342 research outputs found
Attributing sense to some integrals in Regge calculus
Regge calculus minisuperspace action in the connection representation has the
form in which each term is linear over some field variable (scale of area-type
variable with sign). We are interested in the result of performing integration
over connections in the path integral (now usual multiple integral) as function
of area tensors even in larger region considered as independent variables. To
find this function (or distribution), we compute its moments, i. e. integrals
with monomials over area tensors. Calculation proceeds through intermediate
appearance of -functions and integrating them out. Up to a singular
part with support on some discrete set of physically unattainable points, the
function of interest has finite moments. This function in physical region
should therefore exponentially decay at large areas and it really does being
restored from moments. This gives for gravity a way of defining such
nonabsolutely convergent integral as path integral.Comment: 14 pages, presentation improve
Spin foam model from canonical quantization
We suggest a modification of the Barrett-Crane spin foam model of
4-dimensional Lorentzian general relativity motivated by the canonical
quantization. The starting point is Lorentz covariant loop quantum gravity. Its
kinematical Hilbert space is found as a space of the so-called projected spin
networks. These spin networks are identified with the boundary states of a spin
foam model and provide a generalization of the unique Barrette-Crane
intertwiner. We propose a way to modify the Barrett-Crane quantization
procedure to arrive at this generalization: the B field (bi-vectors) should be
promoted not to generators of the gauge algebra, but to their certain
projection. The modification is also justified by the canonical analysis of
Plebanski formulation. Finally, we compare our construction with other
proposals to modify the Barret-Crane model.Comment: 26 pages; presentation improved, important changes concerning the
closure constraint and the vertex amplitude; minor correctio
The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
Some components of the graviton two-point function have been recently
computed in the context of loop quantum gravity, using the spinfoam
Barrett-Crane vertex. We complete the calculation of the remaining components.
We find that, under our assumptions, the Barrett-Crane vertex does not yield
the correct long distance limit. We argue that the problem is general and can
be traced to the intertwiner-independence of the Barrett-Crane vertex, and
therefore to the well-known mismatch between the Barrett-Crane formalism and
the standard canonical spin networks. In a companion paper we illustrate the
asymptotic behavior of a vertex amplitude that can correct this difficulty.Comment: 31 page
From Dimensional Reduction of 4d Spin Foam Model to Adding Non-Gravitational Fields to 3d Spin Foam Model
A Kaluza-Klein like approach for a 4d spin foam model is considered. By
applying this approach to a model based on group field theory in 4d (TOCY
model), and using the Peter-Weyl expansion of the gravitational field,
reconstruction of new non gravitational fields and interactions in the action
are found. The perturbative expansion of the partition function produces graphs
colored with su(2) algebraic data, from which one can reconstruct a 3d
simplicial complex representing space-time and its geometry; (like in the
Ponzano-Regge formulation of pure 3d quantum gravity), as well as the Feynman
graph for typical matter fields. Thus a mechanism for generation of matter and
construction of new dimensions are found from pure gravity.Comment: 11 pages, no figure, to be published in International Journal of
Geometric Methods in Modern Physic
Abelian BF theory and Turaev-Viro invariant
The U(1) BF Quantum Field Theory is revisited in the light of
Deligne-Beilinson Cohomology. We show how the U(1) Chern-Simons partition
function is related to the BF one and how the latter on its turn coincides with
an abelian Turaev-Viro invariant. Significant differences compared to the
non-abelian case are highlighted.Comment: 47 pages and 6 figure
Semiclassical Analysis of the Wigner Symbol with One Small Angular Momentum
We derive an asymptotic formula for the Wigner symbol, in the limit of
one small and 11 large angular momenta. There are two kinds of asymptotic
formulas for the symbol with one small angular momentum. We present the
first kind of formula in this paper. Our derivation relies on the techniques
developed in the semiclassical analysis of the Wigner symbol [L. Yu and R.
G. Littlejohn, Phys. Rev. A 83, 052114 (2011)], where we used a gauge-invariant
form of the multicomponent WKB wave-functions to derive asymptotic formulas for
the symbol with small and large angular momenta. When applying the same
technique to the symbol in this paper, we find that the spinor is
diagonalized in the direction of an intermediate angular momentum. In addition,
we find that the geometry of the derived asymptotic formula for the
symbol is expressed in terms of the vector diagram for a symbol. This
illustrates a general geometric connection between asymptotic limits of the
various symbols. This work contributes the first known asymptotic formula
for the symbol to the quantum theory of angular momentum, and serves as a
basis for finding asymptotic formulas for the Wigner symbol with two
small angular momenta.Comment: 15 pages, 14 figure
Holography for the Lorentz Group Racah Coefficients
A known realization of the Lorentz group Racah coefficients is given by an
integral of a product of 6 ``propagators'' over 4 copies of the hyperbolic
space. These are ``bulk-to-bulk'' propagators in that they are functions of two
points in the hyperbolic space. It is known that the bulk-to-bulk propagator
can be constructed out of two bulk-to-boundary ones. We point out that there is
another way to obtain the same object. Namely, one can use two bulk-to-boundary
and one boundary-to-boundary propagator. Starting from this construction and
carrying out the bulk integrals we obtain a realization of the Racah
coefficients that is ``holographic'' in the sense that it only involves
boundary objects. This holographic realization admits a geometric
interpretation in terms of an ``extended'' tetrahedron.Comment: 12 pages, 2 figures; v2: minor changes; v3: "extended" tetrahedron
interpretation adde
Lens Spaces and Handlebodies in 3D Quantum Gravity
We calculate partition functions for lens spaces L_{p,q} up to p=8 and for
genus 1 and 2 handlebodies H_1, H_2 in the Turaev-Viro framework. These can be
interpreted as transition amplitudes in 3D quantum gravity. In the case of lens
spaces L_{p,q} these are vacuum-to-vacuum amplitudes \O -> \O, whereas for
the 1- and 2-handlebodies H_1, H_2 they represent genuinely topological
transition amplitudes \O -> T^2 and \O -> T^2 # T^2, respectively.Comment: 14 pages, LaTeX, 5 figures, uses eps
3-dimensional Gravity from the Turaev-Viro Invariant
We study the -deformed su(2) spin network as a 3-dimensional quantum
gravity model. We show that in the semiclassical continuum limit the
Turaev-Viro invariant obtained recently defines naturally regularized
path-integral la Ponzano-Regge, In which a contribution from
the cosmological term is effectively included. The regularization dependent
cosmological constant is found to be , where
. We also discuss the relation to the Euclidean Chern-Simons-Witten
gravity in 3-dimension.Comment: 11page
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