784 research outputs found
Deformed General Relativity and Torsion
We argue that the natural framework for embedding the ideas of deformed, or
doubly, special relativity (DSR) into a curved spacetime is a generalisation of
Einstein-Cartan theory, considered by Stelle and West. Instead of interpreting
the noncommuting "spacetime coordinates" of the Snyder algebra as endowing
spacetime with a fundamentally noncommutative structure, we are led to consider
a connection with torsion in this framework. This may lead to the usual
ambiguities in minimal coupling. We note that observable violations of charge
conservation induced by torsion should happen on a time scale of 10^3 s, which
seems to rule out these modifications as a serious theory. Our considerations
show, however, that the noncommutativity of translations in the Snyder algebra
need not correspond to noncommutative spacetime in the usual sense.Comment: 20 pages, 1 figure, revtex; expanded sections 3 and 4 for clarity,
moved material to appendix B, corrected a few minor error
The Case of the Missing Gates: Complexity of Jackiw-Teitelboim Gravity
The Jackiw-Teitelboim (JT) model arises from the dimensional reduction of
charged black holes. Motivated by the holographic complexity conjecture, we
calculate the late-time rate of change of action of a Wheeler-DeWitt patch in
the JT theory. Surprisingly, the rate vanishes. This is puzzling because it
contradicts both holographic expectations for the rate of complexification and
also action calculations for charged black holes. We trace the discrepancy to
an improper treatment of boundary terms when naively doing the dimensional
reduction. Once the boundary term is corrected, we find exact agreement with
expectations. We comment on the general lessons that this might hold for
holographic complexity and beyond.Comment: 31 pages, 5 figure
Particle Metropolis-Hastings using gradient and Hessian information
Particle Metropolis-Hastings (PMH) allows for Bayesian parameter inference in
nonlinear state space models by combining Markov chain Monte Carlo (MCMC) and
particle filtering. The latter is used to estimate the intractable likelihood.
In its original formulation, PMH makes use of a marginal MCMC proposal for the
parameters, typically a Gaussian random walk. However, this can lead to a poor
exploration of the parameter space and an inefficient use of the generated
particles.
We propose a number of alternative versions of PMH that incorporate gradient
and Hessian information about the posterior into the proposal. This information
is more or less obtained as a byproduct of the likelihood estimation. Indeed,
we show how to estimate the required information using a fixed-lag particle
smoother, with a computational cost growing linearly in the number of
particles. We conclude that the proposed methods can: (i) decrease the length
of the burn-in phase, (ii) increase the mixing of the Markov chain at the
stationary phase, and (iii) make the proposal distribution scale invariant
which simplifies tuning.Comment: 27 pages, 5 figures, 2 tables. The final publication is available at
Springer via: http://dx.doi.org/10.1007/s11222-014-9510-
A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions
We define and discuss classical and quantum gravity in 2+1 dimensions in the
Galilean limit. Although there are no Newtonian forces between massive objects
in (2+1)-dimensional gravity, the Galilean limit is not trivial. Depending on
the topology of spacetime there are typically finitely many topological degrees
of freedom as well as topological interactions of Aharonov-Bohm type between
massive objects. In order to capture these topological aspects we consider a
two-fold central extension of the Galilei group whose Lie algebra possesses an
invariant and non-degenerate inner product. Using this inner product we define
Galilean gravity as a Chern-Simons theory of the doubly-extended Galilei group.
The particular extension of the Galilei group we consider is the classical
double of a much studied group, the extended homogeneous Galilei group, which
is also often called Nappi-Witten group. We exhibit the Poisson-Lie structure
of the doubly extended Galilei group, and quantise the Chern-Simons theory
using a Hamiltonian approach. Many aspects of the quantum theory are determined
by the quantum double of the extended homogenous Galilei group, or Galilei
double for short. We study the representation theory of the Galilei double,
explain how associated braid group representations account for the topological
interactions in the theory, and briefly comment on an associated
non-commutative Galilean spacetime.Comment: 38 pages, 1 figure, references update
Aspects of Multilinear Harmonic Analysis Related to Transversality
The purpose of this article is to survey certain aspects of multilinear
harmonic analysis related to notions of transversality. Particular emphasis
will be placed on the multilinear restriction theory for the euclidean Fourier
transform, multilinear oscillatory integrals, multilinear geometric
inequalities, multilinear Radon-like transforms, and the interplay between
them.Comment: 28 pages. Article based on a short course given at the 9th
International Conference on Harmonic Analysis and Partial Differential
Equations, El Escorial, 201
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