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.
- A semiclassical tetrahedron by Carlo Rovelli and Simone Speziale (quantumtetrahedron.wordpress.com)