Spherical and Hyperbolic Spin Networks: The q-extensions of Wigner-Racah 6j Coefficients and General Orthogonal Discrete Basis Sets in Applied Quantum Mechanics

Discrete basis sets continue to have an important role in mathematics and quantum mechanics. Racah recoupling coefficients or their closely related Wigner 6j symbols form a remarkably rich source of such functions, and now their properties are well understood for Euclidean space where \(q = 1\). Here we report a unified treatment of their q-extensions to non-Euclidean spaces: hyperbolic, for real q different from 1, and spherical, for \(q = r^{th}\) root of unity. We calculate the non-Euclidean coefficients as the eigenvectors of a real symmetric tridiagonal matrix. The eigenvectors form a discrete ortho-normal basis set, and the eigenvalues can be interpreted as energy levels. We provide extensive numerical results, and also show the Neville volume formula for a tetrahedron appears to be valid for both hyperbolic and spherical cases in the semiclassical limit. This q-extended volume is used to scale the magnitude of the eigenvectors in a familiar fashion. We also determine for spherical space that the radius, r, has a sharp minimum value to support all x,y given by triangular relations for a quadrilateral (not necessarily planar) with side lengths a, b, c, d and forming a tetrahedron. The ranges of x and y are truncated for \(r < a+b+c+d+2\).

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