389 research outputs found
Pseudo-Cartesian coordinates in a model of Causal Dynamical Triangulations
Causal Dynamical Triangulations is a non-perturbative quantum gravity model,
defined with a lattice cut-off. The model can be viewed as defined with a
proper time but with no reference to any three-dimensional spatial background
geometry. It has four phases, depending on the parameters (the coupling
constants) of the model. The particularly interesting behavior is observed in
the so-called de Sitter phase, where the spatial three-volume distribution as a
function of proper time has a semi-classical behavior which can be obtained
from an effective mini-superspace action. In the case of the three-sphere
spatial topology, it has been difficult to extend the effective semi-classical
description in terms of proper time and spatial three-volume to include genuine
spatial coordinates, partially because of the background independence inherent
in the model. However, if the spatial topology is that of a three-torus, it is
possible to define a number of new observables that might serve as spatial
coordinates as well as new observables related to the winding numbers of the
three-dimensional torus. The present paper outlines how to define the
observables, and how they can be used in numerical simulations of the model.Comment: 26 pages, 15 figure
Quantum Gravity on the Lattice
I review the lattice approach to quantum gravity, and how it relates to the
non-trivial ultraviolet fixed point scenario of the continuum theory. After a
brief introduction covering the general problem of ultraviolet divergences in
gravity and other non-renormalizable theories, I cover the general methods and
goals of the lattice approach. An underlying theme is the attempt at
establishing connections between the continuum renormalization group results,
which are mainly based on diagrammatic perturbation theory, and the recent
lattice results, which apply to the strong gravity regime and are inherently
non-perturbative. A second theme in this review is the ever-present natural
correspondence between infrared methods of strongly coupled non-abelian gauge
theories on the one hand, and the low energy approach to quantum gravity based
on the renormalization group and universality of critical behavior on the
other. Towards the end of the review I discuss possible observational
consequences of path integral quantum gravity, as derived from the non-trivial
ultraviolet fixed point scenario. I argue that the theoretical framework
naturally leads to considering a weakly scale-dependent Newton's costant, with
a scaling violation parameter related to the observed scaled cosmological
constant (and not, as naively expected, to the Planck length).Comment: 63 pages, 12 figure
Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimization on compact sets
We consider the problem of computing the minimum value of a
polynomial over a compact set , which can be
reformulated as finding a probability measure on minimizing . Lasserre showed that it suffices to consider such measures of the form
, where is a sum-of-squares polynomial and is a given
Borel measure supported on . By bounding the degree of by one gets
a converging hierarchy of upper bounds for . When is
the hypercube , equipped with the Chebyshev measure, the parameters
are known to converge to at a rate in . We
extend this error estimate to a wider class of convex bodies, while also
allowing for a broader class of reference measures, including the Lebesgue
measure. Our analysis applies to simplices, balls and convex bodies that
locally look like a ball. In addition, we show an error estimate in when satisfies a minor geometrical condition, and in when is a convex body, equipped with the Lebesgue measure. This
improves upon the currently best known error estimates in and
for these two respective cases.Comment: 30 pages with 10 figures. Update notes for second version: Added a
new section containing numerical examples that illustrate the theoretical
results -- Fixed minor mistakes/typos -- Improved some notation -- Clarified
certain explanations in the tex
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