Numerical work with sage 8:

Interactive quantum tetrahedron volume eigenvalue calculator

This week I have been consolidating some the previous weeks’ work. I have produced an interactive sagemath application which will calculate the volume eigenvalues described in Numerical work with sage 7. I’m hoping to get this installed at

Interactive Eigenvalue calculatorOver the next couple of weeks I will be adding the area eigenvalues and graphical representations of the quantum tetrahedra produced by differing J values.

Numerical work with sage 7 – Eigenvalues of the volume operator in Loop quantum Gravity

This week I have been looking at how to actually  calculate the eigenvalues of the volume operator in Loop Quantum gravity. I’ve still got a lot more work to do on this but so far I can calculate the volume of a 4-valent spinnetwork node when the dimension d of the Hilbert space is 2. In order to generalize this I’ll need to generate appropriate Jacobi Matrices when the dimension d is different from 2, calculate the matrix elements and then I’ll need to find their eigenvalues. All this looks relatively straightforward given the work that has been done so far.  I’ll also look to either interactively enter j1, j2, j3, j4 or use a data file for data entry.

volume spectrum sage 7 fig 1

volume spectrum sage 7 fig 2


So here we see that the case (j1,j2,j3,j4) = (1/2, 1/2, 3/2, 3/2) has eigenvalue 0.463, which is known to be correct in the Loop Quantum Gravity case. I have also checked other 2 dimensional cases including (1/2,1/2,1/2,1/2) where the eigenvalue is 0.310.

Bohr-Sommerfeld Quantization of Space by E. Bianchi and Hal M. Haggard

This week I have been reading quite a number of papers of about the volume spectrum in loop quantum gravity. One of the most useful is this one – because it gives a  very clear outline of how to actually calculate the volume eigenvalues (see Numerical work with sage 7 Eigenvalues of the volume operator in Loop quantum Gravity)

In this paper the authors introduce semiclassical methods into the study of the volume spectrum in loop gravity. They state that the  classical system behind a 4-valent spinnetwork node is a Euclidean tetrahedron. They  investigate the tetrahedral volume dynamics on phase space and apply Bohr-Sommerfeld quantization to find the volume spectrum. Their analysis shows a remarkable quantitative agreement with the volume spectrum computed in loop gravity. It also provides new geometrical insights into the degeneracy of this spectrum and the maximum and minimum eigenvalues of the volume.

space of shapes for a tetrahedron

Notes on the quantum tetrahedron by R. Coquereaux

This week I have been studying quite a complicated paper by Coquereaux. In this is paper the author describes several aspects of the space of paths on ADE Dynkin diagrams, particularly the graph E6 – the polytope corresponding to the E6 Lie algebra by the McKay correspondence is the 3-dimensional tetrahedron.

He looks at the  concept of essential matrices or intertwiners for a graph and describe their module properties with respect to right and left actions of fusion algebras. In the case of the graph E6, he finds that the essential matrices build up a right module with respect to its own fusion algebra but a left module with respect to the fusion algebra of A11. The author also presents a simple construction for the Ocneanu graph which describes the quantum symmetries of the Dynkin diagram E6.


Numerical work with sage 5 – Wigner {3j}, {6j}, {9j} symbol evaluation

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).

propagation kernal toy model

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 .

sage wigner symbols

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

Using Wigner symbol

A semiclassical tetrahedron by Carlo Rovelli and Simone Speziale

In this paper the authors construct a macroscopic semiclassical state state for a quantum tetrahedron. They find that the expectation values of the geometrical operators representing the volume, areas and dihedral angles are peaked around assigned classical values with vanishing relative uncertainties.

The authors procedure to construct the semiclassical state is:

  • Choose a classical geometry A1 . . .A4, θ12, θ13, and compute the corresponding spins using Ai= ji,  j0(θ12) and k0(θ13) respectively using

semiclassical tetrahedron fig 1


semiclassical tetrahedron fig 2

  •  Pick up an auxiliary tetrahedron with ( 1/2 ) ji, j0, k0 as edge lengths, and calculate the two dihedral angles φ and χ using:

semiclassical tetrahedron fig 3

  • Choose a basis in Ij1…j4 , say |ji>ab, and take the linear combination with coefficients given by

semiclassical tetrahedron fig 4 – Gaussians with expectation value the required j0 or k0 and phase given by the corresponding dihedral angle φ or χ.

In the large spin limit, this state  encodes the classical values of the chosen geometry.