228 research outputs found
On the semiclassical limit of 4d spin foam models
We study the semiclassical properties of the Riemannian spin foam models with
Immirzi parameter that are constructed via coherent states. We show that in the
semiclassical limit the quantum spin foam amplitudes of an arbitrary
triangulation are exponentially suppressed, if the face spins do not correspond
to a discrete geometry. When they do arise from a geometry, the amplitudes
reduce to the exponential of i times the Regge action. Remarkably, the
dependence on the Immirzi parameter disappears in this limit.Comment: 32 pages, 5 figure
A Gauge-Independent Mechanism for Confinement and Mass Gap: Part II -- G=SU(2) and D=3
We apply to the case of gauge group G = SU(2) in three dimensions a recently
proposed gauge-independent mechanism for confinement that is based on a
particular form of the dual spin foam framework for lattice gauge theory.
Explicit formulae for interaction factors and their asymptotics are introduced
and their behavior in different sectors of the theory are identified and
analyzed. We arrive at several elementary properties of the dual theory that
represent one scenario by which confinement may be realized at weak coupling.
We conclude with an outlook for further development of this approach.Comment: 18 pages, 3 figure
Holomorphic Factorization for a Quantum Tetrahedron
We provide a holomorphic description of the Hilbert space H(j_1,..,j_n) of
SU(2)-invariant tensors (intertwiners) and establish a holomorphically
factorized formula for the decomposition of identity in H(j_1,..,j_n).
Interestingly, the integration kernel that appears in the decomposition formula
turns out to be the n-point function of bulk/boundary dualities of string
theory. Our results provide a new interpretation for this quantity as being, in
the limit of large conformal dimensions, the exponential of the Kahler
potential of the symplectic manifold whose quantization gives H(j_1,..,j_n).
For the case n=4, the symplectic manifold in question has the interpretation of
the space of "shapes" of a geometric tetrahedron with fixed face areas, and our
results provide a description for the quantum tetrahedron in terms of
holomorphic coherent states. We describe how the holomorphic intertwiners are
related to the usual real ones by computing their overlap. The semi-classical
analysis of these overlap coefficients in the case of large spins allows us to
obtain an explicit relation between the real and holomorphic description of the
space of shapes of the tetrahedron. Our results are of direct relevance for the
subjects of loop quantum gravity and spin foams, but also add an interesting
new twist to the story of the bulk/boundary correspondence.Comment: 45 pages; published versio
A spin foam model for general Lorentzian 4-geometries
We derive simplicity constraints for the quantization of general Lorentzian
4-geometries. Our method is based on the correspondence between coherent states
and classical bivectors and the minimization of associated uncertainties. For
spacelike geometries, this scheme agrees with the master constraint method of
the model by Engle, Pereira, Rovelli and Livine (EPRL). When it is applied to
general Lorentzian geometries, we obtain new constraints that include the EPRL
constraints as a special case. They imply a discrete area spectrum for both
spacelike and timelike surfaces. We use these constraints to define a spin foam
model for general Lorentzian 4-geometries.Comment: 27 pages, 1 figure; v4: published versio
Spin foams with timelike surfaces
Spin foams of 4d gravity were recently extended from complexes with purely
spacelike surfaces to complexes that also contain timelike surfaces. In this
article, we express the associated partition function in terms of vertex
amplitudes and integrals over coherent states. The coherent states are
characterized by unit 3--vectors which represent normals to surfaces and lie
either in the 2--sphere or the 2d hyperboloids. In the case of timelike
surfaces, a new type of coherent state is used and the associated completeness
relation is derived. It is also shown that the quantum simplicity constraints
can be deduced by three different methods: by weak imposition of the
constraints, by restriction of coherent state bases and by the master
constraint.Comment: 22 pages, no figures; v2: remarks on operator formalism added in
discussion; correction: the spin 1/2 irrep of the discrete series does not
appear in the Plancherel decompositio
Asymptotics of 4d spin foam models
We study the asymptotic properties of four-simplex amplitudes for various
four-dimensional spin foam models. We investigate the semi-classical limit of
the Ooguri, Euclidean and Lorentzian EPRL models using coherent states for the
boundary data. For some classes of geometrical boundary data, the asymptotic
formulae are given, in all three cases, by simple functions of the Regge action
for the four-simplex geometry.Comment: 10 pages, Proceedings for the 2nd Corfu summer school and workshop on
quantum gravity and quantum geometry, talk given by Winston J. Fairbair
A finiteness bound for the EPRL/FK spin foam model
We show that the EPRL/FK spin foam model of quantum gravity has an absolutely
convergent partition function if the vertex amplitude is divided by an
appropriate power of the product of dimensions of the vertex spins. This
power is independent of the spin foam 2-complex and we find that insures
the convergence of the state sum. Determining the convergence of the state sum
for the values requires the knowledge of the large-spin
asymptotics of the vertex amplitude in the cases when some of the vertex spins
are large and other are small.Comment: v6: published versio
A Gauge-Independent Mechanism for Confinement and Mass Gap: Part I -- The General Framework
We propose a gauge-independent mechanism for the area-law behavior of Wilson
loop expectation values in terms of worldsheets spanning Wilson loops
interacting with the spin foams that contribute to the vacuum partition
function. The method uses an exact transformation of lattice-regularized
Yang-Mills theory that is valid for all couplings. Within this framework, some
natural conjectures can be made as to what physical mechanism enforces the
confinement property in the continuum (weak coupling) limit. Details for the
SU(2) case in three dimensions are provided in a companion paper.Comment: 16 pages, 4 figure
Second-order amplitudes in loop quantum gravity
We explore some second-order amplitudes in loop quantum gravity. In
particular, we compute some second-order contributions to diagonal components
of the graviton propagator in the large distance limit, using the old version
of the Barrett-Crane vertex amplitude. We illustrate the geometry associated to
these terms. We find some peculiar phenomena in the large distance behavior of
these amplitudes, related with the geometry of the generalized triangulations
dual to the Feynman graphs of the corresponding group field theory. In
particular, we point out a possible further difficulty with the old
Barrett-Crane vertex: it appears to lead to flatness instead of Ricci-flatness,
at least in some situations. The observation raises the question whether this
difficulty remains with the new version of the vertex.Comment: 22 pages, 18 figure
Cosmological Plebanski theory
We consider the cosmological symmetry reduction of the Plebanski action as a
toy-model to explore, in this simple framework, some issues related to loop
quantum gravity and spin-foam models. We make the classical analysis of the
model and perform both path integral and canonical quantizations. As for the
full theory, the reduced model admits two types of classical solutions:
topological and gravitational ones. The quantization mixes these two solutions,
which prevents the model to be equivalent to standard quantum cosmology.
Furthermore, the topological solution dominates at the classical limit. We also
study the effect of an Immirzi parameter in the model.Comment: 20 page
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