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:

- Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint
- A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements
- Sagemath 18: Calculation of the Matrix Elements of Thiemann’s Hamiltonian Constraint in Loop Quantum Gravity
- Matrix Elements of Lorentzian Hamiltonian Constraint in LQG

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.

###### Related articles

- A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements by Gaul and Rovelli (quantumtetrahedron.wordpress.com)
- Sagemath 18: Calculation of the Matrix Elements of Thiemann’s Hamiltonian Constraint in Loop Quantum Gravity (quantumtetrahedron.wordpress.com)
- Numerical work with Sagemath 17: Exploring curvature (quantumtetrahedron.wordpress.com)

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