2,291 research outputs found

    Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

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    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

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    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

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    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

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    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

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    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 Information Theor
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