145 research outputs found
Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity
The Barrett-Crane models of Lorentzian quantum gravity are a family of spin
foam models based on the Lorentz group. We show that for various choices of
edge and face amplitudes, including the Perez-Rovelli normalization, the
amplitude for every triangulated closed 4-manifold is a non-negative real
number. Roughly speaking, this means that if one sums over triangulations,
there is no interference between the different triangulations. We prove
non-negativity by transforming the model into a ``dual variables'' formulation
in which the amplitude for a given triangulation is expressed as an integral
over three copies of hyperbolic space for each tetrahedron. Then we prove that,
expressed in this way, the integrand is non-negative. In addition to implying
that the amplitude is non-negative, the non-negativity of the integrand is
highly significant from the point of view of numerical computations, as it
allows statistical methods such as the Metropolis algorithm to be used for
efficient computation of expectation values of observables.Comment: 13 page
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
The loop-quantum-gravity vertex-amplitude
Spinfoam theories are hoped to provide the dynamics of non-perturbative loop
quantum gravity. But a number of their features remain elusive. The best
studied one -the euclidean Barrett-Crane model- does not have the boundary
state space needed for this, and there are recent indications that,
consequently, it may fail to yield the correct low-energy -point functions.
These difficulties can be traced to the SO(4) -> SU(2) gauge fixing and the way
certain second class constraints are imposed, arguably incorrectly, strongly.
We present an alternative model, that can be derived as a bona fide
quantization of a Regge discretization of euclidean general relativity, and
where the constraints are imposed weakly. Its state space is a natural subspace
of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network
space. The model provides a long sought SO(4)-covariant vertex amplitude for
loop quantum gravity.Comment: 6page
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
The complete LQG propagator: II. Asymptotic behavior of the vertex
In a previous article we have show that there are difficulties in obtaining
the correct graviton propagator from the loop-quantum-gravity dynamics defined
by the Barrett-Crane vertex amplitude. Here we show that a vertex amplitude
that depends nontrivially on the intertwiners can yield the correct propagator.
We give an explicit example of asymptotic behavior of a vertex amplitude that
gives the correct full graviton propagator in the large distance limit.Comment: 16 page
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
Three dimensional loop quantum gravity: physical scalar product and spin foam models
In this paper, we address the problem of the dynamics in three dimensional
loop quantum gravity with zero cosmological constant. We construct a rigorous
definition of Rovelli's generalized projection operator from the kinematical
Hilbert space--corresponding to the quantization of the infinite dimensional
kinematical configuration space of the theory--to the physical Hilbert space.
In particular, we provide the definition of the physical scalar product which
can be represented in terms of a sum over (finite) spin-foam amplitudes.
Therefore, we establish a clear-cut connection between the canonical
quantization of three dimensional gravity and spin-foam models. We emphasize
two main properties of the result: first that no cut-off in the kinematical
degrees of freedom of the theory is introduced (in contrast to standard
`lattice' methods), and second that no ill-defined sum over spins (`bubble'
divergences) are present in the spin foam representation.Comment: Typos corrected, version appearing in Class. Quant. Gra
Graviton propagator in loop quantum gravity
We compute some components of the graviton propagator in loop quantum
gravity, using the spinfoam formalism, up to some second order terms in the
expansion parameter.Comment: 41 pages, 6 figure
3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory
An effective field theory for matter coupled to three-dimensional quantum
gravity was recently derived in the context of spinfoam models in
hep-th/0512113. In this paper, we show how this relates to group field theories
and generalized matrix models. In the first part, we realize that the effective
field theory can be recasted as a matrix model where couplings between matrices
of different sizes can occur. In a second part, we provide a family of
classical solutions to the three-dimensional group field theory. By studying
perturbations around these solutions, we generate the dynamics of the effective
field theory. We identify a particular case which leads to the action of
hep-th/0512113 for a massive field living in a flat non-commutative space-time.
The most general solutions lead to field theories with non-linear redefinitions
of the momentum which we propose to interpret as living on curved space-times.
We conclude by discussing the possible extension to four-dimensional spinfoam
models.Comment: 17 pages, revtex4, 1 figur
3+1 spinfoam model of quantum gravity with spacelike and timelike components
We present a spinfoam formulation of Lorentzian quantum General Relativity.
The theory is based on a simple generalization of an Euclidean model defined in
terms of a field theory over a group. The model is an extension of a recently
introduced Lorentzian model, in which both timelike and spacelike components
are included. The spinfoams in the model, corresponding to quantized
4-geometries, carry a natural non-perturbative local causal structure induced
by the geometry of the algebra of the internal gauge (sl(2,C)). Amplitudes can
be expressed as integrals over the spacelike unit-vectors hyperboloid in
Minkowski space, or the imaginary Lobachevskian space.Comment: 16 pages, 1 figur
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