Classical 6j–symbols and the tetrahedron by Justin Roberts

I have been reading this paper over the weekend together with some other papers on spin networks (which I’ll post about later). The goal of this paper is to prove and explain the classical 6J symbol by using geometric quantization. A classical 6j –symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). It has a deep geometric significance -Ponzano and Regge, expanding on work of Wigner, gave an asymptotic formula relating the value of the 6j – symbol, when the dimensions of the representations are large, to the volume of an Euclidean tetrahedron whose edge lengths are these dimensions.



Quantum tetrahedron and its classical limit by Daniel R. Terno

A great paper which clearly explains the geometry of both the classical and quantum tetrahedron. Its aim was to show that classical information that is retrieved from a quantum tetrahedron is intrinsically fuzzy – in fact for a single tetrahedron it was shown that  the  uncertainty in the  dihedral angles is inversely proportional the surface area. The paper states that a  quantum tetrahedron is a useful toy model to investigate the classical limit of LQG and  in provides boundary states in studies of the graviton propagator. 

Review of  classical and quantum tetrahedra

Six numbers that determine the shape of a tetrahedron, for example,  the six edges of a tetrahedron, or four facial areas and two independent dihedral angles between them. There are number of useful relations between areas, angles and volume in this paper.

If the outward normals to the faces are  labelled as Ji , then lengths are twice the triangular areas,

Ji = 2Ai

Because it is  a closed surface a tetrahedron satisfies the closure condition

Ji = 0

Angles between the triangular faces – the  inner dihedral angles  are related to the outer dihedral angles θij by ,

Jij Ji · Jj = JiJj cos θij

The volume squared of the tetrahedron can be expressed in terms of the area vectors,

V2 = -(1/36 ) J1 · J2 × J3


In the quantized problem the four normals J1, J2, J3, J4 are identified with the generators of SU(2). So that,

J^2 |ji,mi>   = j(j+1) |ji,mi>  and Jz |ji,mi> = mi|ji,mi>

Asymptotically the Casimir Operator

J= sqrt[ J (J+1)] ~ J + 1/2

The Quantum Tetrahedron in 3 and 4 Dimensions by John C. Baez

John Baez is one of my favourite physicists. He has done a lot of work on quantum gravity especially with regard to loops and knot theory. He used to maintain the ‘ This weeks finds in Mathematical Physics’ webpage.  In this paper Baez states that recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the ‘quantum tetrahedron’.  By starting with a classical phase space whose points correspond to geometries of the tetrahedron in R3, he uses geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labelled by the areas of the faces of the tetrahedron together with another quantum number, such as  the area of one of the parallelograms formed by midpoints of the tetrahedron’s edges.


Parallelogram formed by midpoints of the tetrahedron’s edges

By repeating the procedure for the tetrahedron in R4, he also  obtains a Hilbert space with a basis labelled solely by the areas of the tetrahedron’s faces.

Quantum tetrahedra and simplicial spin networks by A.Barbieri

So this what I’ve been reading with my coffee this week. I like this paper mainly because its clearly written and has precise mathematical  detail of the properties such as area, volume and dihedral angle of the classical and quantum tetrahedron.

In this paper Barbieri finds:

A new link between tetrahedra and the group SU(2), this is done by  by associating to each face of a tetrahedron an irreducible unitary SU(2) representation and by imposing that the faces close – from this process the concept of a quantum tetrahedron emergse. The Hilbert space of the quantum tetrahedron is introduced and it is shown that, due to an uncertainty relation, the “geometry of the tetrahedron” exists only in the sense of “mean geometry”.

In the paper a kinematical model of quantum gauge theory is also proposed, which shares the advantages of the Loop Representation approach in handling in a simple way gauge- and diff-invariances at a quantum level, but is completely combinatorial. The concept of quantum tetrahedron finds a natural application in this model, giving a possible interpretation of SU(2) spin networks in terms of geometrical objects.

Looking at a classical tetrahedron

Fig 1

The Area of the tetrahedron faces are given by:

n1 ≡ −e1 ×e2

n2 ≡ −e2 ×e3

n3 ≡ −e3 ×e1

The closure condition so the faces actually make a tetrahedron is:

n4e4 ×e5 = −n1n2n3

The volume relationship is given by;

n1 · n2 × n3 = −(e1 · e2 ×e3)2 = −36V 2

where V is the volume.


ni2 ≡ ni · nand nijni · nj




Why the Quantum Tetrahedron?

Well in loop quantum gravity (LQG), the geometry of the physical space turns out to be quantized. When studying the spectral problem associated with the operators representing geometrical quantities such as area and volume, one finds two families of quantum numbers, which have a direct geometrical interpretation: SU(2) spins, labelling the links of a spin network, and SU(2) intertwiners, labelling its nodes. The spins are associated with the area of surfaces intersected by the link, while the intertwiners are associated with the volume of spatial regions that include the node, and to the angles formed by surfaces intersected by the links . A four-valent link, for instance, can be interpreted as a quantum tetrahedron: an elementary ‘atom of space’ whose face areas, volume and dihedral angles are determined by the spin and intertwiner quantum numbers.

The very same geometrical interpretation for spins and intertwiners can be obtained from a formal quantization of the degrees of freedom of the geometry of a tetrahedron, without any reference to the full quantization of general relativity which is at the base of LQG. In this case, one can directly obtain the Hilbert space H describing a single quantum tetrahedron. The states in H can be interpreted as ‘quantum states of a tetrahedron’, and the
resulting quantum geometry is the same as that defined by LQG.

Atoms of spacetime

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