1,217 research outputs found
Simplicial faces of the set of correlation matrices
This paper concerns the facial geometry of the set of
correlation matrices. The main result states that almost every set of
vertices generates a simplicial face, provided that , where is an absolute constant. This bound is qualitatively
sharp because the set of correlation matrices has no simplicial face generated
by more than vertices.Comment: 12 pages, 2 figure
Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach
We study the continuity of an abstract generalization of the maximum-entropy
inference - a maximizer. It is defined as a right-inverse of a linear map
restricted to a convex body which uniquely maximizes on each fiber of the
linear map a continuous function on the convex body. Using convex geometry we
prove, amongst others, the existence of discontinuities of the maximizer at
limits of extremal points not being extremal points themselves and apply the
result to quantum correlations. Further, we use numerical range methods in the
case of quantum inference which refers to two observables. One result is a
complete characterization of points of discontinuity for matrices.Comment: 27 page
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
Catching homologies by geometric entropy
A geometric entropy is defined as the Riemannian volume of the parameter
space of a statistical manifold associated with a given network. As such it can
be a good candidate for measuring networks complexity. Here we investigate its
ability to single out topological features of networks proceeding in a
bottom-up manner: first we consider small size networks by analytical methods
and then large size networks by numerical techniques. Two different classes of
networks, the random graphs and the scale--free networks, are investigated
computing their Betti numbers and then showing the capability of geometric
entropy of detecting homologies.Comment: 12 pages, 2 Figure
The Holst Spin Foam Model via Cubulations
Spin foam models are an attempt for a covariant, or path integral formulation
of canonical loop quantum gravity. The construction of such models usually rely
on the Plebanski formulation of general relativity as a constrained BF theory
and is based on the discretization of the action on a simplicial triangulation,
which may be viewed as an ultraviolet regulator. The triangulation dependence
can be removed by means of group field theory techniques, which allows one to
sum over all triangulations. The main tasks for these models are the correct
quantum implementation of the Plebanski constraints, the existence of a
semiclassical sector implementing additional "Regge-like" constraints arising
from simplicial triangulations, and the definition of the physical inner
product of loop quantum gravity via group field theory. Here we propose a new
approach to tackle these issues stemming directly from the Holst action for
general relativity, which is also a proper starting point for canonical loop
quantum gravity. The discretization is performed by means of a "cubulation" of
the manifold rather than a triangulation. We give a direct interpretation of
the resulting spin foam model as a generating functional for the n-point
functions on the physical Hilbert space at finite regulator. This paper focuses
on ideas and tasks to be performed before the model can be taken seriously.
However, our analysis reveals some interesting features of this model: first,
the structure of its amplitudes differs from the standard spin foam models.
Second, the tetrad n-point functions admit a "Wick-like" structure. Third, the
restriction to simple representations does not automatically occur -- unless
one makes use of the time gauge, just as in the classical theory.Comment: 25 pages, 1 figure; v3: published version. arXiv admin note:
substantial text overlap with arXiv:0911.213
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