20,276 research outputs found
Isometry groups of non-positively curved spaces: discrete subgroups
We study lattices in non-positively curved metric spaces. Borel density is
established in that setting as well as a form of Mostow rigidity. A converse to
the flat torus theorem is provided. Geometric arithmeticity results are
obtained after a detour through superrigidity and arithmeticity of abstract
lattices. Residual finiteness of lattices is also studied. Riemannian symmetric
spaces are characterised amongst CAT(0) spaces admitting lattices in terms of
the existence of parabolic isometries.Comment: This is the second of a pair of two articles, originally posted on
September 2, 2008 as arXiv:0809.0457v1 For the first part, see
arXiv:0809.0457v
Local causal structures, Hadamard states and the principle of local covariance in quantum field theory
In the framework of the algebraic formulation, we discuss and analyse some
new features of the local structure of a real scalar quantum field theory in a
strongly causal spacetime. In particular we use the properties of the
exponential map to set up a local version of a bulk-to-boundary correspondence.
The bulk is a suitable subset of a geodesic neighbourhood of any but fixed
point p of the underlying background, while the boundary is a part of the
future light cone having p as its own tip. In this regime, we provide a novel
notion for the extended *-algebra of Wick polynomials on the said cone and, on
the one hand, we prove that it contains the information of the bulk counterpart
via an injective *-homomorphism while, on the other hand, we associate to it a
distinguished state whose pull-back in the bulk is of Hadamard form. The main
advantage of this point of view arises if one uses the universal properties of
the exponential map and of the light cone in order to show that, for any two
given backgrounds M and M' and for any two subsets of geodesic neighbourhoods
of two arbitrary points, it is possible to engineer the above procedure such
that the boundary extended algebras are related via a restriction homomorphism.
This allows for the pull-back of boundary states in both spacetimes and, thus,
to set up a machinery which permits the comparison of expectation values of
local field observables in M and M'.Comment: 42 pages, xy package is used, typos corrected, clarifications adde
On the photon Green functions in curved space-time
Quantization of electrodynamics in curved space-time in the Lorenz gauge and
with arbitrary gauge parameter makes it necessary to study Green functions of
non-minimal operators with variable coefficients. Starting from the integral
representation of photon Green functions, we link them to the evaluation of
integrals involving Gamma functions. Eventually, the full asymptotic expansion
of the Feynman photon Green function at small values of the world function, as
well as its explicit dependence on the gauge parameter, are obtained without
adding by hand a mass term to the Faddeev--Popov Lagrangian. Coincidence limits
of second covariant derivatives of the associated Hadamard function are also
evaluated, as a first step towards the energy-momentum tensor in the
non-minimal case.Comment: 22 pages, plain Tex. All sections and appendices have been improve
Coherent states for FLRW space-times in loop quantum gravity
We construct a class of coherent spin-network states that capture proprieties
of curved space-times of the Friedmann-Lama\^itre-Robertson-Walker type on
which they are peaked. The data coded by a coherent state are associated to a
cellular decomposition of a spatial (const.) section with dual graph given
by the complete five-vertex graph, though the construction can be easily
generalized to other graphs. The labels of coherent states are complex SL(2,
\mathbbm{C}) variables, one for each link of the graph and are computed
through a smearing process starting from a continuum extrinsic and intrinsic
geometry of the canonical surface. The construction covers both Euclidean and
Lorentzian signatures; in the Euclidean case and in the limit of flat space we
reproduce the simplicial 4-simplex semiclassical states used in Spin Foams.Comment: 10 pages, 1 figure, published versio
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