Category Archives: Reviews

The 4d Quantum Tetrahedron

We start from the GFT formulation of 4d gravity. Starting from the classical continuum action which is the basis for most model building in LQG and spin foam models, we have the Holst-Palatini action:

4dtetrafig1 Classically equivalent to this is the  Plebanski-Holst action derived from topological BF theory and simplicity constraints.


Looking at the classical discrete phase space  for a single tetrahedron and the classical tetrahedron in 4d.


Another construction is the EPRL model which is a 4d model for Riemannian Plebanski-Holst gravity  in noncommutative bivector variables. Starting from GFT for 4d BF theory with Bi ∈ so(4) we get:


Where φ(B1,..,B4;N) the non-commutative bivector, flux representation of tetrahedron wavefunction of the classical GFT field, including normal, satisfies the gauge covariance closure condition and the connection h gives parallel transport across frames:

4dtetrafig5At the level of Feynman amplitudes, this gives usual BF simplicial path integral – the spin representation of the usual Ooguri spin foam model with SO(4) intertwiners and 15j-symbols

 4d case (riemannian): quantum tetrahedron

The simplicity constraints:


We can impose this simplicity constraint as a NC delta function in bivector/flux representation:


The geometricity operator is related to the  simplicity and covariance or closure constraint since they commute:


The action for Quantum Gravity model:


Looking at the combinatorics of field arguments in vertex the gluing of 5 tetrahedra across common triangles, to form 4-simplex.
This  Feynman diagram is equivalent to stranded graph or a 4d simplicial complex:


We can also give a spin foam formulation of the same amplitudes

The Non-commutative bivector representation of Feynman amplitudes gives the simplicial path integral for discrete Plebanski-Holst gravity:


with non-trivial measure on the connection resulting from parallel transport of simplicity constraints across frames:


Again we can  also give a spin foam formulation of the same amplitudes

Related articles


Review of wigner nj-Symbols

This week I’ve been reviewing the properties of the Wigner nj symbols.


Relation to Clebsh-Gordan coefficients:



Compatibility criteria

6wignerabcvanishes if the following are not true:




Definition in terms of 3j’s






unless the triangle inequalities hold for {a,b;,c};    {a,e;,f}, {d,b,f}and {d,e,c}






Definition by 3j’s


Definition by 6j’s





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Review of the Quantum Tetrahedron – part II

Intrinsic coherent states

A class of wave-packet states is given by the coherent states, which are  states labelled by classical variables (position and momenta) that minimize the spread of both. Coherent states are the basic tool for studying the classical limit in quantum gravity. They connect  quantum theory with classical general relativity. Coherent states in the Hilbert space of the theory can be used in proving the large distance behavior of the vertex amplitude and connecting it to the Einstein’s equations.

Given a classical tetrahedron, we can find a quantum state in Hγ such that all the dihedral angles are minimally spread around the classical values, these are the intrinsic coherent states

Tetrahedron geometry

Consider the geometry of a classical tetrahedron reviewed in Review of the Quantum Tetrahedron – part I . A tetrahedron in flat space can be determined by giving three vectors,tetraequ8.1a, representing three of its sides emanating from a vertex P.

Fig 1

Forming a non-orthogonal coordinate system where the axes are along these vectors and the vectors determine the unit of coordinate length, then ei  is the triad and


is the metric in these coordinates. The three vectors


are normal to the three triangles adjacent to P and their length is the area of these faces. The products


define the matrix hab which is the inverse of the metric h =ea.eb. The volume of the tetrahedron is

tetraequ8.4Extending the range of the index a to 1, 2, 3, 4, and denote all the four normals, normalised to the area, as Ea. These satisfy the closure condition


The dihedral angle between two triangles is given by


Now we move to the quantum theory. Here, the quantities Ea are quantized as


in terms of the four operators La, which are the hermitian generators of the rotation group:


The commutator of two angles is:


From this commutation relation, the Heisenberg relation follows:


Now we want to look for states whose dispersion is small compared with their expectation value: semiclassical states where


SU(2) coherent states
Consider a single rotating particle. How do we write a state for which the dispersion of its angular momentum is minimized? If j is the quantum number of its total angular momentum, a basis of states is



we have the Heisenberg relations


Every state satisfies this inequality. A state |j,j> that saturatestetraequ8.15is one  for whichtetraequ8.16a.

In the large j limit we have


Therefore this state becomes sharp for large j.

The geometrical picture corresponding to this calculation is that the state |j, j> represents a spherical harmonic maximally concentrated on the North pole of the sphere, and the ratio between the spread and the radius decreases with the spin.

Other coherent states  are  obtained rotating the state |j, j> into an arbitrary direction n. Introducing Euler angles θ,Φ  to label rotations,


Then let tetraequ8.22aand define the matrix R in SO(3) of the form  , tetraequ8.22bWith this, define:


The states |j,n> form a family of states, labelled by the continuous parameter n, which saturate the uncertainty relations for the angles. Some of their properties are the following.


For a generic direction n = (nx, ny, nz),therefore:




The expansion of these states in terms of Lz eigenstates is


The most important property of the coherent states is that they provide a resolution of the identity. That is


The left hand side is the identity in Hj. The integral is over all normalized vectors, therefore over a two sphere, with the standard R3 measure restricted to the unit sphere.

Observe that by taking tensor products of coherent states, we obtain coherent states. This follows from the properties of mean values and variance under tensor product.

Livine-Speziale coherent intertwiners

Now  introduce “coherent tetrahedra” states. A classical tetrahedron is defined by the four areas Aa and the four normalized normals na, up to rotations. These satisfy


Therefore consider the coherent state;

tetraequ8.30                               in         reviewpart1equ1.15

and project it down to its invariant part in the projectiontetraequ8.31

The resulting state


is the element of Hγ that describes the semiclassical tetrahedron. The projection can be explicitly implemented by integrating over SO(3);


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Review of the Quantum Tetrahedron – part 1

Over the last few posts the level of mathematics as been rather high  so I’ve decided to review our basic understanding of the Quantum Tetrahedron in this post. This review is based on the work of Carlo Rovelli and Francesca Vidotto. Other parts of this review will look at the graviton propagator and also coherent states.

If we pick a simple geometrical object ,an elementary portion of space, such as a small tetrahedron t, not necessarily regular.


The geometry of a tetrahedron is characterized by the length of its sides, the area of its faces, its volume, the dihedral angles at its edges, the angles at the vertices of its faces, and so on. These are all local functions of the gravitational field, because geometry is the same thing as the gravitational field. These geometrical quantities are related to one another. A set of independent quantities is provided for instance by the six lengths of the sides, but these are not appropriate for studying quantization, because they are constrained by inequalities. The length of the three sides of a triangle, for instance, cannot be chosen arbitrarily: they must satisfy the triangle inequalities. Non-trivial inequalities between dynamical variables, like all global features of phase space, are generally difficult to implement in quantum theory.

Instead, we choose the four vectors


defined for each triangle a as 1/2 of the outward oriented vector product of two edges bounding the triangle.


These four vectors have several nice properties. Elementary geometry shows that they can be equivalently defined in one of the two following ways:

  •  The vectors La are outgoing normals to the faces of the tetrahedron and their norm is equal to the area of the face.
  • The matrix of the components


where M is the matrix formed by the components of three edges of the tetrahedron that emanate from a common vertex.

The vectors La have the following properties:

  •  They satisfy the closure relation:


  • The quantities La determine all other geometrical quantities such as areas, volume, angles between edges and dihedral angles between faces.
  • All these quantities, that is, the geometry of the tetrahedron, are invariant under a common SO(3) rotation of the four La. Therefore a tetrahedron is determined by an equivalence class under rotations of a quadruplet of vectors La’s satisfying


  •  The area Aa of the face a is |La|.
  • The volume V is determined by the properly oriented triple product of any three faces:


We can describe the gravitational field in terms of triads and tetrads. If the tetrahedron is small compared to the scale of the local curvature, so that the metric can be assumed to be locally flat andLa can be identified with the flux of the triad field


across the face a, then


Since the triad is the gravitational field, this gives the explicit relation between La and the gravitational field. Here the triad is defined in the 3d hyperplane determined by the tetrahedron.

The above give all the ingredients for jumping to quantum gravity. The geometry of a real physical tetrahedron is determined by the gravitational field, which is a quantum field. Therefore the normals La can be described by quantum operators, if we take the quantum nature of gravity into account. These will obey commutation relations. The commutation relation can be obtained from the hamiltonian analysis of GR, by promoting Poisson brackets to operators,  but ultimately they are quantization postulates. The simplest possibility -see post Quantum tetrahedra and simplicial spin networks, is:


where lo is a constant proportional h and with the dimension of an area. These commutation relations are realizations of the algebra of SU(2), reflecting again the rotational symmetry in the description of the tetrahedron. This is useful, for instance we see that C as defined  is precisely the generator of common rotations and therefore the closure condition  is an immediate condition of rotational invariance, which is what we want: the geometry is determined by the La up to rotations, which here are gauge.

The constant lo must be related to the Planck scale , which is the only dimensional constant in quantum gravity. Setting,

reviewpart1equ1.13where γ is a dimensionless parameter of the order of unity that fixes the precise scale of the theory.

One consequence is immediate -see post Discreteness of area and volume in quantum gravity, the quantity Aa = |La| behaves as total angular momentum. As this quantity is the area, it follows immediately that the area of the triangles bounding any tetrahedron is quantized with eigenvalues:


This is the gist of loop quantum gravity. The result extends to any surface, not just the area of the triangles bounding a tetrahedron.

Say that the quantum geometry is in a state with area eigenvalues    j1, …, j4. The four vector operators La act on the tensor product H of four representations of SU(2), with respective spins j1, …, j4. That is, the Hilbert space of the quantum states of the geometry of the tetrahedron at fixed values of the area of its faces is


Have to take into account the closure equation, which is a condition the states must satisfy, if they are to describe a tetrahedron. But C is nothing else than the generator of the global diagonal action of SU(2) on the four representation spaces. The states that solve the closure equation, namely


are the states that are invariant under this action, namely the states in the subspace:


The volume operator V is well defined in K because it commutes with C, namely it is rotationally invariant. Therefore we have a well-posed eigenvalue problem for the self-adjoint volume operator on the Hilbert space K. As this space is finite dimensional, it follows that its eigenvalues are discrete. Therefore we have the result that the volume has discrete eigenvalues as well. In other words, there are ‘quanta of volume’ or ‘quanta of space’: the volume of our tetrahedron can grow only in discrete steps.

Eigenvalues of the volume

Computing the volume eigenvalues for a quantum of space whose sides have minimal non vanishing area. Recall that the volume operator V is determined by:


where the operators La satisfy the commutation relations:


If the faces of the quantum of space have minimal area, the Casimir of the corresponding representations have minimal non-vanishing value. Therefore the four operators La act on the fundamental representations j1 = j2 = j3 = j4 = 1/2 . Therefore they are proportionalto the self-adjoint generators of SU(2), which in the fundamental representation are Pauli matrices. That is:


The proportionality constant has the dimension of length square, is of Planck scale and is fixed by comparing with the commutation relations of the Pauli matrices. This gives


The Hilbert space on which these operators act is therefore

H = H½ x H½xH½xH½

This is the space of objects with 4 spinor indices A, B = 0, 1, each being in the ½- representation of SU(2).


The operator La acts on the a-th index. Therefore the volume operator acts as:


Now implementing the closure condition . Let


We only have to look  for subspaces that are invariant under a common rotation for each space Hji , namely we should look for a quantity with four spinor indices that are invariant under rotations. What is the dimension of this space? Remembering that for SU(2) representations:

implies that:


Since the trivial representation appears twice, the dimension of reviewpart1equ1.57kkkkis two. Therefore there must be two independent invariant tensors with four indices. These are easy to guess, because the only invariant objects available are ε(AB) and σ(AB), obtained raising the indices of the Pauli matrices σi:


Therefore two states that spanreviewpart1equ1.57kkkkare:


These form a non orthogonal basis inreviewpart1equ1.57kkkk. These two states span the physical SU(2)-invariant part of the Hilbert space, that gives all the shapes of our quantum of space with a given area. To find the eigenvalues of the volume it suffices to diagonalize the 2×2 matrix V²nm:


An straightforward calculation with Pauli matrices gives:


so that,


and the diagonalization gives the eigenvalues


The sign depends on the fact that this is the oriented volume square, which depends on the relative orientation of the triad of normal chosen. Inserting the value for α, we have finally the eigenvalue of the non oriented volume reviewpart1equ1.65

About 10¹°º quanta of volume of this size fit into a cm³.

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The Quantum Tetrahedron and the 6j symbol in quantum gravity

This week as well as working on calculations and modelling the quantum tetrahedron in Lorentzian 3d quantum gravity I have been reading more about the Wigner{6j} symbol, in particular a great paper: Quantum Tetrahedra by Mauro Carfora .

Tetrahedra and 6j symbols in quantum gravity

The Ponzano–Regge asymptotic formula for the 6j symbol and  Regge Calculus are the basis of all discretized approaches to General Relativity, both at the classical and at the quantum level.

sagemath 13 equ0

In Regge’s approach the edge lengths of a triangulated spacetime are taken as discrete counterparts of the metric, a tensor  which encodes the dynamical degrees of freedom of the gravitational field and appears in the classical Einstein–Hilbert action for General Relativity through its second derivatives combined in the  Riemann scalar curvature.

A Regge spacetime is a piecewise linear  manifold of dimension D dissected into simplices; triangles in D = 2, tetrahedra in D = 3, 4 simplices in D = 4 and so on. Inside each simplex either an Euclidean or a Minkowskian metric can be assigned: manifolds obtained by gluing together D–dimensional simplices acquire a metric of Riemannian or Lorentzian signature 2.The Regge action is given explicitly by ( G = 1)

sagemath 13 equ1

where the sum is over (D − 2)–dimensional simplices (hinges), the Vol(D−2)are their (D − 2)–dimensional volumes expressed in
terms of the edge lengths and the deficit angles. A discretized
spacetime is flat inside each D–simplex, while curvature is concentrated at the hinges.  The limit of the Regge action  when the edge lengths become smaller and smaller gives the usual Einstein–Hilbert action for a spacetime which is smooth everywhere, the curvature being distributed continuously.

Regge equations –the discretized analog of Einstein field equations– can be derived from the classical action by varying it with respect to the dynamical variables, i.e. the set  of edge lengths , according to Hamilton principle of classical field theory – see the post: Background independence in a nutshell: the dynamics of a tetrahedron by Rovelli
Regge Calculus gave rise to an approach to quantization of General Relativity known as Simplicial Quantum Gravity. The quantization procedure most commonly adopted is the Euclidean path–sum approach, which is a discretized version of Feynman’s path integral describing D–dimensional Regge geometries undergoing quantum fluctuations.

The Ponzano–Regge asymptotic formula for the 6j symbol;

sagemath 13 equ0

represents the semiclassical limit of a path–sum over all quantum fluctuations, to be associated with the simplest 3–dimensional ‘spacetime’, an Euclidean tetrahedron T. In fact the argument in the exponential reproducesthe Regge action S3 for T.

In general, we denote by T3(j) (a particular triangulation of a closed 3–dimensional Regge manifoldM3 (of fixed topology) obtained by assigning SU(2) spin variables {j} to the edges of T 3. The assignment must satisfy a number of conditions,  illustrated if we introduce the state functional associated with T3(j), namely

sagemath 13 equ2

where No, N1, N3 are the number of vertices, edges and tetrahedra in T3(j).The Ponzano–Regge state sum is obtained by summing over triangulations corresponding to all assignments of spin variables {j} bounded by the cut–off L.

sagemath 13 equ3

where the cut–off is formally removed by taking the limit in front of the sum.

The state sum ZPR [M3] is a topological invariant of the manifold M3, owing to the fact that its value is actually independent of the particular triangulation, namely does not change under suitable combinatorial transformations. These moves are expressed algebraically in terms of  the
Biedenharn-Elliott identity  –representing the moves (2 tetrahedra) <-> (3 tetrahedra)– and of both the Biedenharn–Elliott identity and the orthogonality conditions  for 6j symbols, which represent the barycentric move together its inverse, namely (1 tetrahedra) <-> (4 tetrahedra).

A well–defined quantum invariant for closed 3–manifolds3 based on representation theory of a quantum deformation of the group SU(2)is given by

sagemath 13 equ4
where the summation is over all {j} labeling highest weight irreducible representations of SU(2)q (q = exp{2i/r}, with {j = 0, 1/2, 1 . . . , r − 1}).

The Wigner 6j symbol and its symmetries – The features of the ‘quantum tetrahedron’

Given three angular momentum operators J1, J2, J3 –associated with three kinematically independent quantum systems– the Wigner–coupled Hilbert space of the composite system is an eigenstate of the total angular momentum

J1 + J2 + J3.= J

and of its projection Jz along the quantization axis. The degeneracy can be completely removed by considering binary coupling schemes such as;

(J1 + J2) + J3 and J1 + (J2 + J3),

and by introducing intermediate angular momentum operators defined by;

(J1 + J2) = J12; J12 + J3 = J
(J2 + J3) = J23; J1 + J23 = J

respectively. In Dirac notation the simultaneous eigenspaces of the two complete sets of commuting operators are spanned by basis vectors

|j1j2j12j3> and |j1j2j3j23>

where j1, j2, j3 denote eigenvalues of the corresponding operators, and j is the eigenvalue of J and m is the total magnetic quantum number with range −j < m < j in integer steps.

The j1, j2, j3 run over {0, 1/2 , 1, 3/ 2 , 2, . . . } (labels of SU(2) irreducible representations), while;

|j1 −j2| < j12 < j1 +j2


|j2 −j3| < j23 <j2 + j3.

The Wigner 6j symbol expresses the transformation between the two schemes;

sagemath 13 equ11
Apart from a phase factor, it follows that the quantum mechanical probability;

sagemath 13 equ12

represents the probability that a system prepared in a state of the coupling scheme;

(J1 + J2) = J12; J12 + J3 = J

will be measured to be in a state of the coupling scheme;

(J2 + J3) = J23; J1 + J23 = J

The 6j symbol may be written as sums of products of four Clebsch–Gordan coefficients or their symmetric counterparts, the Wigner 3j symbols. The relations between 6j and 3j symbols are given by;

sagemath 13 equ13
The 6j symbol is invariant under any permutation of its columns or under interchange the upper and lower arguments in each of any two columns. These algebraic relations involve
3! × 4 = 24 different 6j with the same value. The 6j symbol is naturally endowed with a geometric symmetry, the tetrahedral symmetry. In the three–dimensional picture introduced by Ponzano and Regge the 6j is thought of as a real solid tetrahedron T with edge lengths;

L1 = a+ 1/2, L2 ,2 = b+ 1/2 …L6 = f + 1/2

This implies that the quantities;

q1 = a+b+c, q2 = a+e+f, q3 = b+d+f, q4 = c + d + e

(sums of the edge lengths of each face) are all integer. The conditions addressed are sufficient to guarantee the existence of a non–vanishing 6j symbol, but they are not enough to ensure the existence of a geometric tetrahedron T living in Euclidean 3–space with the given edges. More precisely, T exists only if its square volume V evaluated by means
of the Cayley–Menger determinant, is positive.

Ponzano–Regge asymptotic formula
The Ponzano–Regge asymptotic formula for the 6j symbol reads;

sagemath 13 equ0

where the limit is taken for all entries >> 1 and Lp=  j + 1/2,with {jr} = {a, b, c, d, e, f}. V is the Euclidean volume of the tetrahedron T and theta is the angle between the outer normals to the faces which share the edge r.

From a quantum mechanical viewpoint, the above probability amplitude has the form of a semiclassical (wave) function since the factor sagemath 13 equ14ais slowly varying with respect to the spin variables while the exponential is a rapidly oscillating dynamical phase. This  asymptotic behaviour complies with Wigner’s semiclassical estimate for the probability;

sagemath 13 equ14b
compared with the quantum probability:

sagemath 13 equ12

According to Feynman path sum interpretation of quantum mechanics , the argument of the exponential  must represent a classical action, and  it can be read as

sagemath 13 equ14cfor pairs (p, q) of canonical variables, angular momenta and conjugate angle.

Review of the Quantum Tetrahedron

This week I have been reviewing the state of play with regard to the quantum tetrahedron, from next week I will be looking at how a Bose-Einstein condensate of quantum tetrahedra could from spacetime! This review is based on E. Bianchia’ s lectures.Review - classical geometry of a tetrahedron Review - classical geometry of a tetrahedron - area vectors Review - quantum geometry in intertwiner space Review - volume spectrum Review - volume spectrum table