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:

6wigner1

 

Compatibility criteria

6wignerabcvanishes if the following are not true:

6wigner2

Symmetries

6wigner3

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

6wigner4

Symmetries

6wigner5

Compatibility

6wigner6,

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

 

Orthogonality

6wigner7

 

9j-Symbols

Definition by 3j’s

6wigner8

Definition by 6j’s

6wigner9

Symmetries

6wigner10

 

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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.

mono1
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.

 

curveequ32

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.

mono2

Update:

Eigenenvalent vs spin for 4 valent

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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

mono1

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

mono2

And B the intertwiner basis

mono3

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.

 

mono5

 

 

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