Tag Archives: 6-j symbol

GFT Condensates and Cosmology

This week I have been studying some papers and a seminar by Lorenzo Sindoni and Daniele Oriti on Spacetime as a Bose-Einstein Condensate. I have also been reading a great book ‘The Universe in a Helium Droplet’ by Volovik and a really good PhD Thesis, ‘Appearing Out of Nowhere: The Emergence of Spacetime in Quantum Gravity‘ by Karen Crowther – I be posting about these next time.

Spacetime as a Bose-Einstein Condensate has been discussed in a number of other posts including:

Simple condensates

Within the context of  Group Field Theory (GFT), which is a field theory on an auxiliary group manifold. It incorporates many ideas and structures from LQG and spinfoam models in a second quantized language. Spacetime should emerge from the collective dynamics of the microscopic degrees of freedom. Within Condensates all the quanta are in the same state. These simple quantum states of the full theory, can be put in correspondence with Bianchi cosmologies via symmetry reduction at the quantum level. This leads to an effective dynamics for cosmology which makes  contact with LQC and Friedmann  equations.

Group Field Theories the second quantization language for discrete geometry

Group field theories are quantum field theories over a group manifold. The basic defintiion of a GFT is


which can denoted as:


The theory is formulated in terms of a Fock space and Bosonic statistics is used.


Gauge invariance on the right is required, that is:


GFT quanta: spin network vertices  and quantum tetrahedra

Considering D=4 with group G=SU(2). These quanta have a natural interpretation in terms of 4-valent spin-network vertices.



Via a noncommutative Fourier transform it can be formulated in group variables. Considering  SU(2), we have:



We now  have a second quantized theory that creates quantum tetrahedra

gftfig3 represented as gftequ6.

 Correlation functions of GFT and spinfoams

When computing the correlation functions between boundary states the Feynman rules glue tetrahedra into 4-simplices. This is controlled by the combinatorics of the interaction term. This amplitude is designed to match spinfoam amplitudes. For example,
the interaction kernel can be chosen to be the EPRL vertex in a group representation.


The dynamics can be designed to give rise to the transition amplitudes with sum over 4d geometries included using a discrete path integral for gravity.

By  proceeding as in condensed matter physics and we can design
trial states, parametrised by relatively few variables, and deduce from the dynamics of the fundamental model the optimal induced dynamics.

Now we select some trial states to getthe  effective continuum dynamics. We choose trial states that contain the relevant information about the regime that we want to explore. Fock space suggests several interesting possibilities such as field coherent states;


This is a simple state, but not a state with an exact finite number of particles. It is  inspired by the idea that spacetime is a sort of condensate and can be generalized to other states  such assqueezed, and multimode.

The condensates can be naturally interpreted as homogeneous cosmologies:


Elementary quanta possessing the same wavefunction so that  the metric tensor in the frame of the tetrahedron is the same everywhere. This Vertex or wavefunction homogeneity can be interpreted in terms of homogeneous cosmologies, once a
reconstruction procedure into a 3D group manifold has been specified.The reconstruction procedure is based on the idea that each of these tetrahedra is embedded into a background manifold: the edges are aligned with a basis of left invariant vector fields.


Curved Polyhedra

This week I have been studying a great paper:  SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry by  Haggard, Han, Kaminski, and Riello. This post looks at the section of the paper on curved tetrahedra. This carries on the literature review started in the following posts:

Curved Tetrahedra

Using critical point equations, together with the interpretation
of the holonomies Hab we can reconstruct a curved-tetrahedral geometry at every vertex of the graph. This post shows  how to recover the tetrahedral geometry from the holonomies in a constructive way.

The key equation is the closure condition


we will focus on the derived equation


where Ob ∈ SO(3) is the vectorial (spin 1) representation of Hb ∈ SU(2). The Ob are interpreted as parallel transports along specific, simple paths on the tetrahedron 1-skeleton.


The ordered composition of all the paths associated to a tetrahedron is equivalent to the trivial path, hence the identity on the right-hand side of the closure equation.

Flatly Embedded Surfaces

Consider a 4-dimensional spacetime (M4,  gαβ), with no torsion, constant curvature λ , and tetrad eαI . In this spacetime, consider a bounded 2-surface s that is flatly embedded in M4, i.e. such that the wedge product of its space- and time-like normal fields, nα and uβ respectively, is preserved by parallel transport on the surface s.45 Then, the holonomy around s of the torsionless spin connection  curvedequ10.3a is given in the spinor representation by:


where the subscript + indicates the self-dual part of an object, as is the area of s, and we have defined uI and nI to be the internal spacelike and timelike normals to the surface. In the future pointing time gauge curvedequ10.4a and therefore,


where the last two equalities hold in time-gauge.

Finally we obtain in the future pointing time gauge,

curvedequ10.5 and  in the vectorial representation,


Beyond the properties of area, curvature, and orientation, the shape of s is not defined for the moment. We can  further constrain its geometric degrees of freedom  by requiring each vertex of the graph to be identified with the simplest curved geometrical object with four faces, a homogeneously curved tetrahedron.
The cosmological constant or equivalently the curvature is totally free at each face. So this model cannot be considered a quantization of gravity with a fixed-sign cosmological constant: it is
rather a quantization of gravity with a cosmological constant, the sign of which is determined dynamically, and only semiclassically, by the imposed boundary conditions  the external jab and ξab.

 Constant Curvature Tetrahedra

The faces of the curved tetrahedron are spherical or hyperbolic triangles, with a radius of curvature equal to R. This means that their areas must lie in the interval6 or3respectively.

The spherical case is no problem, since SU(2) group elements have the right periodicity in their argument. By looking at the deformed SU(2)q representations with q and k , one only finds spins up to |k|/2, which translates into γj ≤ 6π/|Λ|.

The hyperbolic case, on the other hand, is more subtle.  These subtleties can give rise—in certain cases determined by the choice of the spins—to non-standard geometries that extend across the two sheets of the two-sheeted hyperboloid.

hypConsider the reconstruction at the vertex 5 of 􀀀Γ5. The closure
equation in the vector representation is


and the special side is 24. We will take the base point to be vertex 4. Because all the holonomies are based at vertex 4, all the nb are defined there, which we notate nb(4). However the property of being flatly embedded means having vanishing extrinsic curvature, and so this makes the normal to a face well-defined at any of its points. The faces 1, 2 and 3  contain vertex 4, and this means that n1(4), n2(4), n3(4) can be directly interpreted as normals to their respective faces, while n4(4) is the vector obtained after parallel-transporting n4 from its face to vertex 4, via the edge 24. That is,

curvedequ10.8where ocb is the vector representation of the holonomy from vertex b to vertex c, along the side cb.

It is possible to give an expression of the cosines of the dihedral angles directly in terms of the holonomies,



which are a sort of normalized, connected two-point functions of the holonomies.

Once we have unambiguously fixed the cosines of the dihedral angles, these can be used to construct the Gram matrix of the tetrahedron


The determinant of ³Gram determines whether the tetrahedron is hyperbolic or spherical. detG > 0 gives spherical geometry whilst
detG < 0 gives hyperbolic geometry, this therefore provides the crucial information


This fixes the sign of the cosmological constant at a given vertex. Consequently, there is no freedom, within a vertex, to change this sign, and a unique correspondence between the spinfoam and geometric data can finally be established. Note that flipping the sign of the cosmological constant does not change the Gram matrix, since it corresponds to flipping all the ±b. This fact is crucial, since it means that sgn () can actually be calculated.

Finally, from the Gram matrix one can fully reconstruct the curved tetrahedron. In practice this amounts to repeatedly applying the spherical (and/or hyperbolic) law of cosines to first calculate the face angles of the tetrahedron and then its side lengths.

Related articles


Exact and asymptotic computations of elementary spin networks

This week I have been following up some work which I was introduced to in Dimitri Marinelli’s PhD thesis ‘Single and collective dynamics of discretized geometries’. Essentially this involves the analysis of the volume operator.  This is really exciting for me as it is in my specialist research area –  the numerical analysis of Quantum geometric operators and their spectra. I’ll be following up the literature survey with numerical work in sagemath.

The paper I’ll look at this week is ‘Exact and asymptotic computations of elementary spin networks: classification of the quantum–classical boundaries’ by  Bitencourt, Marzuoli,  Ragni, Anderson and and Aquilanti.

There has been increasing interest to the issues of exact computations and asymptotics of spin networks. The large–entries regimes – semiclassical limits, occur in many areas of physics and in particular in discretization algorithms of applied quantum mechanics.

The authors extend recent work on the basic building block of spin networks, namely the Wigner 6j symbol or Racah coefficient,  by exploiting its self–dual properties and studying it as a function of two discrete variables. This arises from its original definition as an orthogonal angular momentum recoupling matrix. Progress comes
from recognizing its role in the foundation of the modern theory of classical orthogonal polynomials, as extended to include discrete variables. Features of the imaging of various regimes of these orthonormal matrices are made explicit by computational
advances –based on traditional and new recurrence relations– which allow an interpretation of the observed behaviors in terms of an underlying Hamiltonian formulation.

The paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines – caustics in the plane of the two matrix labels. This analysis marks the evolution of the turning points of relevance for the semiclassical regimes and highlights the key role of the Regge symmetries of the 6j.


The diagrammatic tools for spin networks were developed by the Yutsis school and  in connection with applications to discretized models for quantum gravity after Penrose, Ponzano and Regge.

The basic building blocks of all spin networks are the Wigner 6j symbols or Racah coefficients, which are studied here by exploiting their self dual properties and looking at them as functions of two variables. This approach is natural in view of their origin as matrix elements describing recoupling between alternative angular momentum binary coupling schemes, or between alternative hyperspherical harmonics.

Semiclassical and asymptotic views are introduced to describe the dependence on parameters. They originated from the association due to Racah and Wigner to geometrical features, respectively a dihedral angle and the volume of an associated tetrahedron, which is the starting point of the seminal paper by Ponzano and Regge . Their results provided an impressive insight into the functional dependence of angular momentum functions showing a quantum mechanical picture in terms of formulas which describe classical and non–classical discrete wavelike regimes, as well as the transition between them.

The screen: mirror, Piero and Regge symmetries

The 6j symbol becomes the eigenfunction of the Schrodinger–like equation in the variable q, a continuous generalization of j12:


where Ψ(q) is related to


and p² is related with the square of the volume V of the associated tetrahedron.

exactequtetraThe Cayley–Menger determinant permits to calculate the square of the volume of a generic tetrahedron in terms of squares of its edge lengths according to:


The condition for the tetrahedron with fixed edge lengths to exist as a polyhedron in Euclidean 3-space amounts to require V²> 0, while the V²= 0 and V²< 0 cases were associated by Ponzano and Regge to “flat” and nonclassical tetrahedral configurations respectively.

Major insight is provided by plotting both 6js and geometrical functions -volumes, products of face areas – of the associated tetrahedra in a 2-dimensional j12 -􀀀 j23 plane , in whch the square “screen” of allowed ranges of j12 and j23 is used in all the pictures

  •  The mirror symmetry. The appearance of squares of tetrahedron edges entails that the invariance with respect to the exchange J ↔− 􀀀J implies formally j ↔ – 􀀀j 􀀀-1 with respect to the entries of the 6j symbol.
  • Piero line. In general, an exchange of opposite edges of a tetrahedron corresponds to different tetrahedra and different symbols. In Piero formula, there is a term due to this difference that vanishes when any pair of opposite edges are equal.
  • Regge symmetries. The these arises through connection with the projective geometry of the elementary quantum of space, which
    is associated to the polygonal inequalities -triangular and quadrilateral in the 6j case -, which have to be enforced in
    any spin networks.

The basic Regge symmetry can be written in the following form:


The range of both J12 and J23, namely the size of the screen, is given by 2min (J1, J2, J3, J, J1 +ρ , J2 +ρ ,J3 +ρ, J + ρ).

Features of the tetrahedron volume function

Looking at the volume V as a function of  x=J12 and  y=J23 we get the expressions for the xVmaxand yVmax that correspond to the maximum of the volume for a fixed value of x or y:


The plots of these are  called “ridge” curves on the x,y-screen. Each one marks configurations of the associated tetrahedron when two specific pairs of triangular faces are orthogonal. The corresponding values of the volume (xVmax,xand yVmax,y) are

exactequ8F is the area of the triangle with sides a, b and c.Curves corresponding to V = 0, the caustic curves, obey the equations:





Symmetric and limiting cases

When some or all the j’s are equal, interesting features appear in the screen. Similarly when some are larger than others.


Symmetric cases



exactequfig4Limiting cases

We can discuss the caustics of the 3j symbols as the limiting case of the corresponding 6j where three entries are larger than the other ones:






The extensive images of the exactly calculated 6j’s on the square screens illustrate how the caustic curves studied in this paper separate the classical and nonclassical regions, where they show wavelike and evanescent behaviour respectively. Limiting
cases, and in particular those referring to 3j and Wigner’s d matrix elements can be analogously depicted and discussed. Interesting also are the ridge lines, which separate the images in the screen tending to qualitatively different flattening of the quadrilateral,
namely convex in the upper right region, concave in the upper left and lower right ones, and crossed in the lower left region.

Related articles

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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Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model by Bonzom, Livine,Smerlak and Speziale

This week I have been doing further work on the Quantum Tetrahedron as an Quantum Harmonic Oscillator – which I’ll review in a later post and also looking at the 3d toy model in more detail. In particular I have been studying ‘Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model.’ In this paper the authors consider an exact expression for the 6j-symbol for the isosceles tetrahedron, involving SU(2) group integrals, and use it to write the two-point function of 3d gravity on a single tetrahedron as a group integral. The perturbative expansion of this expression can then be performed with respect to the geometry of the boundary using a simple saddle-point analysis. They can then derive the complete expansion in inverse powers of the length scale and evaluate explicitly the quantum corrections up to second order. They also  use the same method to provide the complete expansion of the isosceles 6j-symbol with the next-to-leading correction to the Ponzano-Regge asymptotics.

Considering for simplicity the Riemannian case, the spinfoam amplitude for a single tetrahedron is the 6j-symbol of the Ponzano-Regge model. Its large spin asymptotics is dominated by exponentials of the Regge action for 3d general relativity. This is a key result, since the quantization of the Regge action is known to reproduce the correct free graviton propagator around flat spacetime.

This paper considers the simplest possible setting given by the 3d toy model introduced in the post Towards the graviton from spinfoams: the 3d toy model and studies analytically the full perturbative expansion of the 3d graviton. The results are based on a reformulation of the Wigner 6j symbol and the graviton propagator as group integrals. The authors compute explicitly
the leading order then both next-to-leading and next-to-next analytically. They also calculate  a formula for the next-to-leading order of the  Ponzano-Regge asymptotics of the 6j-symbol in the case of an isosceles tetrahedron.

Applying the same methods and tools to 4d spinfoam models would  allow a more thorough study of the full non-perturbative spinfoam graviton propagator and its correlations in 4d quantum gravity.

The boundary states and the kernel

Consider a triangulation consisting of a single tetrahedron. To define transition amplitudes in a background independent context for a certain region of spacetime, the main idea is to perform a perturbative expansion with respect to the geometry of the boundary. This classical geometry acts as a background for the perturbative expansion. To do so  – have to specify the values of the intrinsic and extrinsic curvatures of such a boundary, that is the edge lengths and the dihedral angles for a single tetrahedron in spinfoam variables. As in the post Towards the graviton from spinfoams: the 3d toy model, attention is restricted  to a situation in which the lengths of four edges have been measured, so that their values are fixed, say to a unique value jt + 1/2 . These constitute the time-like boundary and we are then interested in the correlations of length fluctuations between the two remaining and opposite edges which are the initial and final spatial slices. This setting is referred to as the time-gauge setting. The two opposite edges e1 and e2 have respectively lengths, j1 + 1/2 and j2 + 1/2 .

towards graviton fig1

Physical setting to compute the 2-point function. The two edges whose correlations of length fluctuations will be
computed are in fat lines, and have length j1 + 1/2 and j2 + 1/2 . These data are encoded in the boundary state of the tetrahedron. In the time-gauge setting, the four bulk edges have imposed lengths jt + 1/2 interpreted as the proper time of a particle propagating along one of these edges. Equivalently, the time between two planes containing e1 and e2 has been measured to be T = (jt + 1/2 )/sqrt(2).

In the spinfoam formalism, and in agreement with 3d LQG, lengths are quantized so that jt, j1 and j2 are half-integers.

The lengths and the dihedral angles are conjugated variables with regards to the boundary geometry, and have to satisfy the classical equations of motion. Here, it simply means that they must have admissible values to form a genuine flat tetrahedron. The dimension of the SU(2)-representation of spin j, dj ≡ 2j +1 is twice the edge length.

towards graviton equ 1

To assign a quantum state to the boundary, peaked on the classical geometry of the tetrahedron. Since jt is fixed, we only need such a state for e1, peaked on the length j1 + 1/2 , and for e2, peaked on        j2 + 1/2 .

Previous work have used a Gaussian ansatz for such states. However, it is more convenient to choose states which admit a well-defined Fourier transform on SU(2). In this perspective, the dihedral angles of the tetrahedron are interpreted as the class angles of SU(2) elements. So the Gaussian ansatz can be replaced for the edges e1 and e2 by the following Bessel state:

towards graviton equ 3a

The role of the cosine  is to peak the variable dual to j, i.e. the dihedral angle, on the value αe. Then the boundary state admits a well-defined Fourier transform, which is a Gaussian on the group SU(2). The SU(2) group elements are parameterized as:

towards graviton equ 4

These states carry the information about the boundary geometry necessary to induce a perturbative expansion around it. Interested in the following correlator,

towards graviton equ 6

W1122 measures the correlations between length fluctuations for the edges e1 and e2 of the tetrahedron, and it can be interpreted as
the 2-point function for gravity , contracted along the directions of e1 and e2. The 6j-symbol emerges from the usual spinfoam models for 3d gravity as the amplitude for a single tetrahedron.

towards graviton equ4043

Perturbative expansion of the isosceles 6j-symbol

The procedure described above can be applied directly to the isosceles 6j-symbol, obtaining the known Ponzano-Regge formula and its corrections.

This is interesting for a number of reasons. The corrections to the Ponzano-Regge formula are a key difference between the spinfoam perturbative expansion  and the one from quantum Regge calculus. The 6j-symbol is also the physical boundary state of 3d gravity
for a trivial topology and a one-tetrahedron triangulation. In 4d, it appears as a building block for the spin-foams amplitudes, such as the 15j-symbol. So for many aspects of spin-foams in 3d and 4d, in particular for the quantum corrections to the semiclassical limits, it good to have a better understanding of this object.

.The expansion of this isosceles 6j-symbol is;

towards graviton equ80

The leading order asymptotics, given by the original Ponzano-Regge formula is:

towards graviton equ80a

It is possible to compute analytically the two-point function – the graviton propagator – at all orders in the Planck length for the 3d toy model -the Ponzano-Regge model for a single isoceles tetrahedron as in Towards the graviton from spinfoams: the 3d toy model.

Related articles

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.