2,291 research outputs found
Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results
We review some recent attempts to extract information about the nature of
quantum gravity, with and without matter, by quantum field theoretical methods.
More specifically, we work within a covariant lattice approach where the
individual space-time geometries are constructed from fundamental simplicial
building blocks, and the path integral over geometries is approximated by
summing over a class of piece-wise linear geometries. This method of
``dynamical triangulations'' is very powerful in 2d, where the regularized
theory can be solved explicitly, and gives us more insights into the quantum
nature of 2d space-time than continuum methods are presently able to provide.
It also allows us to establish an explicit relation between the Lorentzian- and
Euclidean-signature quantum theories. Analogous regularized gravitational
models can be set up in higher dimensions. Some analytic tools exist to study
their state sums, but, unlike in 2d, no complete analytic solutions have yet
been constructed. However, a great advantage of our approach is the fact that
it is well-suited for numerical simulations. In the second part of this review
we describe the relevant Monte Carlo techniques, as well as some of the
physical results that have been obtained from the simulations of Euclidean
gravity. We also explain why the Lorentzian version of dynamical triangulations
is a promising candidate for a non-perturbative theory of quantum gravity.Comment: 69 pages, 16 figures, references adde
The Semiclassical Limit of Causal Dynamical Triangulations
Previous work has shown that the macroscopic structure of the theory of
quantum gravity defined by causal dynamical triangulations (CDT) is compatible
with that of a de Sitter universe. After emphasizing the strictly
nonperturbative nature of this semiclassical limit we present a detailed study
of the three-volume data, which allows us to re-confirm the de Sitter
structure, exhibit short-distance discretization effects, and make a first
detailed investigation of the presence of higher-order curvature terms in the
effective action for the scale factor. Technically, we make use of a novel way
of fixing the total four-volume in the simulations.Comment: 30 pages, 10 figure
Nonperturbative Quantum Gravity
Asymptotic safety describes a scenario in which general relativity can be
quantized as a conventional field theory, despite being nonrenormalizable when
expanding it around a fixed background geometry. It is formulated in the
framework of the Wilsonian renormalization group and relies crucially on the
existence of an ultraviolet fixed point, for which evidence has been found
using renormalization group equations in the continuum.
"Causal Dynamical Triangulations" (CDT) is a concrete research program to
obtain a nonperturbative quantum field theory of gravity via a lattice
regularization, and represented as a sum over spacetime histories. In the
Wilsonian spirit one can use this formulation to try to locate fixed points of
the lattice theory and thereby provide independent, nonperturbative evidence
for the existence of a UV fixed point.
We describe the formalism of CDT, its phase diagram, possible fixed points
and the "quantum geometries" which emerge in the different phases. We also
argue that the formalism may be able to describe a more general class of
Ho\v{r}ava-Lifshitz gravitational models.Comment: Review, 146 pages, many figure
Block-Coordinate Frank-Wolfe Optimization for Structural SVMs
We propose a randomized block-coordinate variant of the classic Frank-Wolfe
algorithm for convex optimization with block-separable constraints. Despite its
lower iteration cost, we show that it achieves a similar convergence rate in
duality gap as the full Frank-Wolfe algorithm. We also show that, when applied
to the dual structural support vector machine (SVM) objective, this yields an
online algorithm that has the same low iteration complexity as primal
stochastic subgradient methods. However, unlike stochastic subgradient methods,
the block-coordinate Frank-Wolfe algorithm allows us to compute the optimal
step-size and yields a computable duality gap guarantee. Our experiments
indicate that this simple algorithm outperforms competing structural SVM
solvers.Comment: Appears in Proceedings of the 30th International Conference on
Machine Learning (ICML 2013). 9 pages main text + 22 pages appendix. Changes
from v3 to v4: 1) Re-organized appendix; improved & clarified duality gap
proofs; re-drew all plots; 2) Changed convention for Cf definition; 3) Added
weighted averaging experiments + convergence results; 4) Clarified main text
and relationship with appendi
Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes
We introduce the notion of the stopping redundancy hierarchy of a linear
block code as a measure of the trade-off between performance and complexity of
iterative decoding for the binary erasure channel. We derive lower and upper
bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and
Bonferroni-type inequalities, and specialize them for codes with cyclic
parity-check matrices. Based on the observed properties of parity-check
matrices with good stopping redundancy characteristics, we develop a novel
decoding technique, termed automorphism group decoding, that combines iterative
message passing and permutation decoding. We also present bounds on the
smallest number of permutations of an automorphism group decoder needed to
correct any set of erasures up to a prescribed size. Simulation results
demonstrate that for a large number of algebraic codes, the performance of the
new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on
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