Tag Archives: General relativity

Polyhedra in spacetime from null vectors by Neiman

This week I have been studying a nice paper about Polyhedra in spacetime.

The paper considers convex spacelike polyhedra oriented in Minkowski space. These are classical analogues of spinfoam intertwiners. There is  a parametrization of these shapes using null face normals. This construction is dimension-independent and in 3+1d, it provides the spacetime picture behind the property of the loop quantum gravity intertwiner space in spinor form that the closure constraint is always satisfied after some  SL(2,C) rotation.These  variables can be incorporated in a 4-simplex action that reproduces the large-spin behaviour of the Barrett–Crane vertex amplitude.

In loop quantum gravity and in spinfoam models, convex polyhedra are fundamental objects. Specifically, the intertwiners between rotation-group representations that feature in these theories can be viewed as the quantum versions of convex polyhedra. This makes the parametrization of such shapes a subject of interest for  LQG.
In kinematical LQG, one deals with the SU(2) intertwiners, which correspond to 3d polyhedra in a local 3d Euclidean frame. These polyhedra are naturally parametrized in terms of area-normal vectors: each face i is associated with a vector xi, such that its norm
equals the face area Ai, and its direction is orthogonal to the face. The area normals must satisfy a ‘closure constraint’:


Minkowski’s reconstruction theorem guarantees a one-to-one correspondence between space-spanning sets of vectors xi that satisfy (1) and convex polyhedra with a spatial orientation. In
LQG, the vectors xi correspond to the SU(2) fluxes. The closure condition  then encodes the Gauss constraint, which also generates spatial rotations of the polyhedron.

In the EPRL/FK spinfoam, the SU(2) intertwiners get lifted into SL(2,C) and are acted on by SL(2,C) ,Lorentz, rotations. Geometrically, this endows the polyhedra with an orientation in the local 3+1d Minkowski frame of a spinfoam vertex. The polyhedron’s
orientation is now correlated with those of the other polyhedra surrounding the vertex, so that together they define a generalized 4-polytope. In analogy with the spatial case, a polyhedron with spacetime orientation can be parametrized by a set of area-normal
simple bivectors Bi. In addition to closure, these bivectors must also satisfy a cross-simplicity


In this paper, the author presents a different parametrization of convex spacelike polyhedra with spacetime orientation. Instead of bivectors Bi, they associate null vectors i to the polyhedron’s
faces. This parametrization does not require any constraints between the variables on different faces. It is unusual in that both the area and the full orientation of each face are functions of the data on all the faces. This construction, like the area-vector and
area-bivector constructions above, is dimension-independent. So we can parametrize d-dimensional convex spacelike polytopes with (d − 1)-dimensional faces, oriented in a (d + 1)-dimensional Minkowski spacetime.  These variables can be to construct an action principle for a Lorentzian 4-simplex. The action principle reproduces the large spin behaviour of the Barrett–Crane spinfoam vertex. In particular, it recovers the Regge action for the classical simplicial gravity, up to a possible sign and the existence of additional,degenerate solutions.

In d = 2, 3 spatial dimensions, the parametrization is  contained in the spinor-based description of the LQG intertwiners. There, the face normals are constructed as squares of spinors. It was observed that the closure constraint in these variables can always be satisfied by acting on the spinors with an SL(2,C) boost.  The simple spacetime picture presented in this paper is new. Hopefully, it will contribute to the geometric interpretation of the modern spinor and twistor variables in LQG.

The parametrization
Consider a set of N null vectors liμ in the (d + 1)-dimensional Minkowski space Rd,1, where i = 1, 2, . . . ,N and d ≥2.  Assume the following conditions on the null vectors liμ.

  • The liμ span the Minkowski space and  N ≥  d + 1.
  • The  liμ  are either all future-pointing or all past-pointing.

The central observation in this paper is that such sets of null vectors are in one-to-one correspondence with convex d-dimensional spacelike polytopes oriented in Rd,1.

Constructing the polytrope

Consider a set {liμ} ,take the sum of the liμ normalized
to unit length:


The unit vector nμ is timelike, with the same time orientation as the liμ. Now take nμ to be the unit normal to the spacelike polytope. To construct the polytope in the spacelike hyperplane ∑ orthogonal to nμ define the projections of the null vectors liμinto this hyperplane:


The spacelike vectors siμ  automatically sum up to zero. Also, since the liμ span the spacetime, the siμ must span the hyperplane ∑ . By the Minkowski reconstruction theorem, it follows that the siμ are the (d − 1)-area normals of a unique convex d-dimensional polytope in . In this way, the null vectors li define a d-polytope oriented in spacetime.

Basic features of the parametrization.

The vectors  are liμ  associated to the polytope’s (d −1)-dimensional faces and are null normals to these faces. The orientation of a spacelike (d − 1)-plane in Rd,1 is in one-to-one correspondence with the directions of its two null normals. So each liμ carries partial information about the orientation of the ith face. The second null normal to the face is a function of all the liμ. It can be expressed as:


where  nμ is given by


Similarly, the area Ai of each face is a function of the
null normals liμ to all the faces:


The total area of the faces has the simple expression:


A (d+1)-simplex action

To construct a (d + 1)-simplex action that reproduces in the d = 3 case the large-spin behaviour of the Barrett–Crane spinfoam vertex.

At the level of degree-of-freedom counting, the shape of a (d +1)-simplex is determined by the (d + 1)(d + 2)/2 areas Aab of its (d − 1)-faces. These areas are directly analogous to the spins that appear in the Barrett–Crane spinfoam. Let us fix a set of values for Aab and consider the action:


Then restrict to the variations where:


The stationary points of the action  have the following properties. For each a, the vectors  labμ define a d-simplex with unit normal naμ
and (d − 1)-face areas Aab.


A (d − 1)-face in a (d + 1)-simplex, shared by two d-simplices a and b. The diagram depicts the 1+1d plane orthogonal to the face. The dashed lines are the two null rays in this normal plane.


The d-simplices automatically agree on the areas of their shared (d −1)-faces. The two d-simplices agree not only on the area of their shared (d − 1)-face, but also on the orientation of its (d − 1)-plane in spacetime. In other words, they agree on the face’s area-normal bivector:


The area bivectors defined  automatically satisfy closure and cross-simplicity:


We conclude that the stationary points are in one-to-one correspondence with the bivector geometries of the Barrett-Crane model with an action of the form:




Encoding Curved Tetrahedra in Face Holonomies by Haggard, Han and Riello

This week I have been studying ‘Encoding Curved Tetrahedra in Face Holonomies’. This paper is closely related to the posts:

In this paper the authors  present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra which will apply to homogeneously curved spaces.

Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron’s faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra.

Generalizing the phase space of shapes associated to flat tetrahedra leads to group valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant.

An example of this is  the relation with the spin-network states of loop quantum gravity. This paper provides a justification for the emergence of deformed gauge symmetries and quantum groups in 3+1 dimensional covariant loop quantum gravity in the
presence of a cosmological constant.

Minkowski’s theorem for curved tetrahedra

In the flat case, Minkowski’s theorem associates to any solution of the so-called closure equation


a tetrahedron in E³, whose faces have area al  and outward normals nl.

The paper’s  main result is that Minkowski’s theorem and the closure equation, admit a natural generalization to curved tetrahedra in S³ and H³. The curved closure equation is


where e denotes the identity in SO(3). This equation encodes
the geometry of curved tetrahedra.

As in the flat case, the variables appearing in the closure equation are associated to the faces of the tetrahedron. The {Ol} will be interpreted as the holonomies of the Levi-Civita connection around
each of the four faces of the tetrahedron.

The holonomy Ol around the l-th face of the spherical tetrahedron, calculated at the base point P contained in the face itself, is given by


where {J} are the three generators of so(3), al is the area of the face, and n(P) is the direction normal to the face in the local frame at which the holonomy is calculated.

The vertices of the geometrical tetrahedron are labelled as shown below:


This numbering induces a topological orientation on the tetrahedron, which must be consistent with the geometrical orientation of the paths around the faces.



The holonomies along the simple paths, {Ol}, can be expressed more explicitly by introducing the edge holonomies {oml}, encoding the parallel transport from vertex l to vertex m along the edge connecting them.


The holonomies {Ol } and the normals appearing in their exponents are defined at vertex 4, which is shared by all three faces. Therefore, indicating with θlm the external dihedral angle between faces l and m,



and similarly,


The Gram Matrix

For a given a tetrahedron, flatly embedded in a space of constant positive, negative or null curvature, the Gram matrix is defined  as the matrix of cosines of its external dihedral angles:


One of the main properties of the Gram matrix is that the sign of its determinant reflects the spherical, hyperbolic, or flat nature of the tetrahedron:


Curved Minkowski Theorem for tetrahedra

Four SO(3) group elements Ol , l= 1,..,4 satisfying the closure equation  can be used to reconstruct a unique generalized  constantly curved convex tetrahedron, provided:

  1. the {Ol } are interpreted as the Levi-Civita holonomies around the faces of the tetrahedron
  2.  the path followed around the faces is of the so-called simple type , and has been uniquely fixed by the choice of one of the two couples of faces (24) or (13),
  3.  the orientation of the tetrahedron is fixed and agrees with that of the paths used to calculate the holonomies,
  4.  the non-degeneracy condition det Gram(Ol )  ≠ 0 is satisfied.

Minkowski’s theorem establishes a one-to-one correspondence between closed non-planar polygons in E³ and convex polyhedra, via the interpretation of the vectors defining the sides of the polygon as area vectors for the polygon. This theorem can be extended to curved polyhedra.

In this paper the authors have given a generalization of Minkowski’s theorem for curved tetrahedra. This establishes a correspondence between non-planar, geodesic quadrilaterals in S³, encoded in four SO(3) group elements {Ol} whose product is the group identity, and flatly embedded tetrahedra in either S³ or H³.

This can be used to  reinterpret the Kapovich-Millson-Treloar symplectic structure of closed polygons on a homogeneous space as the symplectic structure on the space of shapes of curved
tetrahedra with fixed face areas.

Related articles

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.

Closure constraints for hyperbolic tetrahedra by Charles and Livine

This week I been continuing to look at non-euclidean tetrahedra as in the post:

In particular I have been looking the paper ‘Closure constraints for hyperbolic tetrahedra’ in this paper the authors  investigate the generalization of loop gravity‘s twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in at space R³. One then

glues them allowing for both curvature and torsion.

It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Λ ≠ 0, and in particular that deforming the loop gravity phase space with real parameter q  would lead to a generalization of twisted geometries to a hyperbolic curvature.

The Flat tetrahedron

Closure constraints


We define the outward vectors normal to the faces with their norm given by the face area:



These normals satisfy the symmetric relation:


This is the closure constraints for the flat tetrahedron.

The hyperbolic triangle 

The curved tetrahedron  lives in the 3d one-sheet hyperboloid defined as the submanifold of R³’¹ of timelike vectors satisfying:


where κ is the curvature radius of the hyperboloid.

The hyperbolic triangle is  embedded in the 3-
hyperboloid H3  which is a coset of the Lorentz group:

closequ25Studying the closure constraints defining the hyperbolic triangle is a first step towards developing the closure constraints for the hyperbolic tetrahedron. In particular, it will lead us to introduce the SU(2) holonomy around the triangle, which will play the role of a group-valued normal vector to the triangle.


The triangle is defined by three points a, b, c. They are defined by three translations from the hyperboloid origin  Ω= (1, 0, 0,0).  Introduce the three oriented edges of the triangle as 1 = (ab), 2 = (ac) and 3 = (bc). We define the translations along each edge:


These three group elements obviously satisfy the following closure relation:


Compact closure constraint for the hyperbolic tetrahedron

Consider the tetrahedron in hyperbolic space. It is defined by four points, a, b, c and d, defined by their respective triangular matrix la,b,c,d describing their position on the hyperboloid.

closfig6We want to find a closure constraint for this hyperbolic tetrahedron in terms of the hyperbolic triangle normals.


We can construct the three SU(2) holonomies defining the normal rotations to the hyperbolic triangles:



and closequ52

It satisfies a simple closure relation:


From the point of view of differential geometry, the holonomies are the discrete counterparts of curvature and this closure relation is just the Bianchi identity.

Non-Compact closure constraint for the hyperbolic tetrahedron

Consider a hyperbolic tetrahedron formed by the 4 points a, b, c, d, with the four hyperbolic triangles (abc), (bcd), (cda) and (dab). Looking at one triangle, say (abc), its three vertices and defined by three translations la,b,c . The hyperbolic translation vectors along the three edges are SB(2,C) elements, which satisfy
the triangle closure constraint:


Define the SB(2,C) normal to the triangle as:


Choosing a definite path along the tetrahedron vertices


say (abcd) following alphabetical and consider the SB(2,C) normal to the triangles following the path’s order:


These SB(2,C) normals satisfy the following closure relation:



In this paper the authors investigated the question of closure constraints for the hyperbolic tetrahedron in the context of loop
quantum gravity with a non-vanishing cosmological constant.

Their goal in doing this was as a first step towards interpreting
the deformed phase space structure for loop gravity on a given graph defined as discrete 3d hyperbolic geometries to be embedded in a 3 + 1-dimensional space-time.

Numerical indications on the semiclassical limit of the flipped vertex by Magliaro, Perini and Rovelli

This week is I’ve reviewing an old but interesting paper on the flipped vertex. I’m working on replicating and extesnding  the calculations in this paper and will post about them next week.

This is related to the posts

In this paper the authors take  the propagation kernel Wt(x,y) of a one-dimensional nonrelativistic quantum system defined by a hamiltonian operator H:


Then they  consider a semiclassical wave packet centered on its initial values and compute its  evolution under the kernel Wt(x,y):


and see whether or not the initial state  evolves into a semiclassical wave packet centered on the correct final values.

The flipped vertex W(jnm,in) is a function of ten spin variables jnm where n,m = 1,…,5 and five intertwiner variables in:

The process described by one vertex can be seen as the dynamics of a single cell in a Regge triangulation of general relativity. This gives a simple and direct geometrical interpretation to the dynamical variables entering the vertex amplitude and a simple formulation of
the dynamical equations. The boundary of a Regge cell is formed by five tetrahedra joined along all their faces, forming a closed space with the topology of a 3-sphere.


  • Anm be the area of the triangle(nm) that separates the tetrahedra n and m.
  • αn(mp,qr) be the angle between the triangles (mp) and (qr) in the tetrahedron n.
  • Θnm be the angle between the normals to the tetrahedra n and m.

These quantities determine entirely the intrinsic and extrinsic classical geometry of the boundary surface.

The ten spins jnm are the quantum numbers of the areas Anm . The five intertwiners in are the quantum numbers associated to the angles αn(mp,qr) . They are the eigenvalues of the quantity:


In general relativity, the Einstein equations can be seen as constraints on boundary variables Anm ,  αn(mp,qr)   and Θnm. These can be viewed as the ensemble of the initial, boundary and final data for a process happening inside the boundary 3-sphere.

In general finding a solution to these constraints is complicated but one is easy –  that corresponding to a flat space and to the boundary
of a regular 4-simplex. This is given by all equal areas Anm = j0, all equal angles in = i0, and Θnm = Θ, where elementary geometry gives:


A boundary wave packet centered on these values must be correctly propagated by the vertex amplitude, if the vertex amplitude is to give the Einstein equations in the classical limit.

The simplest wave packet is a diagonal gaussian wave packet:


normal is the normalization factor. The constants σ and θ are fixed by the requirement that the state is peaked on the value in = i0  so all angles of the tetrahedron are equally peaked on in = i0 :


The state considered is formed by a gaussian state on the spins, with phases given by the extrinsic curvature and by a coherent tetrahedron state:


for each tetrahedron.

We can also write the wave packet


as an initial state times a final state:


We can then test the classical limit of the vertex amplitude by computing the evolution of the four incoming tetrahedra generated by the vertex amplitude and comparing φ(i) with ψ(i).:


So compare the evolved state flippedequ10with the coherent tetrahedron stateflippedequ7



If the function φ(i) turns out to be  close to the coherent tetrahedron state ψ(i), we can say that the flipped vertex amplitude appears to evolve four coherent tetrahedra into one coherent tetrahedron, consistently with the at solution of the classical Einstein equations.

The flipped vertex  in the present case is:



We compared the two functions ψ(i) -coherent tetrahedron and φ(i)  -evolved state for the cases jn = 2 and jn = 4. The numerical results are shown below:



The agreement between the evolved state and the coherent tetrahedron state is quite good. Besides the overall shape of the state, there is a concordance of the mean values and the widths of the wave packet.

Related articles

Tensorial methods and renormalization in Group Field Theories by Sylvain Carrozza

This week I am going to look at a the PhD thesis, Tensorial methods and renormalization in Group Field Theories by  Sylvain Carrozza .

The thesis looks at the two main ways of understanding the construction of GFT models. One way stems from the quantization program for quantum gravity, in the form of loop quantum gravity and spin foam models. Here GFTs are generating functionals for spin foam amplitudes, in the same way as quantum field theories are generating functionals for Feynman amplitudes. They complete the definition of spin foam models by assigning canonical weights to the different foams contributing to a same transition between boundary
states i.e. spin networks. See the post

A second route for GFTs is given by  discrete approaches to quantum gravity. Starting from  matrix models, which allowed us to define random two dimensional surfaces, and so achieve a quantization of two-dimensional quantum gravity. The natural extensions of matrix models are tensor models. From this perspective GFTs appear as enriched tensor models, which allow to the definition of  finer notions of discrete quantum geometries such as the emergence of the continuum.

The thesis then looks at recent aspects of GFTs and tensor models, particularly those following the introduction of colored models. The main results and tools include the combinatorial and topological properties of coloured graphs.

The thesis has two main results, the first set of results concerns the so-called 1/N expansion of topological GFT models. This applies to GFTs with cut-off, given by the parameter N, in which a particular scaling of the coupling constant allows them to reach an asymptotic many-particle regime at large N. The second set of results concerns full-fledge renormalization. Tensorial Group Field Theories (TGFTs), are refined versions of the cut-off models with new non trivial propagators. They have a built-in notion of scale, which generates a well-defined renormalization group flow, and gives rise to dynamical versions of the 1/N expansions.

The thesis looks at a renormalizable TGFT based on the group SU(2) in three dimensions, and incorporating the closure constraint of spin foam models. This TGFT can be considered a field theory realization of the original Boulatov model for three-dimensional
quantum gravity .

Let’s look at the relationship between  the quantum tetrahedron, GFT and the Boulatov model. If the GFT field ϕ is assumed to represent an elementary building block of geometry, then the geometric data should refer to this building block.The Boulatov model generates Ponzano-Regge amplitudes. In its simplicial version, the boundary states of the Ponzano-Regge model are labeled byclosed graphs with three-valent vertices, whose analogue in the field theory formalism are convolutions of the fields ϕ. Following general QFT procedure, we encode the boundary states of the model into functionals of  a single scalar field ϕ. Using the Boulatov model as an example ϕ(g1, g2, g3) is to be interpreted as a flat triangle, and the variables gi label its edges. It is the role of the constraint to introduce an SU(2) flat discrete connection at the level of the amplitudes, encoded in the elementary line holonomies hℓ. The natural interpretation of the variable gi is as the holonomy from a reference point inside the triangle, to the center of the edge i. Thanks to the flatness assumption, this holonomy is independent of the path one chooses to compute gi. The constraint  encodes the freedom in the choice of reference point. From the discrete geometric perspective, the Boulatov model can therefore naturally be called a second quantization of a flat triangle: the GFT field ϕ is the wave-function of a quantized flat triangle, and the path integral provides an interacting theory for such quantum geometric degrees of freedom.


In four dimensions, the correspondence between group and Lie algebra representation can also be put to use. There, the GFT field represents a quantum tetrahedron, and Lie algebra elements correspond to bivectors associated to its boundary triangles. In addition to the closure constraint – equivalent to the Gauss constraint in group space, additional geometricity conditions have to be imposed to guarantee that the bivectors are built from edge vectors of a geometric tetrahedron. These additional constraints are nothing but the simplicity constraints, and non-commutative δ-functions can be used to implement them.

Related articles

Hamiltonian dynamics of a quantum of space

In this post I follow up on some of the work reviewed in the post:

The action of the quantum mechanical volume operator plays a fundamental role in discrete quantum gravity models, can be given as a second order difference equation. By a complex phase change this can be  turned into a discrete Schrödinger-like equation.

The introduction of discrete potential–like functions reveals the role of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols.

I’ll look at the underlying geometric features. When the spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary quantum of space, and an asymptotic picture emerges of the semiclassical and classical regimes.

The definition of coordinates adapted to Regge symmetry is used to construct a set of discrete orthogonal polynomials, characterizing the oscillatory components of torsion-like modes.


For an elementary spin network as shown below


A quadrilateral and its Regge “conjugate”illustrating the elementary spin network representation of the symmetric coupling scheme: each quadrilateral is dissected into two triangles sharing, as a common side, the diagonal l.


the volume operator K = J1 .J2 × J3,  acts democratically on vectors J1; J2 and J3 plus a fourth one, J4, which closes a not necessarily planar quadrilateral vector diagram J1 + J2 + J3 + J4 = 0.

The matrix elements of K are computed to provide a Hermitian representation, whose features have been studied by many see posts:

By a suitable complex change of phase we can transform the imaginary antisymmetric representation into a real, time-independent Schrödinger equation which governs the Hamiltonian dynamics as a function of a discrete variable denoted l. The Hilbert space spanned by the eigenfunctions of the volume operator can be constructed combinatorially and geometrically, applying polygonal relationships to the two quadrilateral vector diagram which are conjugated by a hidden symmetry discovered by Regge .

Discrete schrodinger equation and Regge symmetry

Eigenvalues k and eigenfunctions Ψl(k) of the volume operator are obtained through the three–terms recursion relationship – see post:

Applying a change of phaseΨl (k)=(-i)lΦl(k) to obtain a real, finite difference Schrodinger–like equation

hamiltonianequ1The Ψl are the eigenfunctions of the volume operator expanded in the J12 = J1 +J2 basis. The matrix elements αl in  are given in terms of geometric quantities, namely

hamiltonianequ2αl is proportional to the product of the areas of the two triangles sharing the side of length l and forming a quadrilateral of sides J1 +½, J2 +½, J3 +½ and J4 +½.

The requirement that the four vectors form a (not necessarily planar) quadrilateral leads to identify the range of l with


which is also the dimension of the Hilbert space where the volume operator acts.

Hamiltonian Dynamics

The Hamiltonian operator for the discrete Schrödinger equation


can be written, in terms of the shift operator


The two-dimensional phase space (l, φ) supports the corresponding classical Hamiltonian function given by

hamiltonianequ7This is  illustrated in below for the two Regge conjugate quadrilaterals of the diagram above.



The quadrilaterals are now allowed to fold along l with φ seen as a torsion angle.

The classical regime occurs when quantum numbers j are large and l can be considered as a continuous variable. This limit for l permits us to draw the closed curves in the (l, k) plane when φ = 0 or φ=π . These curves have the physical meaning of torsional-like potential functions


viewing the quadrilaterals as mechanical systems.

hamiltonianfig4Potential functions U+ and U in  hamiltonianequ8are shown for two cases where the conjugated tetrahedra coincide.

  • left panel:   j1,j2, j3, j4=100,110,130,140 the tangential quadrilateral
  • right panel: j1= j2=j3=j4 =120 the ex-tangential quadrilateral.

During the classical motion, the diagonal l changes its value preserving the energy of the system. The result is a geometric configuration  a tetrahedron changing continuously its shape but preserving its volume as a constant of motion.

Quantum mechanics extends the domain of the canonical variables to regions of phase space classically not allowed. Boundaries of these regions are the so-called potential-energy curves particularly important in applications. They are defined as turning points, namely the points where for each value of energy the classical  changes sign. This happens when the momentum φ is either 0 or π.


The above conditions define closed curved in the l-energy plane. These curves have the physical meaning of torsional like potential functions.

At each value of the possible values E’ of the Hamiltonian are bounded by


and the eigenvalues λk of the quantum system are bounded by



The sagemath code and output for this work is shown below: