1,640 research outputs found
Dual variables and a connection picture for the Euclidean Barrett-Crane model
The partition function of the SO(4)- or Spin(4)-symmetric Euclidean
Barrett-Crane model can be understood as a sum over all quantized geometries of
a given triangulation of a four-manifold. In the original formulation, the
variables of the model are balanced representations of SO(4) which describe the
quantized areas of the triangles. We present an exact duality transformation
for the full quantum theory and reformulate the model in terms of new variables
which can be understood as variables conjugate to the quantized areas. The new
variables are pairs of S^3-values associated to the tetrahedra. These
S^3-variables parameterize the hyperplanes spanned by the tetrahedra (locally
embedded in R^4), and the fact that there is a pair of variables for each
tetrahedron can be viewed as a consequence of an SO(4)-valued parallel
transport along the edges dual to the tetrahedra. We reconstruct the parallel
transport of which only the action of SO(4) on S^3 is physically relevant and
rewrite the Barrett-Crane model as an SO(4) lattice BF-theory living on the
2-complex dual to the triangulation subject to suitable constraints whose form
we derive at the quantum level. Our reformulation of the Barrett-Crane model in
terms of continuous variables is suitable for the application of various
analytical and numerical techniques familiar from Statistical Mechanics.Comment: 33 pages, LaTeX, combined PiCTeX/postscript figures, v2: note added,
TeX error correcte
2-Vector Spaces and Groupoids
This paper describes a relationship between essentially finite groupoids and
2-vector spaces. In particular, we show to construct 2-vector spaces of
Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding
to functors between groupoids in both a covariant and contravariant way, which
are ambidextrous adjoints. This is used to construct a representation--a weak
functor--from Span(Gpd) (the bicategory of groupoids and spans of groupoids)
into 2Vect. In this paper we prove this and give the construction in detail.Comment: 44 pages, 5 figures - v2 adds new theorem, significant changes to
proofs, new sectio
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
Spin Foam Models of Riemannian Quantum Gravity
Using numerical calculations, we compare three versions of the Barrett-Crane
model of 4-dimensional Riemannian quantum gravity. In the version with face and
edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we
show the partition function diverges very rapidly for many triangulated
4-manifolds. In the version with modified face and edge amplitudes due to Perez
and Rovelli, we show the partition function converges so rapidly that the sum
is dominated by spin foams where all the spins labelling faces are zero except
for small, widely separated islands of higher spin. We also describe a new
version which appears to have a convergent partition function without drastic
spin-zero dominance. Finally, after a general discussion of how to extract
physics from spin foam models, we discuss the implications of convergence or
divergence of the partition function for other aspects of a spin foam model.Comment: 23 pages LaTeX; this version to appear in Classical and Quantum
Gravit
A spin foam model for pure gauge theory coupled to quantum gravity
We propose a spin foam model for pure gauge fields coupled to Riemannian
quantum gravity in four dimensions. The model is formulated for the
triangulation of a four-manifold which is given merely combinatorially. The
Riemannian Barrett--Crane model provides the gravity sector of our model and
dynamically assigns geometric data to the given combinatorial triangulation.
The gauge theory sector is a lattice gauge theory living on the same
triangulation and obtains from the gravity sector the geometric information
which is required to calculate the Yang--Mills action. The model is designed so
that one obtains a continuum approximation of the gauge theory sector at an
effective level, similarly to the continuum limit of lattice gauge theory, when
the typical length scale of gravity is much smaller than the Yang--Mills scale.Comment: 18 pages, LaTeX, 1 figure, v2: details clarified, references adde
Categorical Groups, Knots and Knotted Surfaces
We define a knot invariant and a 2-knot invariant from any finite categorical
group. We calculate an explicit example for the Spun Trefoil.Comment: 40 pages, lots of figures. Second version: Added example and
discussion, clarification of the fact that the maps associated with
Reidemeister Moves are well define
Principal 2-bundles and their gauge 2-groups
In this paper we introduce principal 2-bundles and show how they are
classified by non-abelian Cech cohomology. Moreover, we show that their gauge
2-groups can be described by 2-group-valued functors, much like in classical
bundle theory. Using this, we show that, under some mild requirements, these
gauge 2-groups possess a natural smooth structure. In the last section we
provide some explicit examples.Comment: 40 pages; v3: completely revised and extended, classification
corrected, name changed, to appear in Forum Mat
When Do Measures on the Space of Connections Support the Triad Operators of Loop Quantum Gravity?
In this work we investigate the question, under what conditions Hilbert
spaces that are induced by measures on the space of generalized connections
carry a representation of certain non-Abelian analogues of the electric flux.
We give the problem a precise mathematical formulation and start its
investigation. For the technically simple case of U(1) as gauge group, we
establish a number of "no-go theorems" asserting that for certain classes of
measures, the flux operators can not be represented on the corresponding
Hilbert spaces.
The flux-observables we consider play an important role in loop quantum
gravity since they can be defined without recourse to a background geometry,
and they might also be of interest in the general context of quantization of
non-Abelian gauge theories.Comment: LaTeX, 21 pages, 5 figures; v3: some typos and formulations
corrected, some clarifications added, bibliography updated; article is now
identical to published versio
Global Progress Toward Implementing the United Nations Fish Stocks Agreement
This brief examines the progress made in implementing the Fish Stocks Agreement, based on a review of the status of certain highly migratory stocks and the effectiveness of regional fishery management organization (RFMO) measures in meeting specific mandates. It also looks at whether recommendations made in prior reviews have been implemented
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