On the resolution of the big bang singularity in isotropic Loop Quantum Cosmology

In contrast to previous work in the field, we construct the Loop Quantum Cosmology (LQC) of the flat isotropic model with a massless scalar field in the absence of higher order curvature corrections to the gravitational part of the Hamiltonian constraint. The matter part of the constraint contains the inverse triad operator which can be quantized with or without the use of a Thiemann- like procedure. With the latter choice, we show that the LQC quantization is identical to that of the standard Wheeler DeWitt theory (WDW) wherein there is no singularity resolution. We argue that the former choice leads to singularity resolution in the sense of a well defined, regular (backward) evolution through and beyond the epoch where the size of the universe vanishes. Our work along with that of the seminal work of Ashtekar, Pawlowski and Singh (APS) clarifies the role, in singularity resolution, of the three `exotic' structures in this LQC model, namely: curvature corrections, inverse triad definitions and the `polymer' nature of the kinematic representation. We also critically examine certain technical assumptions made by APS in their analysis of WDW semiclassical states and point out some problems stemming from the infrared behaviour of their wave functionsComment: 26 pages, no figure

Entrainment of marginally stable excitation waves by spatially extended sub-threshold periodic forcing

We analyze the effects of spatially extended periodic forcing on the dynamics of one-dimensional excitation waves. Entrainment of unstable primary waves has been studied numerically for different amplitudes and frequencies of additional sub-threshold stimuli. We determined entrainment regimes under which excitation blocks were transformed into consistent 1:1 responses. These responses were spatially homogeneous and synchronized in the entire excitable medium. Compared to primary pulses, pulses entrained by secondary stimulations were stable at considerably shorter periods which decreased at higher amplitudes and greater number of secondary stimuli. Our results suggest a practical methodology for stabilization of excitation in reaction-diffusion media with regions of reduced excitability.Comment: 6 pages, 6 figure

Deformation Quantization of Coadjoint Orbits

A method for the deformation quantization of coadjoint orbits of semisimple Lie groups is proposed. It is based on the algebraic structure of the orbit. Its relation to geometric quantization and differentiable deformations is explored.Comment: Talk presented at the meeting "Noncommutative geometry and Hopf algebras in Field Theory and Particle Physics", Torino, 199

SU(2) Poisson-Lie T duality

Poisson-Lie target space duality is a framework where duality transformations are properly defined. In this letter we investigate the pair of sigma models defined by the double SO(3,1) in the Iwasawa decomposition.Comment: 12 pages, 1 figur

Homogeneous 2+1 dimensional gravity in the Ashtekar formulation

The constraint hypersurfaces defining the Witten and Ashtekar formulations for 2+1 gravity are very different. In particular the constraint hypersurface in the Ashtekar case is not a manifold but consists of several sectors that intersect each other in a complicated way. The issue of how to define a consistent dynamics in such a situation is then rather non-trivial. We discuss this point by working out the details in a simplified (finite dimensional) homogeneous reduction of 2+1 gravity in the Ashtekar formulation

On the deformation quantization of affine algebraic varieties

We compute an explicit algebraic deformation quantization for an affine Poisson variety described by an ideal in a polynomial ring, and inheriting its Poisson structure from the ambient space.Comment: AMS-LaTeX, 20 page

Functional evolution of quantum cylindrical waves

Kucha{\v{r}} showed that the quantum dynamics of (1 polarization) cylindrical wave solutions to vacuum general relativity is determined by that of a free axially-symmetric scalar field along arbitrary axially-symmetric foliations of a fixed flat 2+1 dimensional spacetime. We investigate if such a dynamics can be defined {\em unitarily} within the standard Fock space quantization of the scalar field. Evolution between two arbitrary slices of an arbitrary foliation of the flat spacetime can be built out of a restricted class of evolutions (and their inverses). The restricted evolution is from an initial flat slice to an arbitrary (in general, curved) slice of the flat spacetime and can be decomposed into (i) `time' evolution in which the spatial Minkowskian coordinates serve as spatial coordinates on the initial and the final slice, followed by (ii) the action of a spatial diffeomorphism of the final slice on the data obtained from (i). We show that although the functional evolution of (i) is unitarily implemented in the quantum theory, generic spatial diffeomorphisms of (ii) are not. Our results imply that a Tomanaga-Schwinger type functional evolution of quantum cylindrical waves is not a viable concept even though, remarkably, the more limited notion of functional evolution in Kucha{\v{r}}'s `half parametrized formalism' is well-defined.Comment: Replaced with published versio

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