This post looks at the part of Livine’s great PhD thesis about the quantum tetrahedron.

Below is a table from this thesis looking at the types of structures found in quantum gravity:

I’ll be looking at the quantum tetrahedron, intertwiner and group field structures.

Loop quantum gravity provides a mathematically rigorous quantisation of operators for geometric observables such as area and volume and also shows that the spectra of these operators are discrete in Planck units. This leads to a clear picture of discrete quantum geometry.

- See the post Discreteness of area and volume in quantum gravity

There is a straightforward geometric interpretation of spin network states: the edges e of the graph are dual to elementary surfaces whose area is given by the spin j carried to the edge, and the vertices v are dual to elementary chunks of 3d space bounded by those elementary surfaces and whose volume is determined by the intertwiner living at the vertex.

Spin network states

This interpretation points towards the reconstruction of a discrete geometry dual to the spin network state, with classical polyhedra reconstructed around each vertex whose faces are dual to the edges attached to the vertex and whose exact shape would depend on the explicit intertwiner living at the vertex. This point of view has been particularly developed from the perspective of geometric quantization. It is possible to see intertwiners as quantum polyhedra.

See the post: Polyhedra in loop quantum gravity

In particular, a lot of work has focused on the interpretation of 4-valent intertwiners as quantum tetrahedron. This point of view has been particularly useful to build spinfoam models as quantized 4-dimensional triangulations

The quantum tetrahedron

In order to identify intertwiners as quantum polyhedra and spin network states as discrete geometries, we need to be able to build semi-classical intertwiner states whose shape would be peaked on classical polyhedra and then to glue them together in order to build semi-classical spin network states peaked on classical discrete geometries.

There has been a lot of research work done on developing concepts such as complexifier coherent states introduced by Thiemann, the related holomorphic spin network states and coherent intertwiner states introduced by Livine Speziale.

- See the post Holomorphic factorisation for a quantum tetrahedron

Particular recent lines of research which seems to re-unify these works and viewpoints are the twisted geometry framework and the U(N) framework for intertwiners which actually converge themselves to a unified picture of coherent spin network states as semiclassical discrete geometries . These frameworks are partly inspired from the picture of coherent intertwiners and allow to define explicit variables which control the shape of intertwiners and also parameterize classical polyhedra, thus creating an explicit bridge between the two.

** Related articles**

- Polyhedra in loop quantum gravity by Bianchia , Dona and Speziale (quantumtetrahedron.wordpress.com)