Spin-Foams for All Loop Quantum Gravity by Kaminski ,Kisielowski and Lewandowski

This week I return more directly to work connected to the Hamiltonian constraint operator and look at an important paper on the spin-foam formalism.

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.

Abstract spin-networks

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.

Simple intertwiners

There are three main proposals for the simple intertwiners:

  1.  That of Barrett-Crane (BC) corresponding to the Palatini action
  2. That of Engle-Pereira-Rovelli-Livine (EPRL) corresponding the Holst action with the value of the Barbero-Immirzi parameter γ= ±1
  3. 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.

Related articles

Learning about Quantum Gravity with a Couple of Nodes by Boria, Garay and Vidotto

This week I have looking at  paper about Quantum gravity on a graph with just two nodes. This has great technical information on performing quantum cosmological calculations which I’ll review in another post. In this post I just want to give a brief qualitative outline of the paper.

Loop Quantum Gravity provides a natural truncation of the infinite degrees of freedom of gravity by setting the theory on a finite graph. In this paper the authors review this procedure  and  present the construction of the canonical theory on a simple graph, formed by only two nodes.

They  review the U(N) framework, which provides a powerful tool for the canonical study of this model, and a formulation of the system based on spinors. They also consider  the covariant theory, which allows the  derivation of the  the model from a more complex formulation.

Why graphs?

Discrete gravity

The essential idea behind the graph truncation can be traced to
Regge calculus, which is based on the idea of approximating spacetime with a triangulation, where the metric is everywhere flat except on the triangles. On a fixed spacelike surface, Regge
calculus induces a discrete 3-geometry defined on a 3d triangulation, where the metric is everywhere flat except on the bones. The two-skeleton of the dual of this 3d cellular decomposition is a graph Γ, obtained by taking a point -a node- inside each cell, and connecting it to the node in an adjacent cell by a link, puncturing the triangle shared by the two cells.

Spin networks

In Loop Quantum Gravity, the spinnetwork basis |Γ, j, ni 〉 is an orthonormal basis that diagonalizes the area and volume operators. The states in this basis are labelled by a graph Γand two quantum numbers coloring it: a spin j at each link  and a volume eigenvalue  at each node n. The  Hilbert space HΓ obtained by considering only the states on the  graph Γis precisely the Hilbert space of an SU(2) Yang-Mills theory on this lattice. Penrose’s spin-geometry theorem connects this Hilbert space with the description of the geometry of the cellular decomposition: states in this Hilbert space admit a geometrical interpretation as a quantum version of the 3-geometry. That is, a Regge 3-geometry defined on a triangulation with dual graph Γ can be approximated by semiclassical state in HΓ.


In the canonical quantization of General Relativity, in order to implement Dirac quantization, it’s convenient to choose the densitized inverse triad Ea –Ashtekar‘s electric field and the Ashtekar-Barbero connection Ai as conjugate variables, and then use the flux of Ei and the the holonomy h as fundamental variables for the quantization. The holonomies can be taken along the links of the graph, and the densitized inverse triad can be smeared over the faces of the triangulation. This connects the holonomy triad variables to the discrete geometry picture.

The common point of these different derivations is 3d coordinate gauge invariance. This invariance is the reason for the use of abstract graphs: it removes the physical meaning of the location of the graph on the manifold. Therefore the graph is just a combinatorial object, that codes the adjacency of the nodes. Each node describe a quantum of space, and the graph describes the relations between different pieces of space. The Hilbert subspaces associated to distinct but topologically equivalent embedded graphs are identified, and each graph space contains the Hilbert spaces of all the subgraphs.


Doing physics with few nodes

The restriction to a fixed triangulation or a fixed graph amounts only to a truncation of the theory, by cutting down the theory to an approximate theory with a finite number of degrees of freedom. Truncations are always needed in quantum field theory, in order to extract numbers from the theory -in appropriate physical
regimes even a low-order approximation can be effective.

Discretizing a continuous geometry by a given graph is nothing but coarse graning the theory. The discreteness introduced by this process is different from the fundamental quantum discreteness
of the theory. The first is the discreteness of the abstract graphs; the later is the discreteness of the spectra of the area and volume operator on each given HΓ.

There are also  two different expansions:

  • The graph expansion obtained by a refinement of the graph, valid at scales smaller that the curvature scale R.
  • The semiclassical expansion,  a large-distance limit on each graph, valid at scales larger that the Planck scale Lp


Interesting physics can arise even by considering a simple graph, with few nodes, and comparing our results with classical discrete gravity. This is true for FLRW cosmologies. It has in fact been proven numerically  that the dynamics of a closed universe, with homogeneous and isotropic geometry, can be captured by 5, 16 and 600 nodes –  the regular triangulations of a 3-sphere.


 On the left, a 4d building block of spacetime and, on the right, the evolution of 5, 16 and 500 of these building block (dashed lines), modelling a closed universe, compared whit the continuous analytic
solution (solid line).

The cosmological interpretation

The most striking example, where this kind of approximation applies, is given by cosmology itself. Modern cosmology is based on the cosmological principle, that says that the dynamics of a homogeneous and isotropic space approximates well our universe. The presence of inhomogeneities can be disregarded at a first order approximation, where we consider the dynamics as described at the scale of the scale factor, namely the size of the universe.

Working with a graph corresponds to choosing how many degrees of freedom to describe. A graph with a single degree of freedom is just one node: in a certain sense, this is the case of usual Loop Quantum Cosmology . To add degrees of freedom, we add nodes and links with a colouring. These further degrees of freedom are a natural way to
describe inhomogeneities and anisotropies , present in our universe.

The easiest thing that can be done is to pass from n = 1 to n = 2 nodes. We choose to connect them by L = 4 links, because in this way the dual graph will be two tetrahedra glued together, and this can be viewed as the triangulation of a 3-sphere .


Choosing a graph corresponding to a triangulation is useful when we want to associate an intuitive interpretation to our model. In order to understand how this can be concretely use to do quantum gravity and quantum cosmology, we need to place on the graph the SU(2) variables.

LQG can describe large semiclassical geometries also over a small number of nodes and links. This paper  reviewed a number of constructions in Loop Quantum Gravity, based on the idea of truncating the Hilbert space of the theory down to the states supported on a simple graph with two nodes.


Below there is a diagram of the solutions found using a 2-node graph. The quantum Hamiltonian of the 2-node model is mathematically analogous to the gravitational part of the Hamiltonian in LQC. Following this analogy, we can interpret the results obtained as the classical analogous of the quantum big bounce found in LQC.


The restriction of the full LQG Hilbert space to a simple graph is a truncation of the degrees of freedom of the full theory. It defines an approximation where concrete calculations can be performed. The approximation is viable in physical situations where only a small number of the degrees of freedom of General Relativity are relevant.

A characteristic example is cosmology. The 2-node graphs with 4 links defines the simplest triangulation of a 3-sphere and can accommodate the anisotropic degrees of freedom of a Bianchi IX model, plus some inhomogeneous degrees of freedom.  In this context, a Bohr-Oppenheimer approximation provides a tool to separate heavy and light degrees of freedom, and extract the
FLRW dynamics. This way of deriving quantum cosmology from LQG is different from the usual one: in standard loop quantum cosmology, the strategy is to start from a symmetry-reduced system, and quantize the single or the few degrees of freedom that survive in the symmetry reduction. In this paper  a truncated version of the full quantum theory of gravity is consided.

Semiclassical analysis of Loop Quantum Gravity by Claudio Perini

This week I have been reading the PhD thesis Semiclassical analysis of Loop Quantum Gravity by Claudio Perini .

Semiclassical states for quantum gravity

The concept of semiclassical state of geometry is a key ingredient in the semiclassical analysis of LQG. Semiclassical states are kinematical states peaked on a prescribed intrinsic and extrinsic geometry of space. The simplest semiclassical geometry one can consider is the one associated to a single node of a spin-network with given spin labels. The node is labeled by an intertwiner, i.e. an invariant tensor in the tensor product of the representations meeting at the node. However a generic intertwiner does not admit a semiclassical interpretation because expectation values of non-commuting geometric operators acting on the node do not give the correct classical result in the large spin limit. For example, the 4-valent intertwiners defined with the virtual spin do not have the right semiclassical behavior; one has to take a superposition of them with a specific weight in order to construct semiclassical intertwiners.

The Rovelli-Speziale quantum tetrahedron  is an example of semiclassical geometry; there the weight in the linear superposition of virtual links is taken as a Gaussian with phase. The Rovelli-Speziale quantum tetrahedron is actually equivalent to the Livine-Speziale coherent intertwiner with valence 4 more precisely, the former constitutes the asymptotic expansion of the latter for large spins. Coherent intertwiners are as the geometric quantization of the classical phase space associated to the degrees of freedom of a tetrahedron.

In  recent graviton propagator calculations of semiclassical states
associated to a spin-network graph Γ have been  considered. The states used in the definition of semiclassical n-point functions ) are labeled by a spin jo and an angle ξe per link e of the
graph, and for each node a set of unit vectors n, one for each link surrounding that node. Such variables are suggested by the simplicial interpretation of these states: the graph Γ is in fact assumed to be dual to a simplicial decomposition of the spatial manifold, the vectors n are associated to unit-normals to faces of tetrahedra, and the spin jo is the average of the area of a face. Moreover, the simplicial extrinsic curvature is an angle associated to faces shared by tetrahedra and is identified with the label ξe. Therefore, these
states are labeled by an intrinsic and extrinsic simplicial 3-geometry. They are obtained via a superposition over spins of spin-networks having nodes labeled by Livine-Speziale coherent intertwiners.
The coefficients cj of the superposition over spins are given by a Gaussian times a phase as originally proposed by Rovelli.


Such proposal is motivated by the need of having a state peaked both on the area and on the extrinsic angle. The dispersion is chosen to be given by


so that, in the large jo limit, both variables have vanishing relative dispersions. Moreover, a recent result of Freidel and Speziale strengthens the status of these classical labels: they show that the
phase space associated to a graph in LQG can actually be described in terms of the labels (jo , ξe, ne, n′) associated to links of the graph. The states have good semiclassical properties and a clear geometrical

Within the canonical framework, Thiemann and collaborators have strongly advocated the use of complexifier coherent states . Such states are labeled by a graph Γ and by an assignment of a SL(2,C) group element to each of its links. The state is obtained from the gauge-invariant projection of a product over links of modified heat-kernels for the complexification of SU(2). Their peakedness properties have been studied in detail.

Perini’s thesis presents the  proposal of coherent spin-network states: the proposal is to consider the gauge invariant projection of
a product over links of Hall’s heat-kernels for the cotangent bundle of SU(2). The labels of the state are the ones used in Spin Foams: two normals, a spin and an angle for each link of the graph. This set of labels can be written as an element of SL(2,C) per link of the graph. Therefore, these states coincide with Thiemann’s coherent states with the area operator chosen as complexifier, the SL(2,C) labels written in terms of the phase space variables (jo , ξe, ne, n′e) and the heat-kernel time given as a function of jo.

The author shows that, for large jo , coherent spin-networks reduce to the semiclassical states used in the spinfoam framework. In particular that they reproduce a superposition over spins of spin-networks with nodes labeled by Livine-Speziale coherent intertwiners and coefficients cj given by a Gaussian times a phase as originally proposed by Rovelli. This provides a clear interpretation of the geometry these states are peaked on.

Livine-Speziale coherent intertwiners
In ordinary Quantum Mechanics, SU(2) coherent states are defined as the states that minimize the dispersion


of the angular momentum operator J, acting as a generator of rotations on the representation space Hj ≃C2j+1 of the spin j representation of SU(2). On the usual basis |j,m> formed by simultaneous eigenstates of J2 and J3 we have


so the maximal and minimal weight vectors |j,±j> are coherent states. Starting from |j, j〉, the whole set of coherent states is constructed through the group action


One can take a subset of them labelled by unit vectors on the sphere S²:


where n is a unit vector defining a direction on the sphere S² and g(n) a SU(2) group element rotating the direction z ≡ (0, 0, 1) into the direction n. In other words, a coherent states is a state satisfying


For each n there is a U(1) family of coherent states and they are related one another by a phase factor. The choice of this arbitrary phase is equivalent to a section of the Hopf fiber bundles :

S2 ≃ SU(2)/U(1) → SU(2).

Explicitly, denoting n = (sin θ cos φ, sin θ sin φ, cos θ), a possible section is


where m ≡ (sin φ,−cos φ, 0) is a unit vector orthogonal both to z andn. A coherent state can be expanded in the usual basis as



Coherent states are normalized but not orthogonal, and their scalar product is


where A is the area of the geodesic triangle on the sphere S² with vertices z, n and n. Furthermore they provide an overcomplete basis for the Hilbert space Hj of the spin j irreducible representation of SU(2), and the resolution of the identity can be written as


with d²n the normalized Lebesgue measure on the sphere S².

The Livine-Speziale coherent intertwiners are naturally defined taking the tensor product of V coherent states (V stands for valence of the node) and projecting onto the gauge-invariant subspace:


Here the projection is implemented by group averaging. They are labeled by V spins j and V unit vectors n. The states |j, n〉o carry enough information to describe a classical geometry associated to the node. Interpret the vectors j,n as normal vectors to triangles, normalized to the areas j of the triangles.

 Coherent tetrahedron

The case of 4-valent coherent intertwiners is of particular importance for LQG and especially for Spin Foam Models. In fact it is the lowest valence carrying a non-zero volume and most SFM’s are build over a simplicial triangulation, so that the boundary state space has only 4-valent nodes, dual to tetrahedra. A 4-valent coherent intertwiner with normals satisfying the closure condition can be interpreted as a semiclassical tetrahedron. In fact expectation values of geometric operators associated to a node give the correct classical quantities in the semiclassical regime. This regime is identified with the large spin (large areas) asymptotics. In the following we give some details.

Define n = jn as the normals normalized to the area. In terms of them, the volume (squared) of the tetrahedron is given by the simple relation:


The geometric quantization of these degrees of freedom is based on the identification of generators Ji of SU(2) as quantum operators corresponding to the  ni . This construction gives directly
the same quantum geometry that one finds via a much longer path by quantising the phase space of General Relativity, that is via Loop Quantum Gravity. The squared lengths |ni|² are the SU(2) Casimir
operators C²(j), as in LQG. A quantum tetrahedron with fixed areas lives in the tensor product ⊗Hj .The closure constraint reads:


and imposes that the state of the quantum tetrahedron is invariant under global rotations. The state space of the quantum tetrahedron with given areas is thus the Hilbert space of intertwiners


The operators Ji · Jj are well defined on this space, and so is the operator


Its absolute value |U| can immediately be identified with the quantization of the classical squared volume 36V²,  in agreement with standard LQG results.

To find the angle operators,  introduce the quantities  Jij := Ji+  Jj. Given these quantities, the angle operators θij can be recovered from


The quantum geometry of a tetrahedron is encoded in the operators Ji ², Jij² and U, acting on Ij₁…j. It is a fact that out of the six independent classical variables parametrizing a tetrahedron, only
five commute in the quantum theory. Indeed while we have

[Jk², Ji · Jj] = 0,

it is easy to see that:


A complete set of commuting operators, in the sense of Dirac, is given by the operators Ji² , J². In other words, a basis for Ij…j4 is provided by the eigenvectors of any one of the operators Jij² . We write the corresponding eigenbasis as |j〉ij . These are virtual links here we are introducing them via a geometric quantization of the classical tetrahedron. For instance, the basis |j〉 diagonalises the four triangle areas and the dihedral angle θ or, equivalently, the area A of one internal parallelogram. The relation between different basis is  obtained from SU(2) recoupling theory: the matrix describing the change of basis in the space of intertwiners is given by the usual Wigner 6j-symbol,


so that


Notice that from the orthogonality relation of the 6j-symbol,


we have


The states |j〉 are eigenvectors of the five commuting geometrical operators Ji² , J²    so the average value of the operator corresponding to the sixth classical observable, say J² is on these states maximally spread. This means that a basis state has undetermined classical geometry or, in other words, is not an eigenstate of the geometry.  Then led to consider superpositions of states to be able to study the semiclassical limit of the geometry. Suitable superpositions could be constructed for instance requiring that they minimise the uncertainty relations between non–commuting observables, such as


States minimising the uncertainty above are usually called coherent states. Coherent intertwiners seem not to verify exactly semi290, but they are such that all relative uncertainties h〈Δ²〉Jij/〈J²ij〉, or equivalently 〈Δθij〉/〈θij〉, vanish in the large scale limit. The limit is defined by taking the limit when all spins involved go uniformly to infinity, namely ji = λki with λ → ∞. Because of these good semiclassical properties we can associate to a coherent intertwiner the geometrical interpretation of a semiclassical tetrahedron; an analogous interpretation should be also valid nodes of higher valence.

 Related articles

The Geometry of a four-simplex by Wolfgang Wieland

This post looks at a small part of Wolfgang Wieland’s PhD thesis about the geometry of a four-simplex. In this the author  finds  that the Gauss law and the linear simplicity constraints together imply its geometricity. Geometricity  means that we can introduce a metric compatible with the fluxes, and speak  about the unique length of the bones bounding the triangles of the four simplex. This is the reconstruction, or shape-matching problem.



Lets define some terminology, and see what  a four-simplex actually looks like . Given its corners wieequ1 we can view the four-simplex as the set of points


So  5×4 = 20 numbers determining a four-simplex. If we identify any two four-simplices related by a Poincaré transformation, there are only ten of them left: Ten numbers define a four-simplex up to Poincaré transformations. A four-simplex contains several
sub-simplices: There are the corners, bones, triangles and tetrahedra. The bones are the geodesic lines connecting any two of the corners, they are given by the vectors:



Two bones b(kl) and b(lm) span an oriented triangle τij : If (ijklm) is an even permutation of (12345),  the pair (b(kl); b(lm)) have positive orientation in τij . The triangles τij , τik, τil, and τim bound a tetrahedron, we call it Ti, and vi is its volume, while nμ(i) denotes its normal. In the following we restrict ourselves to the case where nμ(i) is a future oriented timelike vector, i.e. nμ(i)nν(i) = -1, n0(i) > 0.

Assume the four-simplex be non-degenerate, which means
that its four-volume V should not vanish. This is the condition that:


Because of their prominent role in loop quantum gravity, we are most interested in the fluxes through the triangles. We compute them as the integrals:


Stoke’s theorem implies for any tetrahedron the fluxes sum up to zero:


This is nothing but Gauss’s law. If n (i) denotes the normal of the i-th tetrahedron we also have the condition:


which is the linear simplicity constraint that has appeared at several occasions in our theory as a reality condition on phase space.

Reconstruction of a four-simplex from fluxes

For every tetrahedron bounding a four-simplex, the four fluxes sum up to zero and lie perpendicular to the tetrahedron’s normal.  The opposite is also true – given five future oriented normals n (i), that span all of R4, and ten bivectors  subject to both the Gauss law and the linear simplicity constraint, there is a four-simplex Δ, such that the bivectors ∑αβ  (ij) are the fluxes through its triangles and the vectors n (i) are the normals of the tetrahedra. The resulting four-simplex is unique up to rigid translations and inversion at the origin.

The fluxes have the following form:


where a(ij) > 0 is the area of the triangle and Ξji = Ξij ∈ R is the rapidity between the two adjacent normals, i.e.:


Introduce some simplifying abbreviations. First of all we define the three dimensional volume elements and  the three-metric:


Now write the fluxes in the i-th tetrahedron as the spatial pseudo vectors:


The three-dimensional volume vi > 0 of the i-th tetrahedron is the wedge product of three bounding triangles:


This brings the volume formula into the form:


A similar formula exists for the four-volume V of the polytope, it is:


Substitute for the fluxes and  the volume formula turns into:


Combining the formulae for the three- and four-volume  get the following expression for the rapidity:


This implies for the ratio between any two volumina that:


Have defined the three-volume of the tetrahedra


and the four volume:


Now construct the underlying four-simplex. Substitute the  solution of the simplicity constraints into the Gauss’ law and get:



The last identity is the same as :


Now use assumption that any quadruple of normals be linearly independent in R4 – the non-degeneracy of the geometry), then all i must be proportional to the three-volume vi through a universal sign:


We can then also find a set of numbers  εi∈{-1,1} such that:


The numbers εi are unique up to a global sign. Equation (3.218) now have:


This condition is the four-dimensional analogue of Gauss’s law: wieequ205

The Minkowski theorem in four dimensions guarantees that there is a
corresponding four-simplex consisting of five tetrahedra with normals n (i) and volume vi. This four-simplex is unique up to rigid translations and reflection at the origin.

Now compute the fluxes through the triangles of the four-simplex.  The first step is to look at the bones. A bone b(il) connects the i-th corner with the l-th. From the perspective of the j-th tetrahedron, this bone follows the intersection of the triangles τjm and τjk.

Inserting the parametrisation of the fluxes, this gives:


Forming a triangle, the bones must close to zero:


This suggests the following ansatz for the bone b(il):


Find the value of  by demanding that their wedge product gives the fluxes through the triangles:


This fixes  to the value  ±3/2³ , and we can eventually write the bones in terms of the fluxes as:


This concludes the reconstruction.


The relation with loop gravity is the following.  Starting from the Plebanski two-form and the connection smeared over triangles and links. We integrate the Plebanski two-form over the triangles in the frame of the tetrahedron, obtaining:


where h(p →Ti) is a Lorentz holonomy mapping any point in the triangle τij towards the centre of the tetrahedron Ti.

Variables attached to different tetrahedra belong to different frames. To compare them, map into a common frame. This is the bulk holonomy from the centre of the four-simplex – the vertex to the centre of the i-th tetrahedron – the node. The resulting variables are in one-to-one correspondence with the fluxes and normals of this supplement, that is:


so we have the Gauss law together with the linear simplicity constraints impose the geometricity of the underlying four-simplex. In particular, the length l(ij) of each bone b(ij) turns into a unique function of the fluxes. This function is needed to explore the relation between loop quantum gravity and classical Regge calculus.


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Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity by Hal Haggard

This week I have been reading a couple of PhD thesis. The first is ‘The Chiral Structure of Loop Quantum Gravity‘ by Wolfgang  Wieland, the other is Hal Haggard’s PhD thesis, ‘Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity‘ .

I’m going to focus in this post on a small section of  Haggard’s work about  the quantization of space.

In loop gravity quantum states are built upon a spin coloured graph. The nodes of this graph represent three dimensional grains
of space and the links of the graph encode the adjacency of these regions. In the semiclassical description of the grains of space – see the post:

these grains are  interpreted as giving rise to convex polyhedra in this limit. This geometrical picture suggests a natural proposal for the quantum volume operator of a grain of space: it should be an operator that corresponds to the volume of the associated classical polyhedron. This section provides a detailed analysis of such an operator, for a single 4-valent node of the graph. The valency of the node determines the number of faces of the polyhedron and so we will be focusing on classical and quantum tetrahedra.

The space of convex polyhedra with fixed face areas has a natural phase space and symplectic structure.  Using this kinematics we can study the classical volume operator and perform a Bohr-Sommerfeld quantization of its spectrum. This quantization
is in good agreement with previous studies of the volume operator spectrum and provides a simplified derivation.

This section of the thesis looks at:

  • The node Hilbert Spaces Kn for general valency.
  • The classical phase space and  its Poisson structure.
  • The volume operator  in loop gravity.
  •  The quantum tetrahedron
  • Phase space for the classical tetrahedron and its Bohr-Sommerfeld quantization.
  • Wavefunctions for the volume operator


Focus on a single node n and its Hilbert space
Kn. The space Kn is defined as the subspace of the tensor
product Hj1⊗ · · ·  ⊗HjF that is invariant under global SU(2) transformations


We call Kn the space of intertwiners. The diagonal action is generated by the operatorJ,


States of Kn are called intertwiners and can be expanded as:


The components i transform as a tensor under SU(2) transformations in such a way that the condition


is satisfied. These are precisely the invariant tensors captured graphically by spin networks.The finite dimensional intertwiner space Kn can be understood as the quantization of a classical phase space.

Begin with Minkowski’s theorem, which states the following: given F vectors Ar ∈ R³ (r = 1, . . . , F) whose sum is zero



then up to rotations and translations there exists a unique convex polyhedron with F faces associated to these vectors. These convex polyhedra are our semiclassical interpretation of the loop gravity grains of space.

The second step  is to associate a classical phase space to these

We call this the space of shapes and denote it P(A1, . . . ,AF ) or more briefly PF.  Interpret the partial sums halequ6

as generators of rotations about the μk = A1 + · · · + Ak+1 axis.This interpretation follows naturally from considering each of the Ar vectors to be a classical angular momentum, so that really Ar ∈ Λr ≅R³.

There is is a natural Poisson structure on each r and this extends to the bracket


With this Poisson bracket the μk do in fact generate rotations about the axis A1 + · · ·Ak+1. This geometrical interpretation suggests a natural conjugate coordinate, namely the angle φk of the rotation. Let k be the angle between the vectors




The pairs (μk,φk) are canonical coordinates for a classical phase space of dimension 2(F −3). Thisis called  the space of shapes and denote it P(A1, . . . ,AF ) or more briefly PF.

In the final step we  show that that quantization of PF is the Hilbert space Kn of an F-valent node n. The constraint




which is precisely the gauge invariance condition. This is where the classical vector model of angular momentum originates.

 Volume operators in loop gravity

Most of the loop gravity research on the volume operator has been done in the context of the original canonical quantization of general relativity.  In Ashtekar’s formulation of classical general relativity the gravitational field is described in terms of the triad variables E. This triad corresponds to a 3-metric h and is called the electric field. The elementary quantum operator that measures the geometry of space
corresponds to the flux of the electric field through a surface S. When such a surface is punctured by a link of the spin network graph Γ the flux can be parallel transported, back along the link, to the node using the second of Ashtekar’s variables, the
Ashtekar-Barbero connection. This results in an SU(2) operator that acts on the intertwiner space Kn at the node n.The parallel transported flux operator at the node is proportional to the generator of SU(2) transformations


where γ is a free parameter of the theory called the Barbero-Immirzi parameter and Pl is the Planck length.

The volume of a region of space R is obtained by regularizing and quantizing the classical expression


using the operators Er. The total volume is obtained by summing the contributions from each node of the spin network graph Γcontained in the region R.

There are different proposals for the volume operator at a node. The operator originally proposed by Rovelli and Smolin is


A second operator introduced by Ashtekar and Lewandowski is


Both the Rovelli-Smolin and the Ashtekar-Lewandowski proposals have classical versions. This results in two distinct functions on phase space:


A third proposal for the volume operator at a node has emerged,
Dona, Bianchi and Speziale suggest the promotion of the classical volume of the polyhedron associated to {Ar} to an operator


In the case of a 4-valent node all three of these proposals agree.

The volume of a quantum tetrahedron

In the case of a node with four links, F = 4, all the proposals for the volume operator discussed above coincide and match the operator introduced by Barbieri for the volume of a quantum tetrahedron.

– see post Quantum tetrahedra and simplicial spin networks  by A.Barbieri

The Hilbert space K4 of a quantum tetrahedron is the intertwiner space of four representations of SU(2),

halequ19Introduce  basis into this Hilbert space using the recoupling channel Hj1 ⊗Hj2 and call these basis states |k>. The basis vectors are defined as


where the tensor ik is defined in terms of the Wigner 3j-symbols as

halequ21The index k ranges from kmin to kmax in integer steps with


The dimension d of the Hilbert space K4 is finite and given by

The states |k> form an orthonormal basis of eigenstates of the operator Er · Es. This operator measures the dihedral angle between the faces r and s of the quantum tetrahedron.
The operator The operator √Er · Er measures the area of the rth face of the quantum tetrahedron and states in K4 are area eigenstates with eigenvalues halequ24a,


The volume operator introduced by Barbieri is


and because of the closure relation


this operator coincides with the Rovelli-Smolin operator for                  α = 2√2/3. The volume operator introduce by Barbieri can be understood as a special case of the volume of a quantum polyhedron.

In order to compute the spectrum of the volume operator, it is useful to introduce the operator Q defined as


It represents the square of the oriented volume. The matrix elements of this operator are  computed in the post:

The eigenstates |q> of the operator Q,


are also eigenstates of the volume. The eigenvalues of the volume are simply given by the square-root of the modulus of q,


The matrix elements of the operator Q in the basis |k> are given by


The function Δ(a, b, c) returns the area of a triangle with sides of length (a, b, c) and is conveniently expressed in terms of Heron’s formula


This can be done numerically and we can  compare the eigenevalues calculated in this manner to the results of the Bohr-Sommerfeld quantization.

There are a number of properties of the spectrum of Q and therefore of V that can be determined analytically.

  • The spectrum of Q is non-degenerate: it contains d distinct real eigenvalues. This is a consequence of the fact that the matrix elements of Q on the basis |k> determine a d × d Hermitian matrix of the formhalequ33

with real coefficients ai.

  • The non-vanishing eigenvalues of Q come in pairs ±q. A vanishing eigenvalue is present only when the dimension d of the intertwiner space is odd.
  • For given spins j1, . . . , j4, the maximum volume eigenvalue can be estimated  using Gershgorin’s circle theorem and wefind that it scales as halequ35a   where jmax is the largest of the four spins jr.
  • The minimum non-vanishing eigenvalue (volume gap)
    scales as halequ35b

Tetrahedral volume on shape space

The starting point for the  Bohr-Sommerfeld analysis is the volume of a tetrahedron as a function on the shape phase space P(A1, . . . ,A4)≡  P4.

The Minkowski theorem guarantees the existence and uniqueness
of a tetrahedron associated to any four vectors Ar, (r = 1, . . . , 4) that satisfy A1 + · · · + A4 = 0. The magnitudes Ar ≡ |Ar|, (i = 1, . . . , 4) are interpreted as the face areas. A condition for the existence of a tetrahedron is that A1 +A2 +A3  ≥ A4, equality giving a flat -zero volume tetrahedron. The space of tetrahedra with four fixed face areas P(A1,A2,A3,A4) ≡ P4 is a sphere.


Consider the classical volume,

halequ37it will be more straightforward to work with the squared classical volume,


Writing the triple product of Q as the determinant of a matrix M = (A1,A2,A3) whose columns are the vectors A1,A2 and A3 and squaring yields,


Bohr-Sommerfeld quantization of tetrahedra

The Bohr-Sommerfeld quantization condition is expressed in
terms of the action I associated to the each of these orbits:




Just as the Bohr-Sommerfeld approximation can be used for finding the eigenvalues of the volume operator. Semiclassical techniques can
also be used to find the volume wavefunctions.






At the Planck scale, a quantum behaviour of the geometry of space is expected. Loop gravity provides a specific realization of this expectation: it predicts a granularity of space with each grain having a quantum behaviour.

Based on semiclassical arguments applied to the simplest model for a grain of space, a Euclidean tetrahedron, and is closely related to Regge’s discretization of gravity and to more recent ideas about
general relativity and quantum geometry. The spectrum has been computed by applying Bohr-Sommerfeld quantization to the volume of a tetrahedron seen as an observable on the phase space of shapes.
There is quantitative agreement of the spectrum calculated here and
the spectrum of the volume in loop gravity. This result lends credibility to the intricate derivation of the volume spectrum in loop gravity, showing that it matches an elementary semiclassical approach.

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Sagemath 20: Numerical work on the expectation values of the Ricci curvature on semi-classical states in the case of a monochromatic 4 – valent node dual to an equilateral tetrahedron.

This week I have been continuing my research on the the expectation values of the Ricci curvature on Carlo-Speziale semi-classical states, for different values of the spins in the case of a simple graph : a monochromatic 4 – valent node dual to an equilateral tetrahedron.

The Rovelli-Speziale semi-classical tetrahedron is a semiclassical quantum state corresponding to the classical geometry of the tetrahedron determined by the areas A1, . . . ,A4 of its faces and
two dihedral angles θ12, θ34 between A1 and A2 respectively A3 and A4.

It is defined as a state in the intertwiner basis  |j12>



with coefficients cj12 such that


in the large scale limit, for all ij. The large scale limit considered here is taken when all spins are large.
The expression of the coefficients cj12 verifying the requirements is:


where joand ko are given real numbers respectively linked to θ12 and θ34 through the following equations:


σj12 is the variance which is appropriately fixed and the phase φ(jo, ko) is the dihedral angle to join an auxiliary tetrahedron related to the asymptotic of the 6j symbol performing the change
of coupling in the intertwiner basis.
For a classical regular tetrahedron, using the expression the integrated classical curvature scales linearly in terms of the length of its hinges because the angles do not change in the equilateral configuration when the length is rescaled, which means that the
integrated classical curvature scales as  a square root function of the area of a face.



In the plot above we can see that the expected values of R on coherent states and semi-classical regular tetrahedra for large spins scales as a square root function of the spin, this matches the semi-classical evolution we expect.

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Semiclassical Mechanics of the Wigner 6j-Symbol by Aquilanti et al

This week I have been continuing my research on the the expectation values of the Ricci curvature on Carlo-Speziale semi-classical states, for different values of the spins in the case of a simple graph : a monochromatic 4 – valent node dual to an equilateral tetrahedron. I will be posting about this later.

As part of this work I am studying a paper on the semiclassical mechanics of the wigner 6j symbol.  In this paper the semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano- Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is given to symplectic reduction, the reduced phase space of the 6j-symbol and the reduction of Poisson bracket expressions for semiclassical amplitudes. 

The Wigner 6j-symbol or Racah W-coefficient)is a central object in angular momentum theory, with many applications in atomic, molecular and nuclear physics. These usually involve the recoupling of three angular momenta, that is, the 6j symbol contains the unitary matrix elements of the transformation connecting the two bases that arise when three angular momenta are added in two different ways.

More recently the 6j- and other 3nj-symbols have found applications
in quantum computing and in algorithms for molecular
scattering calculations , which make use of their connection with discrete orthogonal polynomials.
The 6j-symbol is an example of a spin network, a graphical representation for contractions between tensors that occur in angular momentum theory. The graphical notation has been developed since the ’60s.

The 6j symbol is the simplest, nontrivial, closed spin network -one that represents a rotational invariant. Spin networks are important in lattice QCD and in loop quantum gravity where they provide a gauge-invariant basis for the field. Applications in quantum gravity are described by Rovelli and Smolin , Baez, Carlip , Barrett and Crane, Regge and Williams , Rovelli and Thiemann .

There are three approaches to the evaluation of rotational SU(2)invariants.

  • The Yutsis school of graphical notation
  • The Clebsch-Gordan school of algebraic manipulation
  •  The chromatic evaluation – which grew out of Penrose’s doctoral work on the graphical representation of tensors and is closely related to knot theory. 

The asymptotics of spin networks and especially the 6j-symbol has played an important role in many areas. Aymptotics refer to the asymptotic expansion for the spin network when all j’s are large, equivalent to a semiclassical approximation since large j is equivalent to small h/2π. The asymptotic expression for the 6j-symbol the leading term in the asymptotic series)was first obtained by Ponzano and Regge. In the  same paper those authors gave the first spin foam model for quantum gravity. The formula of Ponzano and Regge is notable  for its high symmetry and the manner in which it is related to the geometry of a tetrahedron in three-dimensional space. It is also remarkable because the phase of the asymptotic  expression is identical to the Einstein-Hilbert action for three-dimensional gravity integrated over a tetrahedron, in Regge’s simplicial approximation to general relativity. The semiclassical limit of the 6j-symbol thus plays a crucial role in simplicial approaches to the quantization of the gravitational field.

 Spin network notation

The 3j-symbol and Wigner intertwiner


Bras, kets and scalar products








An SU(2) intertwiner is a linear map between two vector spaces that commutes with the action of SU(2) on the two spaces


Tensor products and resolution of identity

The outer product of a ket with a bra is represented in spin network language simply by placing the spin networks for the ket and the bra on the same page as shown below






The 2j symbol and intertwiner

|Ki> is  expressed in terms of the Clebsch-Gordan coefficients by


This vector can also be expressed in terms of the 2j-symbol,which is defined in terms of the usual 3j-symbol by


The invariant vector |Ki> can also be written,







Kets to bras





Bras to kets














Raising and lowering indices





Models for the 6j-symbol

For given values of the six j’s, the 6j-symbol is just a number, but to study its semiclassical limit it is useful to write it as a scalar product <B|A>of wave functions in some Hilbert space. This can be done in many different ways, corresponding to different models of the 6j-symbol.

The 12j-model of the 6j-symbol






The 6j-symbol is represented as a product of six copies of a 2j-symbol and four of a 3j-symbol. Using the definition for the
2j-symbol, the result is














The Triangle and Polygon Inequalities


This is equivalent to the triangle inequalities when n = 3. In general, it represents the necessary and sufficient condition that line segments of the given, nonnegative lengths can be fitted together to form a polygon with n sides

The 4j-model of the 6j-symbol





The 6j-symbol is proportional to
a scalar product



The Ponzano-Regge Formula



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