I’ve been working on weighed digraphs, Dynkin diagrams and various other graph theory concepts over the last week. The reason for this is that as we saw in “Towards the graviton from spinfoams: the 3d toy model by Simone Speziale” in the Ponzano–Regge (PR) model each tetrahedron has an amplitude given by Wigner’s {6j} symbol for the recoupling theory of SU(2).

This use of Wigner symbols is a common feature of Quantum Gravity models. Fortunately, Sagemaths provides a simple and straightforward way to evaluate Wigner symbols {3j}, {j6}, or {9j}. I am still working on evaluating {10j}, {15j} and {20j} Wigner symbols which are used in 3d and 4d Loop Quantum Gravity .

The table below taken from Quantum Gravity By Rovelli shows how the Wigner symbols are used in various quantum gravity models.

###### Related articles

- Towards the graviton from spinfoams: the 3d toy model by Simone Speziale (quantumtetrahedron.wordpress.com)