Numerical work with Vpython and sagemath12: The Quantum Tetrahedron

In a number of recent posts including:

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:

physical boundary fig 1

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

Background independence fig3

via an equilateral quantum tetrahedron

Background independence fig1

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

Background independence fig2

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

physical boundary equ 4

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.


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