6 research outputs found
The EPRL intertwiners and corrected partition function
Do the SU(2) intertwiners parametrize the space of the EPRL solutions to the
simplicity constraint? What is a complete form of the partition function
written in terms of this parametrization? We prove that the EPRL map is
injective for n-valent vertex in case when it is a map from SO(3) into
SO(3)xSO(3) representations. We find, however, that the EPRL map is not
isometric. In the consequence, in order to be written in a SU(2) amplitude
form, the formula for the partition function has to be rederived. We do it and
obtain a new, complete formula for the partition function. The result goes
beyond the SU(2) spin-foam models framework.Comment: RevTex4, 15 pages, 5 figures; theorem of injectivity of EPRL map
correcte
The kernel and the injectivity of the EPRL map
In this paper we prove injectivity of the EPRL map for |\gamma|<1, filling
the gap of our previous paper.Comment: 17 pages, 3 figure
One vertex spin-foams with the Dipole Cosmology boundary
We find all the spin-foams contributing in the first order of the vertex
expansion to the transition amplitude of the Bianchi-Rovelli-Vidotto Dipole
Cosmology model. Our algorithm is general and provides spin-foams of
arbitrarily given, fixed: boundary and, respectively, a number of internal
vertices. We use the recently introduced Operator Spin-Network Diagrams
framework.Comment: 23 pages, 30 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
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.
Spin-Foams for All Loop Quantum Gravity
The simplicial framework of Engle-Pereira-Rovelli-Livine spin-foam models is
generalized to match the diffeomorphism invariant framework of loop quantum
gravity. The simplicial spin-foams are generalized to arbitrary linear 2-cell
spin-foams. The resulting framework admits all the spin-network states of loop
quantum gravity, not only those defined by triangulations (or cubulations). In
particular the notion of embedded spin-foam we use allows to consider knotting
or linking spin-foam histories. Also the main tools as the vertex structure and
the vertex amplitude are naturally generalized to arbitrary valency case. The
correspondence between all the SU(2) intertwiners and the SU(2)SU(2)
EPRL intertwiners is proved to be 1-1 in the case of the Barbero-Immirzi
parameter , unless the co-domain of the EPRL map is trivial and
the domain is non-trivial.Comment: RevTex4, 23 pages, 8 figures; important references added; minor
corrections, version published in Class.Quant.Grav; theorem of injectivity of
EPRL map correcte