In this paper 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 allows the consideration of knotting or linking spin-foam histories. 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)xSU(2) EPRL intertwiners is shown to be 1-1 in the case of the Barbero-Immirzinparameter γ≥ 1.
Recent spin-foam models of the gravitational field
Spin-foams were introduced as histories of quantum spin-network states of loop quantum gravity (LQG) . That idea gave rise to spin foam models (SFM). The spin-foam model of 3-dimensional gravity is derived from a discretization of the BF action.
For the 4-dimensional gravity there were several approaches, including the Barrett-Crane spinfoam model. The BC model is mathematically elegant, but it was shown not to have sufficiently many degrees of freedom to ensure the correct classical limit.
A suitable modification was found by two teams: (i) Engle, Pereira, Rovelli and Livine and (ii) Freidel and Krasnov who found systematic derivations of a spin-foam model of gravitational field using as the starting point a discretization of the Holst action. This led to a spin-foam model valid for the Barbero-Immirzi parameter belonging to the interval −1 ≤γ ≤ 1 which is a promising candidate for a path integral formulation of LQG, The values of the Barbero-Immirzi parameter predicted by various black hole models also belong to this range.
The aim of the authors of this paper is to develop spin-foam models and most importantly the EPRL model so that they can be used to define spin-foam histories of an arbitrary spin network state of LQG. The notion of embedded spin-foam we use, allows to consider knotting or linking spin-foam histories. Since the knots and links may play a role in LQG, it is important to keep these topological degrees of freedom in a spin-foam approach.
The authors also characterize the structure of a general spin-foam vertex. They encode all the information about the vertex structure in the spin-network induced on the boundary of the neighborhood of
a given vertex. They also define the generalize the EPRL spin-foams and the Engle-Pereira-Rovelli-Livine intertwiner to general spin foams.
Given a compact group G, a G-spin-network is a triple ( γ,ρ,ι ), an oriented, piecewise linear 1-complex (a graph) equipped with two colourings: ρ and ι defined below
The colouring ρ, maps the set (γ) of the 1-cells (edges) in γ into the set of irreducible representations of G. That is, to every edge e we assign an irreducible representation e defined in the Hilbert space He.
The colouring ι maps each vertex v ∈γ(o) (where (γ0) denotes the set of 0-cells of γ) into the subspace
Spin-foams of spin-networks and of spin-network functions
The motivation of the spin-foam approach to LQG is to develop an analog of the Feynman path integral, the paths, should be suitably defined histories of the spin-network states.
A foam is an oriented linear 2-cell complex with boundary. Briefly, each foam consists of 2-cells (faces), 1-cells (edges), and 0-cells (vertices).
The faces are polygons, their sides are edges, the ends of the edges are vertices. Faces and edges are oriented, and the orientation of an edge is independent of the orientation of the face it is contained in. Each edge is contained in several faces, each vertex is contained in several edges.
Given a foam , a spin-foam κ is defined by introducing two colourings:
To every internal vertex of a spin-foam ( κ,ρ,ι ) we can naturally assign a number by contracting the invariants which colour the incoming and outgoing edges and we call it the spin-foam vertex trace at a vertex v.
The spin-foam vertex trace has a clear interpretation in terms of the spin-networks.
Example of spin-foam evolution
The recent spin-foam models of Barrett-Crane , Engle-Pereira-Rovelli-Livine and Freidel-Krasnov fall into a common scheme. Combine it with the assumption that the spin-foams are histories of the embedded spin-network functions defined within the framework
of the diffeomorphism invariant quantization of the theory of connections. The resulting scheme reads as follows:
- First, ignoring the simplicity constraints, one quantizes the theory using the Hilbert space
- Next, the simplicity constraints are suitably quantized to become linear quantum constraints imposed on the elements.
- The amplitude of each spin-foam ( κ,ρ,ι ) is defined by using the spin-foam trace.
There are three main proposals for the simple intertwiners:
- That of Barrett-Crane (BC) corresponding to the Palatini action
- That of Engle-Pereira-Rovelli-Livine (EPRL) corresponding the Holst action with the value of the Barbero-Immirzi parameter γ= ±1
- That of Freidel-Krasnov (FK) also corresponding to the Holst action with the value of the Barbero-Immirzi parameter γ= ±1,
The authors conclusion is that we do not have to reformulate LQG in terms of thebpiecewise linear category and triangulations to match it with the EPRL SFM. The generalized spinfoam framework is compatible with the original LQG framework and accommodates all the spinnetwork states and the diffeomorphism covariance.
This papers generalization goes in two directions:
The first one is from spin-foams defined on simplicial complexes to spin-foams defined on the arbitrary linear 2-cell complexes. The main tools of the SF models needed to describe 4- dimensional spacetime are available: spin-foam, boundary spin-network, characterization of vertex, vertex amplitude, the scheme of the SF models of 4-dim gravity, the EPRL intertwiners.
The second direction is from abstract spin-foams to embedded spin-foams, histories of the embedded spin-network.2 For example the notion of knots and links is again available in that framework.
One may for example consider a spin-foam history which turns an unknotted embedded circle into a knotted circle.
The most important result is the characterization of a vertex of a generalized spin-foam. The structure of each vertex can be completely encoded in a spin-network induced locally on the boundary of the vertex neighbourhood. The spin-network is used for the natural generalization of the vertex amplitude used in the simplicial spin-foam models. The characterization of the vertices
leads to a general construction of a general spin-foam. The set of all the possible vertices is given by the set of the spin-networks and and set of all spin-foams can be obtained by gluing the vertices
and the “static” spin-foams.
In the literature, the 3j, 6j, 10j and 15j symbols are used extensively in the context of the BC and EPRL intertwiners and amplitudes. From the point of view of this paper, they are just related to a specific
choice of the basis in the space of intertwiners valid in the case of the simplicial spin-foams and spin-networks.
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