Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications

The quantum theory of angular momentum and the associated Racah–Wigner algebra of the Lie group SU(2) have been widely used in many branches of theoretical and applied physics, chemical physics, and mathematical physics. This paper starts with an account of the basics of such a theory, which represents the most exhaustive framework in dealing with interacting many-angular momenta quantum systems. We then outline the essential features of this algebra, that can be encoded, for each fixed number N = (n + 1) of angular momentum variables, into a combinatorial object, the spin network graph, where vertices are associated with finite-dimensional, binary coupled Hilbert spaces while edges correspond to either phase or Racah transforms (implemented by 6j symbols) acting on states in such a way that the quantum transition amplitude between any pair of vertices is provided by a suitable 3nj symbol. Applications of such a combinatorial setting—both in fully quantum and in semiclassical regimes—are briefly discussed providing evidence of a unifying background structure.