acknowledge the financial support provided by the Spanish Ministry of Science and Innovation (Ministerio de Ciencia e Innovación), the Spanish Research Council, CSIC (Consejo Superior de Investigaciones Cientĺficas), The BBVA Foundation (Fundación BBVA), The CONSOLIDER-CPAN project, the Spanish Society for Gravitation and Relativity (SEGRE), The Universidad Carlos III of Madrid (UC3M), and the European Science Foundation (ESF). The ESF, through the Quantum Gravity and Quantum Geometry network, provided full support for a number of young participants that have contributed to these proceedings: Dario Benedetti (Albert Einstein Institute, Potsdam, Germany), Norbert Bodendorfer (Institute for Theoretical Physics III, FAU Erlangen Nürnberg, Germany), Mariam Bouhmadi López (CENTRA, Centro Multidisciplinar de Astrofĺsica, Lisbon), Timothy Budd (Institute for Theoretical Physics, Utrecht University, The Netherlands), Miguel Campiglia (Institute for Gravitation and the Cosmos, Penn State University, USA), Gianluca Delfino (School of Mathematical Sciences, University of Nottingham, UK), Maite Dupuis (Institute for Theoretical Physics III, FAU Erlangen Nürnberg, Germany), Michał Dziendzikowski (Institute of Theoretical Physics, Warsaw University, Poland), Muxin Han (Centre de Physique Théorique de Luminy, Marseille, France), Philipp Höhn (Institute for Theoretical Physics, Utrecht University, The Netherlands), Jacek Puchta (Centre de Physique Théorique de Luminy, Marseille, France), James Ryan (Albert Einstein Institute, Potsdam, Germany), Lorenzo Sindoni (Albert Einstein Institute, Golm, Germany), David Sloan (Institute for Theoretical Physics, Utrecht University, The Netherlands), Johannes Tambornino (Laboratoire de Physique, ENS Lyon, France), Andreas Thurn (Institute for Theoretical Physics III, FAU Erlangen Nürnberg, Germany), Francesca Vidotto (Laboratoire de Physique Subatomique et de Cosmologie, Grenoble, France), and Matteo Smerlak (Albert Einstein Institute, Golm, Germany). We would like to conclude this preamble by thanking all the attendants of the conference for their high and enthusiastic participation. The presence of a large number of past and present Loop Quantum Gravity practitioners, as well as a significant number of top researchers in other approaches to quantum gravity, provided ample opportunities for fruitful scientific exchanges and a very lively atmosphere. It is encouraging to see that, 25 years after the inception of Loop Quantum Gravity, there is a vibrant young community of researchers entering the field. Let us hope that, with their help, the quantization of general relativity can be successfully accomplished in the near future. The Editors
Journal of Physics Conference Series 01/2012; 360(1).
[Show abstract][Hide abstract] ABSTRACT: We discuss the thermodynamic limit in the canonical area ensemble used in
loop quantum gravity to model quantum black holes. The computation of the
thermodynamic limit is the rigorous way to obtain a smooth entropy from the
counting entropy given by a direct determination of the number of microstates
compatible with macroscopic quantities (the energy in standard statistical
mechanics or the area in the framework presented here). As we will show in
specific examples the leading behavior of the smoothed entropy for large
horizon areas is the same as the counting entropy but the subleading
contributions differ. This is important because these corrections determine the
concavity or convexity of the entropy as a function of the area.
Classical and Quantum Gravity 06/2011; 28(21). · 3.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We give a complete and detailed description of the computation of black hole
entropy in loop quantum gravity by employing the most recently introduced
number-theoretic 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.
Physical review D: Particles and fields 01/2011; 82(8).
[Show abstract][Hide abstract] ABSTRACT: We use mathematical methods based on generating functions to study the
statistical properties of the black hole degeneracy spectrum in loop quantum
gravity. In particular we will study the persistence of the observed effective
quantization of the entropy as a function of the horizon area. We will show
that this quantization disappears as the area increases despite the existence
of black hole configurations with a large degeneracy. The methods that we
describe here can be adapted to the study of the statistical properties of the
black hole degeneracy spectrum for all the existing proposals to define black
hole entropy in loop quantum gravity.
Physical review D: Particles and fields 01/2011; 83(10).
[Show abstract][Hide abstract] ABSTRACT: We give a comprehensive review of the quantization of midisuperspace models.
Though the main focus of the paper is on quantum aspects, we also provide an
introduction to several classical points related to the definition of these
models. We cover some important issues, in particular, the use of the principle
of symmetric criticality as a very useful tool to obtain the required
Hamiltonian formulations. Two main types of reductions are discussed: those
involving metrics with two Killing vector fields and spherically symmetric
models. We also review the more general models obtained by coupling matter
fields to these systems. Throughout the paper we give separate discussions for
standard quantizations using geometrodynamical variables and those relying on
loop quantum gravity inspired methods.
Living Reviews in Relativity 10/2010; · 22.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We will briefly review the classical formulation of the 3-handle 1 × 2 and 3-sphere 3 Gowdy cosmological models coupled with massless scalar fields and their exact (non-perturbative) quantization by defining suitable Schrödinger functional representations in terms of appropriate probability spaces. We will pay special attention to the construction of closed expressions for the corresponding quantum time evolution propagators.
Journal of Physics Conference Series 06/2009; 175(1):012010.
[Show abstract][Hide abstract] ABSTRACT: 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 Perez. 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. Comment: 5 pages. Accepted for publication in Physical Review D
[Show abstract][Hide abstract] ABSTRACT: We show that, for space-times with inner boundaries, there exists a natural area operator different from the standard one used in loop quantum gravity. This new flux-area operator has equidistant eigenvalues. We discuss the consequences of substituting the standard area operator in the Ashtekar-Baez-Corichi-Krasnov definition of black hole entropy by the new one. Our choice simplifies the definition of the entropy and allows us to consider only those areas that coincide with the one defined by the value of the level of the Chern-Simons theory describing the horizon degrees of freedom. We give a prescription to count the number of relevant horizon states by using spin components and obtain exact expressions for the black hole entropy. Finally we derive its asymptotic behavior, discuss several issues related to the compatibility of our results with the Bekenstein-Hawking area law and the relation with Schwarzschild quasi-normal modes. Comment: 25 pages
[Show abstract][Hide abstract] ABSTRACT: Einstein‐Rosen waves can be exactly quantized. The Hamiltonian operator is a nonlinear, bounded function of the free Hamiltonian corresponding to an axisymmetric massless scalar field propagating in a 2+1 dimensional Minkowskian background. In this short review we will discuss the possibility of constructing true coherent states for this model. We also provide quantitative information about the viability of using the coherent states corresponding to the dynamics of the auxiliary, free Hamiltonian appearing in the description of the system, to study its full physical dynamics. For time periods of arbitrary length we show that this is only possible for states that are close, in a precise mathematical sense, to the vacuum.
[Show abstract][Hide abstract] ABSTRACT: In this article, we formulate the study of the unitary time evolution of systems consisting of an infinite number of uncoupled time-dependent harmonic oscillators in mathematically rigorous terms. We base this analysis on the theory of a single one-dimensional time-dependent oscillator, for which we first summarize some basic results concerning the unitary implementability of the dynamics. This is done by employing techniques different from those used so far to derive the Feynman propagator. In particular, we calculate the transition amplitudes for the usual harmonic oscillator eigenstates and define suitable semiclassical states for some physically relevant models. We then explore the possible extension of this study to infinite dimensional dynamical systems. Specifically, we construct Schrödinger functional representations in terms of appropriate probability spaces, analyze the unitarity of the time evolution, and probe the existence of semiclassical states for a wide range of physical systems, particularly, the well-known Minkowskian free scalar fields and Gowdy cosmological models.
[Show abstract][Hide abstract] ABSTRACT: We discuss some issues related to the computation of black hole entropy in loop quantum gravity from the novel point of view provided by the recent number-theoretical methods introduced by the authors and their collaborators. In particular we give exact expressions, in the form of integral transforms, for the black hole entropy in terms of the area. We do this by following several approaches based both on our combinatorial techniques and also on functional equations similar to those employed by Meissner in his pioneering work on this subject. To put our results in perspective we compare them with those of Meissner. We will show how our methods confirm some of his findings, extend the validity of others, and correct some mistakes. At the end of the paper we will discuss the delicate issue of the asymptotics of black hole entropy. Comment: 25 pages
Classical and Quantum Gravity 10/2008; · 3.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We discuss two different types of issues concerning the quantization of Einstein-Rosen waves. First of all we study in detail the possibility of using the coherent states corresponding to the dynamics of the auxiliary, free Hamiltonian appearing in the description of the model to study the full dynamics of the system. For time periods of arbitrary length we show that this is only possible for states that are close, in a precise mathematical sense, to the vacuum. We do this by comparing the quantum evolutions defined by the auxiliary and physical Hamiltonians on the class of coherent states. In the second part of the paper we study the structure of n-point functions. As we will show their detailed behavior differs from the one corresponding to standard perturbative quantum field theories. We take this as a manifestation of the fact that the correct approximation scheme for physically interesting objects in these models does not lead to a power series expansion in the relevant coupling constant but to a more complicated asymptotic behavior.
Classical and Quantum Gravity 09/2008; · 3.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider the quantum dynamics of a minimally coupled massless scalar field in de Sitter spacetime. The classical evolution is represented by a canonical transformation on the phase space for the field theory. By studying the corresponding Bogoliubov transformations, we show that the symplectic map that encodes the evolution between two instants of time cannot be unitarily implemented on any Fock space built from a SO(4)-symmetric complex structure. We will also show that, in contrast with some effectively lower dimensional examples arising from quantum general relativity such as Gowdy models, it is impossible to find a time-dependent conformal redefinition of the massless scalar field leading to a quantum unitary dynamics.
Classical and Quantum Gravity 06/2008; 25(14):145008. · 3.56 Impact Factor