This week I have been continuing my work on the Hamiltonian constraint in Loop Quantum Gravity, The main paper I’ve been studying this week is ‘Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint’. Fortunately enough linking covariant and canonical LQG was also the topic of a recent seminar by Zipfel in the ilqgs spring program.

The authors of this paper emphasize that spin-foam models could realize a projection on the physical Hilbert space of canonical Loop Quantum Gravity (LQG). As a test the authors analyze the one-vertex expansion of a simple Euclidean spin-foam. They find that for fixed Barbero-Immirzi parameter γ= 1 the one vertex-amplitude in the KKL prescription annihilates the Euclidean Hamiltonian constraint of LQG. Since for γ = 1 the Lorentzian part of the Hamiltonian constraint does not contribute this gives rise to new solutions of the Euclidean theory. Furthermore, they find that the new states only depend on the diagonal matrix elements of the volume. This seems to be a generic property when applying the spin-foam projector.

To circumvent the problems of the canonical theory, a

covariant formulation of Quantum Gravity, the so-called spin-foam model was introduced. This model is mainly based on the observation that the Holst action for GR defines a constrained BF-theory. The strategy is first to quantize discrete BF-theory and then to implement the so called simplicity constraints. The main building block of the model is a linear two-complex κ** **embedded into 4-dimensional space-time M whose boundary is given by an initial and final gauge invariant spin-network, Ψi respectively Ψf , living on the initial respectively final spatial hyper surface of a

foliation of M. The physical information is encoded in the spin-foam amplitude.

where Af , Ae and Av are the amplitudes associated to the internal faces, edges and vertices of κ and B contains the boundary amplitudes.

Each spin-foam can be thought of as generalized

Feynman diagram contributing to the transition amplitude from an ingoing spin-network to an outgoing spin-network. By summing over all possible two-complexes one obtains the complete transition amplitude between ψi and ψf .

The main idea in this paper is that if f spin-foams provide a rigging map the physical inner product would be given by

and the rigging map would correspond to

Since all constraints are satised in H*phys* the physical scalar product must obey

for all ψout, ψn ∈ H*kin*.

As a test the authors consider an easy spin-foam amplitude and show that

where κ is a two-complex with only one internal vertex such that Φ is a spin-network induced on the boundary of κ and **H**n is the Hamiltonian constraint acting on the node n.

** Hamiltonian constraint**

The classical Hamiltonian constraint is

where,

The constraint can be split into its Euclidean part H = Tr[F∧e] and Lorentzian part HL = C- H.

Using,

where V is the volume of an arbitrary region ∑ containing the point x. Smearing the constraints with lapse function N(x) gives

This expression requires a regularization in order to obtain a well-defined operator on H*kin*. Using a triangulation T of the manifold into elementary tetrahedra with analytic links adapted to the graph Γ of an arbitrary spin-network.

Three non-planar links define a tetrahedron . Now decompose H[N] into a sum of one term per each tetrahedron of the triangulation,

To define the classical regularized Hamiltonian constraint as,

The connection A and the curvature are regularized by the holonomy h in SU (2), where in the fundamental representation m = ½. This gives,

which converges to the Hamiltonian constraint if the triangulation is sufficiently fine.

As seen in the post

This can be generalized with a trace in an arbitrary irreducible representation m leading to

this converges to H[N] as well.

**Properties**

The important properties of the Euclidean Hamiltonian constraint are;

when acting on a spin-network state, the operator reduces to a sum over terms each acting on individual nodes. Acting on nodes of valence n the operator gives

The Hamiltonian constraint on diffeomorphism invariant states is independent from the refinement of the triangulation.

Since the Ashtekar-Lewandowsk volume operator annihilates coplanar nodes and gauge invariant nodes of valence three H does not act on the new nodes – the so called extraordinary nodes.

**Action on a trivalent node**

To ompute the action of the operator on a trivalent node where all links are outgoing, denote a trivalent node by |n(ji,jj , jk)> ≡ |n3>, whereas ji, jj ,jk are the spins of the adjacent links ei, ej , ek:

To quantize [N] the holonomies and the volume are replaced by their corresponding operators and the Poisson bracket is replaced by a commutator. Since the volume operator vanishes on a gauge invariant trivalent node only need to compute;

so, h(m) creates a free index in the m-representation located at the node , making it non-gauge invariant and a new node on the link ek:

so we get,

where the range of the sums over a, b is determined by the Clebsch-Gordan conditions and

The complete action of the operator on a trivalent state |n(ji, jj , jk)> can be obtained by contracting the trace part with εijk. So, H projects on a linear combination of three spin networks which differ by exactly one new link labelled by m between each couple of the oldlinks at the node.

**Action on a 4-valent node**

The computation for a 4-valent node |n4> is

where i labels the intertwiner – inner link.

The holonomy h(m) changes the valency of the node and the Volume therefore acts on the 5-valent non-gauge invariant node. Graphically this corresponds to

and finally we get:

This can be simplied to;

**SPIN-FOAM**

Using the definition of an Euclidean spin-foam models as suggested by Kaminski, Kisielowski and Lewandowski (KKL) and since we,re only interested in the evaluation of a spin-foam amplitude we choose a combinatorial definition of the model:

Consider an oriented two-complex defined as the union of the set of faces (2-cells) F, edges (1-cells) E and vertices (0-cells) V such that every edge e is a 1-face of at least one face f (e ∈ ∂f) and every vertex V is a 0-face of at least one edge e (v ∈∂e).

We call edges which are contained in more than one face f internal and denote the set of all internal edges by * Eint.* All vertices adjacent to more than one internal edge are also called internal and denote the set of these vertices by

**. The boundary ∂κ is the union of all external vertices (-nodes) n ∉ Vint and external edges (links) l ∉**

*Vint**.*

**Eint**A spin-foam is a triple (κ, ρ; I) consisting of a proper foam whose faces are labelled by irreducible representations of a Lie-group G, in this case here SO(4) and whose internal edges are labeled by intertwiners I. This induces a spin-network structure ∂(κ, ρ; I) on the boundary of κ.

The BF partition function can be rewritten as

Av defines an SO (4) invariant function on the graph Γ induced on the boundary of the vertex v

In the EPRL model the simplicity constraint is imposed weakly so,

It follows immediately that

defines the EPRL vertex amplitude with

Expanding the delta function in terms of spin-network function and integrating over the group elements gives

In order to evaluate the fusion coefficients by graphical calculus it is convenient to work with 3j-symbols instead of Clebsch-Gordan coefficients. When replacing the Clebsch-Gordan coefficients we have to multiply by an overall factor .

**Spin-foam projector**

Given any couple of ingoing and outgoing kinematical states ψout, ψin, the Physical scalar product can be

formally defined by

where η is a projector – Rigging map onto the Kernel of the Hamiltonian constraint.

Suppose that the transition amplitude Z

can be expressed in terms of a sum of spin-foams, then

this can be interpreted as a function on the boundary graph ∂κ;

in the EPRL sector.

Restricting the boundary elements h ∈ SU (2) ⊂ SO (4) then;

where |S>N is a normalized spin-network function on SU(2). This finally implies;

**NEW SOLUTIONS TO THE EUCLIDEAN HAMILTONIAN CONSTRAINT**

Compute new solutions to the Euclidean Hamiltonian constraint by employing spin-foam methods. Show that

in the Euclidean sector with γ= 1 and s = 1, where κ is a 2-complex with only one internal vertex.

**Trivalent nodes**

Consider the simplest possible case given by an initial and final state |Θ>, characterized by two trivalent nodes joined by three links:

the only states produced by the Hamiltonian acting on a node, are given by a linear combination of spin-networks that differ from the original one by the presence of an extraordinary (new) link. In particular the term will be non vanishing only if |s> is of the kind:

The simplest two-complex κ(Θ,s) with only one internal vertex defining a cobordism between |Θ> and |s> is a tube Θ x[0,1] with an additional face between the internal vertex and the new link m,

Since the space of three-valent intertwiners is one-dimensional and all labelings jf are fixed by the states |Θ> , |s> the first sum is trivial.

Γv= s and therefore we have,

where the sign factor is due to the orientation of s, The fusion coefficients contribute four 9j symbols since,

The full amplitude is

and

This yields,

The last two terms are equivalent to the first term when exchanging jk↔ jj. The EPRL spin-foam reduces just to the SU(2) BF amplitude that is just the single 6j in the first line.

Now using the definition of a 9j in terms of three 6j’s,

The 9j’s involved in this expression can be reordered using the permutation symmetries giving,

the statesare solutions of the Euclidean Hamiltonian constraint

The spin-foam amplitude selects only those terms which depend on the diagonal elements on the volume. This simplifies the calculation since we do not have to evaluate the volume explicitly.

**Four valent nodes**

The case with ψ in = ψout = |n4>where

The matrix element <s| |n4> is non-vanishing if |s>is of the form

Choose again a complex of the form

with one additional face jl.

The vertex trace can be evaluated by graphical calculus

The fusion coefficients give two 9j symbols for the two trivalent edges and two 15j- symbols for the two four-valent edges.The fusion coefficients reduce to 1 when γ=1.Taking the scalar product we obtain,

Taking the scalar product with the Hamiltonian gives,

Summing over a and using the orthogonality relation gives,

The three 6j’s in the two terms define a 9j ,summing over the indexes and b respectively gives:

the final result is,

As for the trivalent vertex the spin-foam amplitude just takes those elements into account which depend on the diagonal Volume elements.

**CONCLUSIONS**

LQG is grounded on two parallel constructions; the canonical and the covariant ones. In this paper the authors construct a simple spin-foam amplitude which annihilates the Hamiltonian constraint .They found that in the euclidean sector with signature s = 1 and Barbero-Immirzi parameter γ = 1 the Euclidean Hamiltonian constraint is annihilated by a spin-foam amplitude Z for a simple two-complex with only one internal vertex. The one vertex amplitudes of BF theory are explicit analytic solutions of the Hamiltonian theory.

Also the 6j symbol associated to every face is annihilated by the Euclidean scalar constraint. This is a generalization of the work by Bonzom-Freidel in the context of 3d gravity where they found that the 6jis annihilated by a suitable quantization of the 3d scalar constraint F = 0 . The spin-foam amplitude diagonalizes the Volume.

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