9-j symbol


In physics, Wigner's 9-j symbols were introduced by Eugene Paul Wigner in 1937. They are related to recoupling coefficients in quantum mechanics involving four angular momenta

Recoupling of four angular momentum vectors

Coupling of two angular momenta and is the construction of simultaneous eigenfunctions of and, where, as explained in the article on Clebsch–Gordan coefficients.
Coupling of three angular momenta can be done in several ways, as explained in the article on Racah W-coefficients. Using the notation and techniques of that article, total angular momentum states that arise from coupling the angular momentum vectors,,, and may be written as
Alternatively, one may first couple and to and and to, before coupling and to :
Both sets of functions provide a complete, orthonormal basis for the space with dimension spanned by
Hence, the transformation between the two sets is unitary and the matrix elements of the transformation are given by the scalar products of the functions.
As in the case of the Racah W-coefficients the matrix elements are independent of the total angular momentum projection quantum number :

Symmetry relations

A 9-j symbol is invariant under reflection about either diagonal as well as even permutations of its rows or columns:
An odd permutation of rows or columns yields a phase factor, where
For example:

Reduction to 6j symbols

The 9-j symbols can be calculated as sums over triple-products of 6-j symbols where the summation extends over all admitted by the triangle conditions in the factors:

Special case

When the 9-j symbol is proportional to a 6-j symbol:

Orthogonality relation

The 9-j symbols satisfy this orthogonality relation:
The triangular delta is equal to 1 when the triad satisfies the triangle conditions, and zero otherwise.

3''n''-j symbols

The 6-j symbol is the first representative,, of -j symbols that are defined as sums of products of of Wigner's 3-jm coefficients. The sums are over all combinations of that the -j coefficients admit, i.e., which lead to non-vanishing contributions.
If each 3-jm factor is represented by a vertex and each j by an edge, these -j symbols can be mapped on certain 3-regular graphs with vertices and nodes. The 6-j symbol is associated with the K4 graph on 4 vertices, the 9-j symbol with the utility graph on 6 vertices, and the two distinct 12-j symbols with the Q3 and Wagner graphs on 8 vertices.
Symmetry relations are generally representative of the automorphism group of these graphs.