This week I have been reading about a recent generalisation of the quantum tetrahedron – the quantum polyhedron. The authors state that interwiners are the building blocks of spin-network states and that the space of intertwiners is the quantization of a classical symplectic manifold. They show that a theorem by Minkowski allows them to interpret conﬁgurations in this space as bounded convex polyhedra in R3. Given that a polyhedron is uniquely described by the areas and normals to its faces, theys are able to give formulas for the edge lengths, the volume and the adjacency of faces of polyhedra. At the quantum level, this correspondence allows them to identify an intertwiner with the state of a quantum polyhedron – generalizing the notion of quantum tetrahedron.
In the loop quantum gravity, they find that coherent spin-network states with nodes of arbitrary valence represent a collection of semiclassical polyhedra. They also introduce an
operator that measures the volume of a quantum polyhedron and examine its relation
with the standard volume operator of loop quantum gravity.