10 research outputs found
Holonomy observables in Ponzano-Regge type state sum models
We study observables on group elements in the Ponzano-Regge model. We show
that these observables have a natural interpretation in terms of Feynman
diagrams on a sphere and contrast them to the well studied observables on the
spin labels. We elucidate this interpretation by showing how they arise from
the no-gravity limit of the Turaev-Viro model and Chern-Simons theory.Comment: 15 pages, 2 figure
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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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
A new look at loop quantum gravity
I describe a possible perspective on the current state of loop quantum
gravity, at the light of the developments of the last years. I point out that a
theory is now available, having a well-defined background-independent
kinematics and a dynamics allowing transition amplitudes to be computed
explicitly in different regimes. I underline the fact that the dynamics can be
given in terms of a simple vertex function, largely determined by locality,
diffeomorphism invariance and local Lorentz invariance. I emphasize the
importance of approximations. I list open problems.Comment: 15 pages, 5 figure
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