This week I have been continuing my research on the the expectation values of the Ricci curvature on Carlo-Speziale semi-classical states, for different values of the spins in the case of a simple graph : a monochromatic 4 – valent node dual to an equilateral tetrahedron.
The Rovelli-Speziale semi-classical tetrahedron is a semiclassical quantum state corresponding to the classical geometry of the tetrahedron determined by the areas A1, . . . ,A4 of its faces and
two dihedral angles θ12, θ34 between A1 and A2 respectively A3 and A4.
It is defined as a state in the intertwiner basis |j12>
with coefficients cj12 such that
in the large scale limit, for all ij. The large scale limit considered here is taken when all spins are large.
The expression of the coefficients cj12 verifying the requirements is:
where joand ko are given real numbers respectively linked to θ12 and θ34 through the following equations:
σj12 is the variance which is appropriately fixed and the phase φ(jo, ko) is the dihedral angle to join an auxiliary tetrahedron related to the asymptotic of the 6j symbol performing the change
of coupling in the intertwiner basis.
For a classical regular tetrahedron, using the expression the integrated classical curvature scales linearly in terms of the length of its hinges because the angles do not change in the equilateral configuration when the length is rescaled, which means that the
integrated classical curvature scales as a square root function of the area of a face.
In the plot above we can see that the expected values of R on coherent states and semi-classical regular tetrahedra for large spins scales as a square root function of the spin, this matches the semi-classical evolution we expect.
- Sagemath 19 Part 2: the Spectrum of the Ricci Operator for a Monochromatic 4 – Valent Node Dual to an Equilateral Tetrahedron (quantumtetrahedron.wordpress.com)
- Sagemath 19 part 1: The spectrum of the Ricci operator for a monochromatic 4 – valent node dual to an equilateral tetrahedron (quantumtetrahedron.wordpress.com)