On rotation distance between binary coupling trees and applications for 3nj-coefficients Original Research Article

Generalized recoupling coefficients or 3nj-coefficients for a Lie algebra (with su(2), the Lie algebra for the quantum theory of angular momentum, as generic example) can always be expressed as multiple sums over products of Racah coefficients (i.e. 6j-coefficients). In general there exist many such expressions; we say that such an expression is optimal if the number of Racah coefficients in such a product (and, correlated, the number of summation indices) is minimal. The problem of finding an optimal expression for a given 3nj-coefficient is equivalent to finding a shortest path in a graph Gn. The vertices of this graph Gn consist of binary coupling trees, representing the coupling schemes in the bra/kets of the 3nj-coefficients. This is the graph of rooted (unordered) binary trees with labelled leaves, and has order (2n − 1)!!. As the order increases so rapidly, finding a shortest path is computationally achievable only for n < 11. We present some mathematical tools to compute or estimate the length of such shortest paths between binary coupling trees. The diameter of Gn is determined explicitly up to n < 11, and it is shown to grow like n log(n). Thus for n large enough, the number of Racah coefficients in the expansion of a 3nj-coefficient is of order nlog(n). We also show that this problem in Racah—Wigner theory is equivalent to a problem in mathematical biology, where one is concerned with the quantitative comparison of classifications or dendrograms. From this context, some algorithms for approximating the shortest path can be deduced.