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.