In a number of recent posts including:

- Physical boundary state for the quantum tetrahedron by Livine and Speziale
- Background independence in a nutshell: the dynamics of a tetrahedron by Rovelli
- Towards the graviton from spinfoams: The 3d toy model by speziale

I have been studying the ‘time-fixed’ quantum tetrahedron in which a quantum tetrahedron is used to model the evolution of a state a into state b as shown below:

In this toy model the quantum tetrahedron evolves from a flat shape:

via an equilateral quantum tetrahedron

towards a long stretched out quantum tetrahedron in the limit of large T.

This can be displayed on a phase space diagram,

Where “Minkowski vacuum states” are the states which minimize the energy.

Using sagemath I am able to model the quantum tetrahedron during this evolution by varying the lenght of c – which is used to measure the time T variable;

By using Vpython I am able to model this same system as shown in these animated gifs, the label in the diagram indicated the value of T.

**An animated gif showing the edge a**

An animated gif showing a different view of the quantum tetrahedron:

**Analyzing the {6j} kernel**

An important object in the modelling of the quantum tetrahedron is the Wigner {6j} symbol. As we saw in the post: Physical boundary state for the quantum tetrahedron by Livine and Speziale The quantum dynamics can be studied as by Ponzano-Regge, by associating with the tetrahedron the amplitude

Using sagemath I was able to evaluate the value of this in some extremal cases: when b=0 and b=2j.

**Case b=0**

**Case b=2j**

In the next post I will be looking at the mathematics behind the Wigner{6j} symbol in more detail.

###### Related articles

- Physical boundary state for the quantum tetrahedron by Livine and Speziale (quantumtetrahedron.wordpress.com)
- It’s not a movie – the universe really is a hologram, according to new science (rawstory.com)
- User:Anirudh215/sandbox/Generalized relative entropy (en.wikipedia.org)
- On a short course of quantum mechanics. (idestphysika.wordpress.com)

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