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

Abstract

Generalized recoupling coefficients or 3 nj-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. 6 j-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 3 nj-coefficient is equivalent to finding a shortest path in a graph G n . The vertices of this graph G n consist of binary coupling trees, representing the coupling schemes in the bra/kets of the 3 nj-coefficients. This is the graph of rooted (unordered) binary trees with labelled leaves, and has order (2 n − 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 G n 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 3 nj-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.