Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations

The basic ingredients of the quantum theory of orbital and spin angular momentum (vector coefficients, 3nj symbols) encounter continuing relevance in wide areas beyond the traditional ones (molecular, atomic and nuclear spectroscopies and dynamics). This paper offers insight on the connection at the most elementary of levels with the diagrammatic approaches to projective geometry. In particular here we exhibit how the Fano, Desargues and related incidence configurations emerge in the Racah and in the Biedenharn-Elliott identities, corresponding respectively to the hexagonal and pentagonal relationships that provide the basis for the construction of 3nj symbols and of spin networks. It is shown that the treatment, although mostly confined to the quadrangulation of the real projective plane, permits however the introduction of networks involving seven and ten spins, and preludes to developments towards computational and asymptotic approaches for quantum and semi-classical applications to spectroscopy and dynamics.