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
formula_1
Recoupling of four angular momentum vectors.
Coupling of two angular momenta formula_2 and formula_3 is the construction of simultaneous eigenfunctions of formula_4 and formula_5, where formula_6, 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 formula_2, formula_3, formula_9, and formula_10 may be written as
Alternatively, one may first couple formula_2 and formula_9 to formula_14 and formula_3 and formula_10 to formula_17, before coupling formula_14 and formula_17 to formula_20:
Both sets of functions provide a complete, orthonormal basis for the space with dimension formula_22 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 (formula_24):
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 formula_27, 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 formula_31 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 ("j"1, "j"2, "j"3) 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 ("K"3,3), and the two distinct (non-isomorphic) 12-"j" symbols with the "Q"3 and Wagner graphs on 8 vertices.
Symmetry relations are generally representative of the automorphism group of these graphs.