# Review of wigner nj-Symbols

This week I’ve been reviewing the properties of the Wigner nj symbols.

3j-Symbols

Relation to Clebsh-Gordan coefficients:

Compatibility criteria

vanishes if the following are not true:

Symmetries

6j-Symbols
Definition in terms of 3j’s

Symmetries

Compatibility

,

unless the triangle inequalities hold for {a,b;,c};    {a,e;,f}, {d,b,f}and {d,e,c}

Orthogonality

9j-Symbols

Definition by 3j’s

Definition by 6j’s

Symmetries

# Sagemath 19 part 2: The spectrum of the Ricci operator for a monochromatic 4 – valent node dual to an equilateral tetrahedron

This week I have  continued working on the spectrum of the Ricci operator for a monochromatic 4 – valent node dual to an equilateral tetrahedron. This blog post reports on work in progress,

In my last post I indicated how far I had got in porting the code for the monochromatic 4-valent node from Mathematica to sagemath. This is essentially complete now and I have just got to output a graph of eigenvalues of curvature versus spin.

Sagemath code for the spectrum of the Ricci Operator for a monochromatic 4-valent node dual to an equilateral  tetrahedron

The curvature is written as a combination of the length of a hinge and the deficit angle around it.

As yet the code is unoptimised. Below is a  sample of the output so far:

So now the eigenvalues have been found I can start to make use of them in calculations of the Hamiltonian Constraint Operator.

Update:

# Sagemath 19 part 1: The spectrum of the Ricci operator for a monochromatic 4 – valent node dual to an equilateral tetrahedron

This week thanks to some colleagues I have  been working on the spectrum of the Ricci operator for a monochromatic 4 – valent node dual to an equilateral tetrahedron. This blog post reports on working in progress,

I have been reviewing the papers seen in the posts:

Basically I am porting Mathematica code over to sagemath so that I can then use it it the calculation of the matrix elements of the  LQG Hamiltonian Constraint operator discussed in the  in the posts:

So far I have written code for a number of operators, but I still have the same number still to do, After this I’ll need to join them together.

The matrix defining the operator Q {e1, e2, e3} used in the definition of the volume operator

H ithe matrix defining the operator δij.Yi.Yj used to define the length operator expressed in the intertwiner basis

And B the intertwiner basis

When complete I’ll be able to produce graphs such as those below which is a plot of the spectrum of R as a function of the spin. This can then e used in The numerical investigation of the LQG Hamiltonian Constraint Operator.