Racah coefficient
Wigner's 6-j symbols were introduced by Eugene Paul Wigner in 1940 and published in 1965. They are defined as a sum over products of four Wigner 3-j symbols,
The summation is over all six mi allowed by the selection rules of the 3-j symbols.
They are closely related to the Racah W-coefficients, which are used for recoupling 3 angular momenta, although Wigner 6-j symbols have higher symmetry and therefore provide a more efficient means of storing the recoupling coefficients. Their relationship is given by:
The 6-j symbol is invariant under any permutation of the columns:
The 6-j symbol is also invariant if upper and lower arguments are interchanged in any two columns:
These equations reflect the 24 symmetry operations of the automorphism group that leave the associated tetrahedral Yutsis graph with 6 edges invariant: mirror operations that exchange two vertices and a swap an adjacent pair of edges.
The 6-j symbol
is zero unless j1, j2, and j3 satisfy triangle conditions, i.e.,
In combination with the symmetry relation for interchanging upper and lower arguments this shows that triangle conditions must also be satisfied for the triads (j1, j5, j6), (j4, j2, j6), and (j4, j5, j3). Furthermore, the sum of each of the elements of a triad must be an integer. Therefore, the members of each triad are either all integers or contain one integer and two half-integers.
When j6 = 0 the expression for the 6-j symbol is:
The triangular delta {j1 j2 j3} is equal to 1 when the triad (j1, j2, j3) satisfies the triangle conditions, and zero otherwise. The symmetry relations can be used to find the expression when another j is equal to zero.
The 6-j symbols satisfy this orthogonality relation:
A remarkable formula for the asymptotic behavior of the 6-j symbol was first conjectured by Ponzano and Regge and later proven by Roberts. The asymptotic formula applies when all six quantum numbers j1, ..., j6 are taken to be large and associates to the 6-j symbol the geometry of a tetrahedron. If the 6-j symbol is determined by the quantum numbers j1, ..., j6 the associated tetrahedron has edge lengths Ji = ji+1/2 (i=1,...,6) and the asymptotic formula is given by,
...
Wikipedia