17,735 research outputs found
Semiclassical Limits of Extended Racah Coefficients
We explore the geometry and asymptotics of extended Racah coeffecients. The
extension is shown to have a simple relationship to the Racah coefficients for
the positive discrete unitary representation series of SU(1,1) which is
explicitly defined. Moreover, it is found that this extension may be
geometrically identified with two types of Lorentzian tetrahedra for which all
the faces are timelike.
The asymptotic formulae derived for the extension are found to have a similar
form to the standard Ponzano-Regge asymptotic formulae for the SU(2) 6j symbol
and so should be viable for use in a state sum for three dimensional Lorentzian
quantum gravity.Comment: Latex2e - 26 pages, 6 figures. Uses AMS-fonts, AMS-LaTeX, epsf.tex
and texdraw. Revised version with improved clarity and additional result
Matrix geometries and fuzzy spaces as finite spectral triples
A class of real spectral triples that are similar in structure to a
Riemannian manifold but have a finite-dimensional Hilbert space is defined and
investigated, determining a general form for the Dirac operator. Examples
include fuzzy spaces defined as real spectral triples. Fuzzy 2-spheres are
investigated in detail, and it is shown that the fuzzy analogues correspond to
two spinor fields on the commutative sphere. In some cases it is necessary to
add a mass mixing matrix to the commutative Dirac operator to get a precise
agreement for the eigenvalues.Comment: 39 pages, final versio
LANDSAT survey of near-shore ice conditions along the Arctic coast of Alaska
The author has identified the following significant results. Winter and spring near-shore ice conditions were analyzed for the Beaufort Sea 1973-77, and the Chukchi Sea 1973-76. LANDSAT imagery was utilized to map major ice features related to regional ice morphology. Significant features from individual LANDSAT image maps were combined to yield regional maps of major ice ridge systems for each year of study and maps of flaw lead systems for representative seasons during each year. These regional maps were, in turn, used to prepare seasonal ice morphology maps. These maps showed, in terms of a zonal analysis, regions of statistically uniform ice behavior. The behavioral characteristics of each zone were described in terms of coastal processes and bathymetric configuration
An algebraic interpretation of the Wheeler-DeWitt equation
We make a direct connection between the construction of three dimensional
topological state sums from tensor categories and three dimensional quantum
gravity by noting that the discrete version of the Wheeler-DeWitt equation is
exactly the pentagon for the associator of the tensor category, the
Biedenharn-Elliott identity. A crucial role is played by an asymptotic formula
relating 6j-symbols to rotation matrices given by Edmonds.Comment: 10 pages, amstex, uses epsf.tex. New version has improved
presentatio
Personal propulsion unit Patent
Lightweight propulsion unit for movement of personnel and equipment across lunar surfac
Asymptotics of 10j symbols
The Riemannian 10j symbols are spin networks that assign an amplitude to each
4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This
amplitude is a function of the areas of the 10 faces of the 4-simplex, and
Barrett and Williams have shown that one contribution to its asymptotics comes
from the Regge action for all non-degenerate 4-simplices with the specified
face areas. However, we show numerically that the dominant contribution comes
from degenerate 4-simplices. As a consequence, one can compute the asymptotics
of the Riemannian 10j symbols by evaluating a `degenerate spin network', where
the rotation group SO(4) is replaced by the Euclidean group of isometries of
R^3. We conjecture formulas for the asymptotics of a large class of Riemannian
and Lorentzian spin networks in terms of these degenerate spin networks, and
check these formulas in some special cases. Among other things, this conjecture
implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the
Riemannian ones.Comment: 25 pages LaTeX with 8 encapsulated Postscript figures. v2 has various
clarifications and better page breaks. v3 is the final version, to appear in
Classical and Quantum Gravity, and has a few minor corrections and additional
reference
Finiteness and Dual Variables for Lorentzian Spin Foam Models
We describe here some new results concerning the Lorentzian Barrett-Crane
model, a well-known spin foam formulation of quantum gravity. Generalizing an
existing finiteness result, we provide a concise proof of finiteness of the
partition function associated to all non-degenerate triangulations of
4-manifolds and for a class of degenerate triangulations not previously shown.
This is accomplished by a suitable re-factoring and re-ordering of integration,
through which a large set of variables can be eliminated. The resulting
formulation can be interpreted as a ``dual variables'' model that uses
hyperboloid variables associated to spin foam edges in place of representation
variables associated to faces. We outline how this method may also be useful
for numerical computations, which have so far proven to be very challenging for
Lorentzian spin foam models.Comment: 15 pages, 1 figur
Spin Foam Models of Matter Coupled to Gravity
We construct a class of spin foam models describing matter coupled to
gravity, such that the gravitational sector is described by the unitary
irreducible representations of the appropriate symmetry group, while the matter
sector is described by the finite-dimensional irreducible representations of
that group. The corresponding spin foam amplitudes in the four-dimensional
gravity case are expressed in terms of the spin network amplitudes for
pentagrams with additional external and internal matter edges. We also give a
quantum field theory formulation of the model, where the matter degrees of
freedom are described by spin network fields carrying the indices from the
appropriate group representation. In the non-topological Lorentzian gravity
case, we argue that the matter representations should be appropriate SO(3) or
SO(2) representations contained in a given Lorentz matter representation,
depending on whether one wants to describe a massive or a massless matter
field. The corresponding spin network amplitudes are given as multiple
integrals of propagators which are matrix spherical functions.Comment: 30 pages, 9 figures, further remarks and references added. Version to
appear in Class. Quant. Gra
Effective action and semiclassical limit of spin foam models
We define an effective action for spin foam models of quantum gravity by
adapting the background field method from quantum field theory. We show that
the Regge action is the leading term in the semi-classical expansion of the
spin foam effective action if the vertex amplitude has the large-spin
asymptotics which is proportional to an exponential function of the vertex
Regge action. In the case of the known three-dimensional and four-dimensional
spin foam models this amounts to modifying the vertex amplitude such that the
exponential asymptotics is obtained. In particular, we show that the ELPR/FK
model vertex amplitude can be modified such that the new model is finite and
has the Einstein-Hilbert action as its classical limit. We also calculate the
first-order and some of the second-order quantum corrections in the
semi-classical expansion of the effective action.Comment: Improved presentation, 2 references added. 15 pages, no figure
On the causal Barrett--Crane model: measure, coupling constant, Wick rotation, symmetries and observables
We discuss various features and details of two versions of the Barrett-Crane
spin foam model of quantum gravity, first of the Spin(4)-symmetric Riemannian
model and second of the SL(2,C)-symmetric Lorentzian version in which all
tetrahedra are space-like. Recently, Livine and Oriti proposed to introduce a
causal structure into the Lorentzian Barrett--Crane model from which one can
construct a path integral that corresponds to the causal (Feynman) propagator.
We show how to obtain convergent integrals for the 10j-symbols and how a
dimensionless constant can be introduced into the model. We propose a `Wick
rotation' which turns the rapidly oscillating complex amplitudes of the Feynman
path integral into positive real and bounded weights. This construction does
not yet have the status of a theorem, but it can be used as an alternative
definition of the propagator and makes the causal model accessible by standard
numerical simulation algorithms. In addition, we identify the local symmetries
of the models and show how their four-simplex amplitudes can be re-expressed in
terms of the ordinary relativistic 10j-symbols. Finally, motivated by possible
numerical simulations, we express the matrix elements that are defined by the
model, in terms of the continuous connection variables and determine the most
general observable in the connection picture. Everything is done on a fixed
two-complex.Comment: 22 pages, LaTeX 2e, 1 figur
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