Tag Archives: Euclidean space

Calculations on Quantum Cuboids and the EPRL-FK path integral for quantum gravity

This week I have been studying a really great paper looking at Quantum Cuboids and path-integral calculations for the EPRL vertex in LQG and also beginning to write some calculational software tools for performing these calculations using Sagemath.

In this work the authors investigate the 4d path integral for Euclidean quantum gravity on a hypercubic lattice, as given by the EPRL-FK model. To tackle the problem, they restrict the path to a set of quantum geometries that reflects the lattice symmetries. In particular, the sum over intertwiners is restricted to quantum cuboids, that is,  coherent intertwiners which describe a cuboidal
geometry in the large-j limit.

Using asymptotic expressions for the vertex amplitude, several interesting properties of the state sum are found.

  • The value of coupling constants in the amplitude functions determines whether geometric or non-geometric configurations dominate the path integral.
  • There is a critical value of the coupling constant α, which separates two phases.  In one phase the main contribution
    comes from very irregular and crumpled states. In the other phase, the dominant contribution comes from a highly regular configuration, which can be interpreted as flat Euclidean space, with small non-geometric perturbations around it.
  • States which describe boundary geometry with high
    torsion have exponentially suppressed physical norm.

The symmetry-restricted state sum

Will work on a regular hypercubic lattice in 4d. On this lattice consider only states which conform to the lattice symmetry. This is a condition on the intertwiners, which  corresponds to cuboids.
A cuboid is completely determined by its three edge lengths, or equivalently by its three areas.


All internal angles are π/2 , and the condition of regular cuboids on all dual edges of the lattice result in a high degree of symmetries on the labels: The area and hence the spin on each two parallel squares of the lattice which are translations perpendicular to the squares, have to be equal.

The high degree of symmetry will make all quantum geometries flat. The analysis carried out here is therefore not suited for describing local curvature.


The plan of the paper is as follows:

  • Review of the EPRL-FK spin foam model
  • Semiclassical regime of the path integral
  • Construction of the quantum cuboid intertwiner
  • Full vertex amplitude, in particular describe its asymptotic expression for large spins
  • Numerical investigation of the quantum path integral

The spin foam state sum  employed is the Euclidean EPRL-FK model with Barbero-Immirzi parameter γ < 1. The EPRL-FK model is defined on an arbitrary 2-complexes. A 2-complex 􀀀 is determined by its vertices v, its edges e connecting two vertices, and faces f which are bounded by the edges.

The path integral is formulated as a sum over states. A state in this context is given by a collection of spins –  irreducible representations
jf ∈ 1/2 N of SU(2) to the faces, as well as a collection of intertwiners ιe on edges.

The actual sum is given by


where Af , Ae and Av are the face-, edge- and vertex- amplitude functions, depending on the state. The sum has to be carried out over all spins, and over an orthonormal orthonormal basis in the intertwiner space at each edge.

The allowed spins jf in the EPRL-FK model are such
that jf are both also half-integer spins.

The face amplitudes are either


The edge amplitudes Ae are usually taken to be equal to 1.

In Sagemath code this looks like:


Coherent intertwiners

In this paper, the space-time manifold used is  M∼ T³×[0, 1] is the product of the 3-torus T3 and a closed interval. The space is compactified toroidally. M is covered by 4d hypercubes, which
form a regular hypercubic lattice H.There is a vertex for each hypercube, and two vertices are connected by an edge whenever two hypercubes intersect ina 3d cube. The faces of 􀀀 are dual to squares in H, on which four hypercubes meet.The geometry will be encoded in the state, by specification of spins jf
and intertwiners ιe.


Intertwiners ιe can be given a geometric interpretation in terms of polyhedra in R³. Given a collection of spins j1, . . . jn and vectors n1, . . . nn which close closure. Can define the coherent polyhedron


The geometric interpretation is that of a polyhedron, with face areas jf and face normals ni. The closure condition ensures that such a polyhedron exists.

We are interested in the large j-regime of the quantum cuboids. In this limit, these become classical cuboids  which are completely specified by their three areas. Therefore, a
semiclassical configuration is given by an assignment of
areas a = lp² to the squares of the hypercubic lattice.

Denote the four directions in the lattice by x, y, z, t. The areas satisfy


The two constraints which reduce the twisted geometric
configurations to geometric configurations are given by:


For a non-geometric configuration, define the 4-volume of a hypercube as:


Define the four diameters to be:


then we have, V4 = dxdydzdt

We also define the non- geometricity as:


as a measure of the deviation from the constraints.

In sagemath code this looks like:


Quantum Cuboids

We let’s look at  the quantum theory. In the 2-complex, every edge has six faces attached to it, corresponding to the six faces of the cubes. So any intertwiner in the state-sum will be six-valent, and therefore can be described by a coherent polyhedron with six faces. In our setup, we restrict the state-sum to coherent cuboids, or quantum cuboids. A cuboid is characterized by areas on opposite sides of the cuboid being equal, and the respective normals being negatives of one another


The state ιj1,j2,j3 is given by:


The vertex amplitude for a Barbero-Immirzi parameter γ < 1 factorizes as Av = A+vAv with


with the complex action


where, a is the source node of the link l, while b is its target node.

Large j asymptotics
The amplitudes A±v possess an asymptotic expression for large jl. There are two distinct stationary and critical points, satisfying the equations.

cuboidequ15for all links ab . Using the convention shown below

cuboidfig4having fixed g0 = 1, the two solutions Σ1 and Σ2 are


The amplitudes A±satisfy, in the large j limit,


In the large j-limit, the norm squared of the quantum cuboid states is given by:


For the state sum, in the large-j limit on a regular hypercubic lattice:


In sagemath code this looks like:






Related articles

Causal cells: spacetime polytopes with null hyperfaces by Neiman

This week I have been reading a paper about polyhedra and 4-polytopes in Minkowski spacetime – in particular, null polyhedra
with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces.

A 3D projection of an tesseract performing an ...
A 3D projection of a tesseract performing an isoclinic rotation.


The paper presents the basic properties of several classes of null faced 4-polytopes: 4-simplices, tetrahedral diamonds and 4-parallelotopes. A most regular representative of each class is proposed.

The most-regular parallelotope is of particular interest: its edges, faces and hyperfaces are all congruent, and it features both null hyperplanes and null segments. A tiling of spacetime with copies of this polytope can be viewed alternatively as a lattice with null edges, such that each point is at the intersection of four light rays in a tetrahedral pattern. This construct may have relevance for discretizations of curved spacetime and for quantum gravity.

In this paper, the author studies the properties of some special 4-polytopes in spacetime. The main qualitative difference between spacetime and Euclidean space is the existence of null i.e. lightlike directions. So, there exist line segments with vanishing length, plane elements with vanishing area, and hyperplane elements with vanishing volume. 3d null hyperplane elements are especially interesting. In relativistic physics, null hypersurfaces play the role of causal boundaries between spacetime regions. They also function as characteristic surfaces for the differential equations of relativistic field theory.

Important examples of null hypersurfaces include the lightcone of an event and the event horizon of a black hole.

SVG version of http://en.wikipedia.org/wiki/Im...

The prime example of a closed null hypersurface is a causal diamond – the intersection of two light cones originating from two timelike-separated points.


Null Hyperplanes 

Null 3d polyhedra or polyhedra with vanishing volume reside in null hyperplanes, such as the hyperplane t = z. let’s look at the geometry of these hyperplanes. The normal ℓμ to the hyperplane, ℓμ (1, 0, 0, 1) is a null vector, i.e. ℓμμ = 0.  As a result, it is also tangent to the hyperplane. It’s integral lines  form null geodesics. The hyperplane
is  foliated into light rays. All intervals within the hyperplane are spacelike, except the null intervals along the rays.

Null Polyhedra

In 3d Euclidean space, each area element has a normal vector n. When discussing polyhedra, it is convenient to define the norm of n to equal the area of the corresponding face. The orientation of the normals is chosen to be outgoing. Not every set of area normals {ni}
describes the faces of some polyhedron. For this to be true, the normals must sum up to zero:


This can be understood as the requirement that the flux of any constant vector field through the polyhedron vanishes. In loop quantum gravity, this condition encodes the local SO(3) rotation symmetry.


Null tetrahedra

The simplest null polyhedron is a tetrahedron. Up to reflections along the null axis, null tetrahedra come in two distinct types: (1,3) and (2,2). The pairs of numbers denote how many of the tetrahedron’s four faces are past-pointing and future- pointing, respectively.


Null-faced 4-simplices

Null-faced 4-simplices have hyperfaces which have zero volume. A 4-simplex has five tetrahedral hyperfaces, which in this case will be null tetrahedra,

The scalar products  ημν(i)μ(j)νof the null volume normals are directly related to the spacetime volume of the 4-simplex and to the areas of the 2d faces. To express the spacetime volume, we must choose a set of four volume normals ℓ(i)μ. The time-orientation of the
normals should be correlated with the past/future status of their hyperfaces.  Next, we construct a symmetric 4 × 4 matrix  Lij =(3!)²ημν(i)μ(j)ν of their scalar products. The diagonal elements of Lij are zero. Elements corresponding to past- future pairs ij are positive, while those for past-past and future-future pairs are negative.
The spacetime volume can then be found as:


The area of the face at the intersection of the i’th and j’th hyperplanes can be found as:


Can also define the 4-volume directly in terms of triangle areas:



Null parallelepipeds

The six faces of a null parallelepiped are spacelike parallelograms. There are three pairs of opposing faces, such that each pair is parallel and congruent. In a given pair of opposing faces, one is past-pointing, and the other future-pointing.



Tetrahedral diamonds

Beginning with an arbitrary spacelike tetrahedron, situated at t = 0 hyperplane, this is the base tetrahedron. For each of the base tetrahedron’s four faces,  the lightcross of two null hyperplanes orthogonal to it are drawn. The tetrahedral diamond is then defined by the convex hull of the intersections of these null hyperplanes.


The 4-volume of a tetrahedral diamond can be found as twice the volume of a 4-simplex, with the spacelike tetrahedron as its base and the inscribed radius r as its height. The result is:


where V is the base tetrahedron’s volume.


Related posts

State Sums and Geometry by Frank Hellmann

This week I have been reading Frank Hellmann’s PhD thesis on The geometry of state sums. Here I’ll review the section on the geometric states arising in the representation theory of SU(2), Spin(4) and SL(2,C).


The Geometry of Representations

We scan construct a state sum invariant using the representation theory of SU(2). A similar construction can be given for Spin(4),
the covering group of SO(4), and SL(2,C), the covering group of SO(3, 1)+ the identity connected component of SO(3, 1).

The Geometry of SU(2)

One way to the representation theory of SU(2) is by the quantization of the sphere

Coherent States

Coherent states are defined as the eigenstates of Lie algebra elements. Given a 3-dimensional unit vector n ∈ S² the associated coherent states αj(n) are defined by:


This fixes the states j(n) up to a phase. Every state in the fundamental representation is proportional to a coherent state.

Every normalized state α’ is a coherent state α’(n(α’)).
In particular we have that


The anti-linear map J transforms a coherent state to one associated
to the opposite direction:


The coherent states satisfy an exponential property for every αj(n) there is an α(n) in the fundamental representation such that


Lie algebra elements transform under the vector representation of SU(2), their transformation behaviour under rotations that stabilize n is,


The modulus square of the inner product between two coherent states in arbitrary representations is


This shows that in the large quantum number limit j → ∞ coherent states become orthogonal.

A resolution of the identity is given by integrating over
the sphere:


Coherent states give the geometry of a sphere associated to particular representations j. Extending this we can give a geometric interpretation to the invariant subspace of the tensor product of representations Inv(j1⊗j2. . .⊗jn)

Next we look at three-valent intertwiners which are associated to triangles and  four-valent intertwiners and their relationship with tetrahedra.


Coherent Triangles

The shape space of triangles can be described as a constrained space of three spheres of radius ji. This space has the product symplectic structure of that of the sphere used in  geometric quantisation. Given three vectors jana of length ja that are on the spheres the constraint

stateequ2.8 forces them to describe the edge vectors of a triangle with edge lengths ja.

To quantize this state space we take the quantized unconstrained state space


 Coherent Tetrahedra

Just as three-valent intertwiners can be interpreted as quantized triangles we can interpret four-valent intertwiners as quantized tetrahedra.

Consider a set of four vectors in directions na of lengths ja satisfying closure


The space of such closing vectors that are non-degenerate is the shape space of non-degenerate tetrahedra.

 Tetrahedra from Closing Vectors

Four vectors jana of length ja that span 3-dimensional space and satisfy stateequ2.10 are the outward normals of a non-degenerate tetrahedron with areas  ja embedded in R³ which is unique up to translation.

Again the constraint generates rotations of the tetrahedron and the non-degenerate sector of the reduced phase space is the shape space of tetrahedra. Implementing the constraint quantum mechanically again we obtain the over-parametrisation of the space of four-valent intertwiners by:


The Geometry of Spin(4)

For the 4-dimensional models need to understand the geometry of the representation theory of Spin(4). To do so we will consider the
Lie algebra spin(4) which is isomorphic to so(4), the Lie algebra of 4-dimensional rotations. We can understand this Lie algebra as arising from bivectors.  It decomposes into a left and right sector under the action of the Hodge star. This allows us to give the representation theory of Spin(4) in terms of SU(2) representations and define coherent bivectors. We will look at a necessary and sufficient set of conditions for a set of bivectors to define a geometric 4-simplex σ4.

Bivectors in Rare elements of Λ²(R4), that is, 2-dimensional antisymmetric tensors BIJ = −BJI , I, J = 0, . . . , 3.  Λ²(R4) is 6-dimensional.

We define the norm of a bivector by |B|² = ½BIJ BIJ . The Lie algebra so(4) is the algebra of antisymmetric matrices with Lie product
given by the commutator. The Lie algebra structure constants are:


 Geometric Bivectors

We call a set of bivectors geometric if they are the bivectors of the faces of a geometric 4-simplex σ4i in R4. Using a, b = 1, . . . , 5 to denote the tetrahedral faces σ3i of the 4-simplex we denote its outward facing normal vectors by Na. The bivectors can then be written as



The associated to the triangles of a non-degenerate geometric 4 simplex satisfy the geometricity conditions:


The Geometry of SL(2,C)

In order to work with theories with Lorentz symmetry we need to consider representations of SO(3, 1). More specifically the identity connected component SO(3, 1)+ and its double cover SL(2,C). That is, the part of SO(3, 1) that takes future pointing normals to future pointing normals.

The corresponding Lorentzian generators are then


the structure constants are given by the same calculation as in the spin(4) case:








Polyhedral quantum geometry

This week I’ve been reading Spinfoams: Simplicity Constraints and Correlation Functions PhD thesis by Ding. I’m reviewing the  section on Polyhedral quantum geometry  which is relevant to my work on the quantum tetrahedron.

Consider the truncation of the LQG Hilbert space HLQG and restrict ourself to a single graph Hilbert space H(Γ) and decompose it in terms of SU(2)-invariant spaces Hn associated to each node n. Here I’ll briefly review the state that this node space Hn is the quantization of the space of shapes of the geometry of solids figures tetrahedra, or more general polyhedra . See the posts:

Let’s start with the classical phase space of shapes of a flat polyhedron in Rwith fixed area. A classically flat three-dimensional polyhedron can be described by a set of L vectors Al, l = 1…L, satisfying the following closure constraint:


Here the L vectors Al can be interpreted as the vectorial areas of the L triangles in the boundary of the polyhedron, in the sense that the norm al = |Al| is the area of the polygon l and normalized vector nl = Al/|Al| is the normal when embedded in to a R3 Euclidean space.
To introduce a symplectic structure, one can associate to each normal Aia generator of the algebra of SO(3).

ding132 A quantum representation of this Poisson algebra is precisely defined by the generators of SU(2) on the space Hn for a 4-valent node n. The operator corresponding to the area al = |Al| is the Casimir of the representation jl, therefore the space quantizes
the space of the shapes of the tetrahedron with areas jl(jl +1). Furthermore, the Hamiltonian flow of G, generates the rotations of the tetrahedron in R3.

By imposing


and factoring out the orbits of this flow, one obtains the intertwiner space Kn.

In this way, one gives an intertwiner a geometrical interpretation in terms of quantum polyhedron.

There is a  relation among spinfoam formalism, kinematical Hilbert space and polyhedral quantum geometry. For example the boundary space of the simplicial EPRL spinfoam model can be obtained from simplicity constraints, which is the simplicial truncation of LQG kinematical Hilbert space and the boundary state has a geometrical interpretation in terms of quantum tetrahedron geometry. This consistent picture can be generalized into an arbitrary-valence spinfoam formalism. It is also possible to compute the two-point correlation function of Lorentian EPRL spinfoam model and show it matches the one from Regge geometry.

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Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint by Alesci, Thiemann, and Zipfel

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 fi nd 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  de fines  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 satis ed in Hphys the physical scalar product must obey


for all ψout, ψn ∈ Hkin.

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 Hn is the Hamiltonian constraint acting on the node n.

 Hamiltonian constraint

The classical Hamiltonian constraint is





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



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-de fined operator on Hkin. 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 de fine 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.


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 di ffeomorphism invariant states is independent from the refi nement 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 Hmdeltaon 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  Hmdelta[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 simpli ed to;



Using  the de finition 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 de finition of the model:

Consider an oriented two-complex  de fined 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 Vint. The boundary ∂κ is the union of all external vertices (-nodes) n ∉ Vint and external edges (links) l ∉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 de fines 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


defi nes 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 cc.

Spin-foam projector

Given any couple of ingoing and outgoing kinematical states ψout, ψin, the Physical scalar product can be
formally de fined 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 fi nally implies;





Compute new solutions to the Euclidean Hamiltonian constraint by employing spin-foam methods. Show thatlinkingequ3.15


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


The simplest two-complex κ(Θ,s) with only one internal vertex de fining 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 fi rst 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




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 defi nition 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 statessphysare 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| Hm|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 defi ne 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.


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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A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements by Gaul and Rovelli

This week I have been studying a paper which presents a generalized version of the Hamiltonian constraint operator in nonperturbative loop quantum gravity.

I have been continuing my collaborative work on the matrix elements of the Hamiltonian Constraint in LQG, as seen the posts:

The generalization is based on admitting arbitrary irreducible SU(2) representations in the regularization of the operator, in contrast to the original definition where only the fundamental representation is taken.

This leads to a quantization ambiguity and to a family of operators with the same classical limit. The authors calculate the action of the Euclidean part of the generalized Hamiltonian constraint on trivalent states, using the graphical notation of Temperley-Lieb recoupling theory.


Loop quantum gravity is a canonical approach to the quantization of general relativity. One of the issues in this approach is the identification of the physically correct . Hamiltonian constraint operator (HCO), encoding the dynamics of classical general relativity. In this paper, the authors analyze one of the quantization ambiguities entering the definition of the HCO, and a corresponding variant of Thiemann’s HCO. In particular, they study a family of operators Hm labelled by irreducible SU(2) representations  m ( = 2j), all having the same classical limit, namely the classical Hamiltonian constraint of general relativity. Thiemann’s HCO corresponds to the fundamental representation m = 1. In a nutshell, the HCO requires a gauge invariant point-splitting-like regularization, which is obtained by using the trace of the holonomy of the gravitational connection. Theimann’s  HCO  operator H¹ adds a link of colour 1 to the nodes of the spin network states. The operators Hm  for arbitrary m act on the spin network states by adding a link of color m.

The generalized Hamiltonian

Classical Theory

The construction of Thiemann’s HCO starts from the classical Lorentzian Hamiltonian constraint C of density weight one. Using real Ashtekar-Barbero variables, this can be written as: 1106equ1 Here a, b are tensorial indices on the compact spatial manifold ∑ , and i, j, k are su(2) indices. Square brackets denote antisymmetrization. The inverse densitized triad (an su(2) valued vector density) has components defined by 1106equ1a, where eai is the triad on ∑, and the real SU(2) Ashtekar-Barbero connection is 1106equ1b On  ∑ we have the induced  metric qab, whose inverse satisfies 1106equ1c , as well as the extrinsic curvature kab. Using the triad, one obtains kia  by transforming one spatial index into an internal one. lambdaiais the spin connection compatible with the triad. The AE form a canonically conjugate pair, whose fundamental Poisson brackets arevariables AEpoission, G being G with Newtons constant Gn. Finally, the components of the curvature of the connection are given by Fabk.

The Hamiltonian constraint can then be written as


A more convenient starting point for the quantization is given
by the polynomial expression for the densitized Hamiltonian constraint



where V is the volume of  ∑,


K is the integrated trace of the densitized extrinsic curvature of ∑ ,


Denote the Euclidean Hamiltonian constraint with H, and the Lorentzian term with T.

Using the relations,




because the integrated extrinsic curvature is the time derivative of the volume, i.e. it can be written as the Poisson bracket of volume and (Euclidean) Hamiltonian constraint at lapse equal to one – the Hamiltonian constraint can be entirely expressed in terms of the volume and the connection.

In this paper the authors restrict themselves to the study of the Euclidean constraint. Its smeared form it is,


where N(x) is the lapse function.

The regularization of H[N] is obtained by approximating Fab and Ac with holonomies of the connection around small loops.

Fix an arbitrary triangulation T of the manifold ∑ into elementary tetrahedra with analytic edges. Consider a tetrahedron , and a vertex v of this tetrahedron.

1106efig1Decompose the smeared Euclidean constraint  into a sum of one term per each tetrahedron of the triangulation,


Finally, consider the holonomy he of the connection along edges e, and  define the classical regularized Euclidean Hamiltonian constraint as




A straightforward calculation, using the expansions,




shows that for a fixed value of the connection and triad, the expression converges to the Hamiltonian constraint.  Write this as,


The expression 1106equ12 can immediately be transformed into a quantum operator, yielding the regularized HCO.


The authors now introduce an alternative regularization. Given an irreducible representation of spin j and colour m = 2j, can write,1106equ17

where R(m) is the matrix representing U in the representation m. Replacing the trace Tr with the trace Trm in the regularization of the constraint, gives,


using again the expansions of the holonomy,


and using


then if,




converges to the Hamiltonian constraint for a sufficiently fine triangulation, that is,


Quantum Theory

The HCO operator is defined by adapting the triangulation T to the
graph γ of the basis state ψ on which the operator is going to act,

By replacing the classical quantities with quantum operators, and the Poisson brackets with commutators, can  obtain the operator associated to a single tetrahedron:


Further  manipulations  give the generalized HCO as,


p is one, whenever  is a tetrahedron having three edges coinciding with three edges of the spin network state, that meet at the
vertex v. In the other cases p equals zero.

The action ofHm on trivalent vertices

In the following the authors calculate explicitly the action of generalized Euclidean Hamiltonian constraint operator on trivalent vertices.

Calculations are performed in the spin network basis. Graphical techniques –  Penrose’ graphical binor calculus are introduced in the connection representation, for example to represent spin network states or operators. This method is equivalent to the graphical description in the loop representation, which satisfies the basic axioms of the tangle-theoretic ormulation of Temperley-Lieb recoupling theory.


The Hamiltonian constraint operator Hmacts independently on
single vertices . Look at its  action on a single trivalent vertex for an arbitrary but fixed color m ,


The trivalent vertex is denoted by |v(p, q, r)i ≡ |vi,
where  p, q and r are the colours of the adjacent edges ei, ei, ek.

Proceed by applying the operators  successively, performing the summation over i, j, k at the end.

The operator hsattaches an open colour-m loop segment to the edge ek, creating a new vertex on it, and altering the colour between the two vertices, i.e.


The action of  V

In the next step, the volume operator acts on the state hs. In the generalization of the HCO we adopt the Ashtekar-Lewandowski volume operator. Its action on function ψ, it is given by,


The first sum extends over the set V(γ) of vertices of the underlying graph, while the sum in  extends over all triples (eI , eJ , eK) of edges adjacent to a vertex.

The essential part of the volume operator  is given by W [ijk] – the ‘square’ of the volume. Its action is described in terms of the grasping of any three distinct edges eI , eJ and eK adjacent to vn.


Regarding the computations of the matrix elements for the generalized HCO, the authors obtain,


The action of W on a non-gauge-invariant 3-vertex can generally be expressed as,




This basis is realized by a rescaling, or vertex normalization respectively. The virtual internal edge is multiplied by √Δ, and each of the two virtual nodes is divided by an appropriate √ Δ , giving


With this normalization, we have,


Evaluating this by using the grasping operation and closing
the open network with itself, gives,


The closed network in this expression is simplified by applying the reduction formula to the upper right three triangle-like vertices,               {(β, c,m), (c, 2, c), (α, c,m)}, reducing it to a 9j−symbol.

Evaluating further until only fundamental, or ‘minimal’ closed networks remain, gives


Defining t = (β+α)/2 and e = (β − α)/2 = ±1,
the non-zero matrix elements are


Taking the absolute value and the square root, the double base transformation needs to be reverted to return to the original
basis in which the whole calculation is performed. In all, this is written as


or explicitly in terms of the matrix elements,


The action of the volume operator is in general not diagonal. Unfortunately, the complexity of the problem for arbitrary m and colourings  of the vertex, prevents an  explicit calculation  . Nevertheless, as soon as specific colouring  of a vertex are chosen, the complete calculation can be performed. It is
just the general expression that is lacking at present.

The relation between the vertex operator ˆV and the
square root of the local grasp √i W reads in the trivalent case

using this we obtain for the non-diagonal action of the volume operator,


Completing the action of Hm

To finish the computation of the action of the generalized Euclidean HCO HmΔ , the relevant part of the operator




Performing this computation diagrammatically,gives



Calculating the trace part of the generalized HCO we  get:





The full action of the generalized HCO on |v(p, q, r)>  is given (up to constant factors) by contracting the trace part with εijk. This leads in the complete action of Hmdelta to a sum of three terms. They are distinguished from each other by the assignment of colour-m segments between mutually distinct pairs of edges adjacent to the vertex. The corresponding amplitudes are determined  by cyclic permutations of argument pairs. For a
generic 3-vertex |v(p, q, r)>,


The compact final form for the action of the generalized local HCO Hmdelta  on a 3-valent vertex is,





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