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


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