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
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:
n4 ≡e4 ×e5 = −n1 − n2 −n3
The volume relationship is given by;
n1 · n2 × n3 = −(e1 · e2 ×e3)2 = −36V 2
where V is the volume.
ni2 ≡ ni · ni and nij ≡ni · nj