Mexi Lazos 2012

Los sistemas cuánticos poliméricos son los modelos de mecánica cuántica cuantizados de una manera similar a la gravedad cuántica de lazos. Estos sistemas han permitido un estudio más simple de algunas características de la cuantización de lazos entre ellos algunos aspectos térmicos. Presentamos ejemplos de la termodinámica de varios sistemas polimerizados: un gas ideal, un sólido cristalino y un gas de átomos de hidrógeno, en este último caso se encontró una modificación a la ecuación para la fracción de ionización del hidrógeno en la recombinación.

La mecánica cuántica polimérica es un modelo que mimifica las variables de flujo y holonomía de la Gravedad Cuántica por Lazos. Ésta coincide con los resultados de la mecánica cuántica estándar cuando un parámetro de escala de longitud es considerado "pequeño". En esta plática presentaremos los principales aspectos en la construcción del análogo de la representación polimérica para un oscilador de Fermi y como su descripción mecánica cuántica usual es recuperada.

El esquema de cuantización de la teoría de lazos de la gravedad  ha sido implementado en sistemas mecánicos sencillos, a esta descripción se le ha llamado mecánica cuántica polimérica (M.C.P.) y asemeja a un sistema cuántico definido en una red. En esta plática discutiremos los propagadores de la partícula libre y de la partícula en una caja en M.C.P.

Presentamos un ansatz sugerido por las teorías modificadas de Krasnov para gravedad, el cual resuelve la constricción Hamiltoniana a nivel clásico y cuántico.

The Plebanski formulation of complex general relativity is given in terms of variables valued in the complexification of the so(3) Lie algebra.Therefore, it is genuinely a gauge theory that is also diffeomorphism-invariant. For this reason, the way that the Levi-Civita connection emerges from this formulation is not direct because both the internal (gauge) and the spacetime connections  are geometrical structures a priori not related, there is not a natural link between them. Any possible relationship must be put in by hand or must come from extra hypotheses. We analyze the correct relationship between these connections and show how they are related.

We perform the Lorentz-covariant Hamiltonian analysis of two Lagrangian action principles that describe general relativity as a constrained BF theory and that include the Immirzi parameter. The relation between these two Lagrangian actions has been already studied through a map among the fields involved. The main difference between these is the way the Immirzi parameter is included, since in one of them the Immirzi parameter is included explicitly in the BF terms, whereas in the other (the CMPR action) it is in the constraint on the B fields. In this work we continue the analysis of their relationship but at the Hamiltonian level. Particularly, we are interested in seeing how the above difference appears in the constraint structure of both action principles. We find that they both possess the same number of first-class and second-class constraints and satisfy a very similar (off-shell) Poisson-bracket algebra on account of the type of canonical variables employed. The two algebras can be transformed into each other by making a suitable change of variables.

Using the master constraint programme and uniform discretization technique, we loop quantize the spherically symmetric model coupled to a matter scalar field, propose a trial state for the vacuum of the theory and find a candidate for the physical states by minimizing the expectation value of the master constraint on the trial states. Then we use that state to find the free propagator of the matter field in the lowest order. This, in turn, leads to a propagator which signals Lorentz violation through modifying the dispersion relation.

We present a geometric-variational model of discrete mechanics and field theory. It is a version of the discrete multisymplectic framework proposed by Marsden et al based on the work of Veselov. Our version is natural in spin foam models; in particular, some of its variables were first introduced by Reisenberger in a simplicial gravity model. Our framework and that proposed by Marsden et al lead to the same space of solutions, and the induced geometric structures in the space of solutions agree. Then we apply our framework to 2d gravity treated as a constrained BF theory with gauge group U(1). We compare our results with previous results of Rovelli et al.

Since on AdS spacetimes there are no (temporally) asymptotically free states, the standard S-Matrix does not exist here. This problem can be circumvented by constructing a new kind of S-matrix using the General Boundary Formulation (GBF) of Quantum Theory. Therein, quantum states live in Hilbert spaces associated to boundaries of spacetime regions. Amplitudes for these can then be calculated, and moreover there is a probability interpretation compatible with standard Quantum Theory. In this short talk, we sketch the construction of an S-Matrix for AdS for Schrödinger-Feynman Quantization and Holomorphic Quantization, and comment on its invariance under actions of the AdS isometry group SO(2,d).

We consider the issue of singularity resolution within loop quantum cosmology for the homogeneous and anisotropic Bianchi IX model. We show that the modifications which come from Loop Quantum Cosmology imply a non-chaotic effective behavior in the vacuum Bianchi IX model. Later, we present results of numerical evolutions of the effective equations. To address the issue of singularity resolution we examine the time evolution of geometrical and curvature invariants that yield information about the semiclassical spacetime geometry. Finally, We discuss generic behavior found for a variety of initial conditions.

We briefly review the loop quantum cosmology of Bianchi IX model. We show that how the Hamiltonian constraint operator is obtained and then present the effective equations for the model which provide some corrections to the classical equations of motion. The correction terms include the holonomy correction effects and those ones which come from the inverse triad operators in the Hamiltonian constraint and could play an important role to avoid the classical chaotic behavior of the model and also resolve the singularity.

We study the generalized Holst action, the usual Holst action plus boundary term that makes it well defined (differentiable and finite) under asymptotically flat conditions, and add to it the topological terms: Pontryagin, Euler, Nieh-Yan. For this generalization of General Relativity we study whether the action is well defined, and what is the contribution of these topological terms to the symplectic structure and conserved quantities.