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.

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.

###### Related articles

- Numerical work with sage 5 – Wigner {3j}, {6j}, {9j} symbol evaluation (quantumtetrahedron.wordpress.com)
- Polyhedra in loop quantum gravity by Bianchia , Dona and Speziale (quantumtetrahedron.wordpress.com)

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