The tetrahedron and its Regge conjugate

This week I have been reading the PhD thesis ‘Single and collective dynamics of discretized geometries’ by Dimitri Marinelli. In this post I’ll look at a small portion about Regge calculus, the  tetrahedron and its Regge conjugate.

Regge Calculus is a dynamical theory of space-time introduced in 1961 by Regge as a discrete approximation for the Einstein theory of gravity. The basic idea is to replace a smooth space-time with a collection of simplices. The collective dynamics of these geometric objects is driven by the Regge action and the dynamical variables are their edge lengths – which play the role of the metric tensor of General Relativity. Simplices are the n-dimensional generalization of triangles and tetrahedra. Regge Calculus inspired and is at the base of almost all the present discretized models for a quantum theory of gravity for at least two reasons:

  • It is a discretized model, so it represents a possible atomistic system typical of quantum systems
  • There is a deep connection between the Regge action, the asymptotic of the 6j symbol and a path integral formulation of gravity.

Let’s see  how the Regge transformation acts on a tetrahedral shape. The formulas


and the association between 6j symbol and an Euclidean tetrahedron tell us that any Regge transformation acts on four edges of a tetrahedron keeping a pair of opposite edges unchanged. The Regge-transformed tetrahedra is called `conjugate’.

Using the Ponzano-Regge formula for the 6j,

reggeequ2.18we can immediately say that the volume of a tetrahedron and that of a Regge transformed one must coincide.



The volume of a tetrahedron is also invariant under the Regge transformation of four consecutive edges.

The volume of a tetrahedron, being a function of six parameters, can be expressed in several ways. For the tetrahedron below:

tetrahedron with dihedral angleThe ‘orientated’ volume reads, 


where AABC and AACD are respectively the areas of the triangles ABC and ACD, lAC is the length of the common edge and β is the dihedral angle between these two faces.

The importance of the Regge symmetry is that it constrains the shape dynamics of a single tetrahedron,  it relates different tetrahedra equating their quantum representations and it is the key tool to understand the classical motion of a four-bar linkage mechanical systems and its link to the the quantum dynamics of tetrahedra.

This thesis also contains a section on the Askey scheme which I’ll be following up in future posts:

askey scheme






State Sums and Geometry by Frank Hellmann

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


The Geometry of Representations

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

The Geometry of SU(2)

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

Coherent States

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


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

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


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


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


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


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


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

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


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

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


Coherent Triangles

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

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

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


 Coherent Tetrahedra

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

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


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

 Tetrahedra from Closing Vectors

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

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


The Geometry of Spin(4)

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

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

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


 Geometric Bivectors

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



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


The Geometry of SL(2,C)

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

The corresponding Lorentzian generators are then


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