108 research outputs found
Coupling of hard dimers to dynamical lattices via random tensors
We study hard dimers on dynamical lattices in arbitrary dimensions using a
random tensor model. The set of lattices corresponds to triangulations of the
d-sphere and is selected by the large N limit. For small enough dimer
activities, the critical behavior of the continuum limit is the one of pure
random lattices. We find a negative critical activity where the universality
class is changed as dimers become critical, in a very similar way hard dimers
exhibit a Yang-Lee singularity on planar dynamical graphs. Critical exponents
are calculated exactly. An alternative description as a system of
`color-sensitive hard-core dimers' on random branched polymers is provided.Comment: 12 page
Complete loop quantization of a dimension 1+2 Lorentzian gravity theory
De Sitter Chern-Simons gravity in D = 1 + 2 spacetime is known to possess an
extension with a Barbero-Immirzi like parameter. We find a partial gauge fixing
which leaves a compact residual gauge group, namely SU(2). The compacticity of
the residual gauge group opens the way to the usual LQG quantization
techniques. We recall the exemple of the LQG quantization of SU(2) CS theory
with cylindrical space topology, which thus provides a complete LQG of a
Lorentzian gravity model in 3-dimensional space-time.Comment: Loops11 - Madrid - 2011 (4 pages, Latex
Bubbles and jackets: new scaling bounds in topological group field theories
We use a reformulation of topological group field theories in 3 and 4
dimensions in terms of variables associated to vertices, in 3d, and edges, in
4d, to obtain new scaling bounds for their Feynman amplitudes. In both 3 and 4
dimensions, we obtain a bubble bound proving the suppression of singular
topologies with respect to the first terms in the perturbative expansion (in
the cut-off). We also prove a new, stronger jacket bound than the one currently
available in the literature. We expect these results to be relevant for other
tensorial field theories of this type, as well as for group field theory models
for 4d quantum gravity.Comment: v2: Minor modifications to match published versio
Effective Hamiltonian Constraint from Group Field Theory
Spinfoam models provide a covariant formulation of the dynamics of loop
quantum gravity. They are non-perturbatively defined in the group field theory
(GFT) framework: the GFT partition function defines the sum of spinfoam
transition amplitudes over all possible (discretized) geometries and
topologies. The issue remains, however, of explicitly relating the specific
form of the group field theory action and the canonical Hamiltonian constraint.
Here, we suggest an avenue for addressing this issue. Our strategy is to expand
group field theories around non-trivial classical solutions and to interpret
the induced quadratic kinematical term as defining a Hamiltonian constraint on
the group field and thus on spin network wave functions. We apply our procedure
to Boulatov group field theory for 3d Riemannian gravity. Finally, we discuss
the relevance of understanding the spectrum of this Hamiltonian operator for
the renormalization of group field theories.Comment: 14 page
Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model
A dual formulation of group field theories, obtained by a Fourier transform
mapping functions on a group to functions on its Lie algebra, has been proposed
recently. In the case of the Ooguri model for SO(4) BF theory, the variables of
the dual field variables are thus so(4) bivectors, which have a direct
interpretation as the discrete B variables. Here we study a modification of the
model by means of a constraint operator implementing the simplicity of the
bivectors, in such a way that projected fields describe metric tetrahedra. This
involves a extension of the usual GFT framework, where boundary operators are
labelled by projected spin network states. By construction, the Feynman
amplitudes are simplicial path integrals for constrained BF theory. We show
that the spin foam formulation of these amplitudes corresponds to a variant of
the Barrett-Crane model for quantum gravity. We then re-examin the arguments
against the Barrett-Crane model(s), in light of our construction.Comment: revtex, 24 page
Degenerate Plebanski Sector and Spin Foam Quantization
We show that the degenerate sector of Spin(4) Plebanski formulation of
four-dimensional gravity is exactly solvable and describes covariantly embedded
SU(2) BF theory. This fact ensures that its spin foam quantization is given by
the SU(2) Crane-Yetter model and allows to test various approaches of imposing
the simplicity constraints. Our analysis strongly suggests that restricting
representations and intertwiners in the state sum for Spin(4) BF theory is not
sufficient to get the correct vertex amplitude. Instead, for a general theory
of Plebanski type, we propose a quantization procedure which is by construction
equivalent to the canonical path integral quantization and, being applied to
our model, reproduces the SU(2) Crane-Yetter state sum. A characteristic
feature of this procedure is the use of secondary second class constraints on
an equal footing with the primary simplicity constraints, which leads to a new
formula for the vertex amplitude.Comment: 34 pages; changes in the abstract and introduction, a few references
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Commuting Simplicity and Closure Constraints for 4D Spin Foam Models
Spin Foam Models are supposed to be discretised path integrals for quantum
gravity constructed from the Plebanski-Holst action. The reason for there being
several models currently under consideration is that no consensus has been
reached for how to implement the simplicity constraints. Indeed, none of these
models strictly follows from the original path integral with commuting B
fields, rather, by some non standard manipulations one always ends up with non
commuting B fields and the simplicity constraints become in fact anomalous
which is the source for there being several inequivalent strategies to
circumvent the associated problems. In this article, we construct a new
Euclidian Spin Foam Model which is constructed by standard methods from the
Plebanski-Holst path integral with commuting B fields discretised on a 4D
simplicial complex. The resulting model differs from the current ones in
several aspects, one of them being that the closure constraint needs special
care. Only when dropping the closure constraint by hand and only in the large
spin limit can the vertex amplitudes of this model be related to those of the
FK Model but even then the face and edge amplitude differ. Curiously, an ad hoc
non-commutative deformation of the variables leads from our new model
to the Barrett-Crane Model in the case of Barbero-Immirzi parameter goes to
infinity.Comment: 41 pages, 4 figure
Feynman diagrammatic approach to spin foams
"The Spin Foams for People Without the 3d/4d Imagination" could be an
alternative title of our work. We derive spin foams from operator spin network
diagrams} we introduce. Our diagrams are the spin network analogy of the
Feynman diagrams. Their framework is compatible with the framework of Loop
Quantum Gravity. For every operator spin network diagram we construct a
corresponding operator spin foam. Admitting all the spin networks of LQG and
all possible diagrams leads to a clearly defined large class of operator spin
foams. In this way our framework provides a proposal for a class of 2-cell
complexes that should be used in the spin foam theories of LQG. Within this
class, our diagrams are just equivalent to the spin foams. The advantage,
however, in the diagram framework is, that it is self contained, all the
amplitudes can be calculated directly from the diagrams without explicit
visualization of the corresponding spin foams. The spin network diagram
operators and amplitudes are consistently defined on their own. Each diagram
encodes all the combinatorial information. We illustrate applications of our
diagrams: we introduce a diagram definition of Rovelli's surface amplitudes as
well as of the canonical transition amplitudes. Importantly, our operator spin
network diagrams are defined in a sufficiently general way to accommodate all
the versions of the EPRL or the FK model, as well as other possible models. The
diagrams are also compatible with the structure of the LQG Hamiltonian
operators, what is an additional advantage. Finally, a scheme for a complete
definition of a spin foam theory by declaring a set of interaction vertices
emerges from the examples presented at the end of the paper.Comment: 36 pages, 23 figure
Group field theory and simplicial quantum gravity
We present a new Group Field Theory for 4d quantum gravity. It incorporates
the constraints that give gravity from BF theory, and has quantum amplitudes
with the explicit form of simplicial path integrals for 1st order gravity. The
geometric interpretation of the variables and of the contributions to the
quantum amplitudes is manifest. This allows a direct link with other simplicial
gravity approaches, like quantum Regge calculus, in the form of the amplitudes
of the model, and dynamical triangulations, which we show to correspond to a
simple restriction of the same.Comment: 14 pages, no figures; RevTex4; v2: definition of the model modified,
discussion extended and improve
Quantization of Lorentzian 3d Gravity by Partial Gauge Fixing
D = 2+1 gravity with a cosmological constant has been shown by Bonzom and
Livine to present a Barbero-Immirzi like ambiguity depending on a parameter. We
make use of this fact to show that, for positive cosmological constant, the
Lorentzian theory can be partially gauge fixed and reduced to an SU(2)
Chern-Simons theory. We then review the already known quantization of the
latter in the framework of Loop Quantization for the case of space being
topogically a cylinder. We finally construct, in the same setting, a quantum
observable which, although non-trivial at the quantum level, corresponds to a
null classical quantity.Comment: Notation defect fixed on pages 5 (bottom) and 6 (around Eqs. 3.1)--
19 pages, Late
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