1,613 research outputs found
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
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