20 research outputs found
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
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
Operator Spin Foam Models
The goal of this paper is to introduce a systematic approach to spin foams.
We define operator spin foams, that is foams labelled by group representations
and operators, as the main tool. An equivalence relation we impose in the set
of the operator spin foams allows to split the faces and the edges of the
foams. The consistency with that relation requires introduction of the
(familiar for the BF theory) face amplitude. The operator spin foam models are
defined quite generally. Imposing a maximal symmetry leads to a family we call
natural operator spin foam models. This symmetry, combined with demanding
consistency with splitting the edges, determines a complete characterization of
a general natural model. It can be obtained by applying arbitrary (quantum)
constraints on an arbitrary BF spin foam model. In particular, imposing
suitable constraints on Spin(4) BF spin foam model is exactly the way we tend
to view 4d quantum gravity, starting with the BC model and continuing with the
EPRL or FK models. That makes our framework directly applicable to those
models. Specifically, our operator spin foam framework can be translated into
the language of spin foams and partition functions. We discuss the examples: BF
spin foam model, the BC model, and the model obtained by application of our
framework to the EPRL intertwiners.Comment: 19 pages, 11 figures, RevTex4.
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
Laplacians on discrete and quantum geometries
We extend discrete calculus for arbitrary (-form) fields on embedded
lattices to abstract discrete geometries based on combinatorial complexes. We
then provide a general definition of discrete Laplacian using both the primal
cellular complex and its combinatorial dual. The precise implementation of
geometric volume factors is not unique and, comparing the definition with a
circumcentric and a barycentric dual, we argue that the latter is, in general,
more appropriate because it induces a Laplacian with more desirable properties.
We give the expression of the discrete Laplacian in several different sets of
geometric variables, suitable for computations in different quantum gravity
formalisms. Furthermore, we investigate the possibility of transforming from
position to momentum space for scalar fields, thus setting the stage for the
calculation of heat kernel and spectral dimension in discrete quantum
geometries.Comment: 43 pages, 2 multiple figures. v2: discussion improved, references
added, minor typos correcte
Coarse graining methods for spin net and spin foam models
We undertake first steps in making a class of discrete models of quantum
gravity, spin foams, accessible to a large scale analysis by numerical and
computational methods. In particular, we apply Migdal-Kadanoff and Tensor
Network Renormalization schemes to spin net and spin foam models based on
finite Abelian groups and introduce `cutoff models' to probe the fate of gauge
symmetries under various such approximated renormalization group flows. For the
Tensor Network Renormalization analysis, a new Gauss constraint preserving
algorithm is introduced to improve numerical stability and aid physical
interpretation. We also describe the fixed point structure and establish an
equivalence of certain models.Comment: 39 pages, 13 figures, 1 tabl