In physics, Wigner's 9-j symbols were introduced by Eugene Wigner in 1937. They are related to recoupling coefficients in quantum mechanics involving four angular momenta:
( 2 j 3 + 1 ) ( 2 j 6 + 1 ) ( 2 j 7 + 1 ) ( 2 j 8 + 1 ) { j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } {\displaystyle {\sqrt {(2j_{3}+1)(2j_{6}+1)(2j_{7}+1)(2j_{8}+1)}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}}
= ⟨ ( ( j 1 j 2 ) j 3 , ( j 4 j 5 ) j 6 ) j 9 | ( ( j 1 j 4 ) j 7 , ( j 2 j 5 ) j 8 ) j 9 ⟩ . {\displaystyle =\langle ((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}|((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}\rangle .}
Recoupling of four angular momentum vectors Coupling of two angular momenta j 1 {\displaystyle \mathbf {j} _{1}} and j 2 {\displaystyle \mathbf {j} _{2}} is the construction of simultaneous eigenfunctions of J 2 {\displaystyle \mathbf {J} ^{2}} and J z {\displaystyle J_{z}} , where J = j 1 + j 2 {\displaystyle \mathbf {J} =\mathbf {j} _{1}+\mathbf {j} _{2}} , 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 j 1 {\displaystyle \mathbf {j} _{1}} , j 2 {\displaystyle \mathbf {j} _{2}} , j 4 {\displaystyle \mathbf {j} _{4}} , and j 5 {\displaystyle \mathbf {j} _{5}} may be written as
| ( ( j 1 j 2 ) j 3 , ( j 4 j 5 ) j 6 ) j 9 m 9 ⟩ . {\displaystyle |((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}m_{9}\rangle .}
Alternatively, one may first couple j 1 {\displaystyle \mathbf {j} _{1}} and j 4 {\displaystyle \mathbf {j} _{4}} to j 7 {\displaystyle \mathbf {j} _{7}} and j 2 {\displaystyle \mathbf {j} _{2}} and j 5 {\displaystyle \mathbf {j} _{5}} to j 8 {\displaystyle \mathbf {j} _{8}} , before coupling j 7 {\displaystyle \mathbf {j} _{7}} and j 8 {\displaystyle \mathbf {j} _{8}} to j 9 {\displaystyle \mathbf {j} _{9}} :
| ( ( j 1 j 4 ) j 7 , ( j 2 j 5 ) j 8 ) j 9 m 9 ⟩ . {\displaystyle |((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}m_{9}\rangle .}
Both sets of functions provide a complete, orthonormal basis for the space with dimension ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) ( 2 j 4 + 1 ) ( 2 j 5 + 1 ) {\displaystyle (2j_{1}+1)(2j_{2}+1)(2j_{4}+1)(2j_{5}+1)} spanned by
| j 1 m 1 ⟩ | j 2 m 2 ⟩ | j 4 m 4 ⟩ | j 5 m 5 ⟩ , m 1 = − j 1 , … , j 1 ; m 2 = − j 2 , … , j 2 ; m 4 = − j 4 , … , j 4 ; m 5 = − j 5 , … , j 5 . {\displaystyle |j_{1}m_{1}\rangle |j_{2}m_{2}\rangle |j_{4}m_{4}\rangle |j_{5}m_{5}\rangle ,\;\;m_{1}=-j_{1},\ldots ,j_{1};\;\;m_{2}=-j_{2},\ldots ,j_{2};\;\;m_{4}=-j_{4},\ldots ,j_{4};\;\;m_{5}=-j_{5},\ldots ,j_{5}.}
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 ( m 9 {\displaystyle m_{9}} ):
| ( ( j 1 j 4 ) j 7 , ( j 2 j 5 ) j 8 ) j 9 m 9 ⟩ = ∑ j 3 ∑ j 6 | ( ( j 1 j 2 ) j 3 , ( j 4 j 5 ) j 6 ) j 9 m 9 ⟩ ⟨ ( ( j 1 j 2 ) j 3 , ( j 4 j 5 ) j 6 ) j 9 | ( ( j 1 j 4 ) j 7 , ( j 2 j 5 ) j 8 ) j 9 ⟩ . {\displaystyle |((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}m_{9}\rangle =\sum _{j_{3}}\sum _{j_{6}}|((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}m_{9}\rangle \langle ((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}|((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}\rangle .}
Symmetry relations A 9-j symbol is invariant under reflection about either diagonal as well as even permutations of its rows or columns:
{ j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = { j 1 j 4 j 7 j 2 j 5 j 8 j 3 j 6 j 9 } = { j 9 j 6 j 3 j 8 j 5 j 2 j 7 j 4 j 1 } = { j 7 j 4 j 1 j 9 j 6 j 3 j 8 j 5 j 2 } . {\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{4}&j_{7}\\j_{2}&j_{5}&j_{8}\\j_{3}&j_{6}&j_{9}\end{Bmatrix}}={\begin{Bmatrix}j_{9}&j_{6}&j_{3}\\j_{8}&j_{5}&j_{2}\\j_{7}&j_{4}&j_{1}\end{Bmatrix}}={\begin{Bmatrix}j_{7}&j_{4}&j_{1}\\j_{9}&j_{6}&j_{3}\\j_{8}&j_{5}&j_{2}\end{Bmatrix}}.}
An odd permutation of rows or columns yields a phase factor ( − 1 ) S {\displaystyle (-1)^{S}} , where
S = ∑ i = 1 9 j i . {\displaystyle S=\sum _{i=1}^{9}j_{i}.}
For example:
{ j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = ( − 1 ) S { j 4 j 5 j 6 j 1 j 2 j 3 j 7 j 8 j 9 } = ( − 1 ) S { j 2 j 1 j 3 j 5 j 4 j 6 j 8 j 7 j 9 } . {\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=(-1)^{S}{\begin{Bmatrix}j_{4}&j_{5}&j_{6}\\j_{1}&j_{2}&j_{3}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=(-1)^{S}{\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\j_{5}&j_{4}&j_{6}\\j_{8}&j_{7}&j_{9}\end{Bmatrix}}.}
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 x admitted by the triangle conditions in the factors:
{ j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 } = ∑ x ( − 1 ) 2 x ( 2 x + 1 ) { j 1 j 4 j 7 j 8 j 9 x } { j 2 j 5 j 8 j 4 x j 6 } { j 3 j 6 j 9 x j 1 j 2 } {\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=\sum _{x}(-1)^{2x}(2x+1){\begin{Bmatrix}j_{1}&j_{4}&j_{7}\\j_{8}&j_{9}&x\end{Bmatrix}}{\begin{Bmatrix}j_{2}&j_{5}&j_{8}\\j_{4}&x&j_{6}\end{Bmatrix}}{\begin{Bmatrix}j_{3}&j_{6}&j_{9}\\x&j_{1}&j_{2}\end{Bmatrix}}} .
Special case When j 9 = 0 {\displaystyle j_{9}=0} the 9-j symbol is proportional to a 6-j symbol:
{ j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 0 } = δ j 3 , j 6 δ j 7 , j 8 ( 2 j 3 + 1 ) ( 2 j 7 + 1 ) ( − 1 ) j 2 + j 3 + j 4 + j 7 { j 1 j 2 j 3 j 5 j 4 j 7 } . {\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&0\end{Bmatrix}}={\frac {\delta _{j_{3},j_{6}}\delta _{j_{7},j_{8}}}{\sqrt {(2j_{3}+1)(2j_{7}+1)}}}(-1)^{j_{2}+j_{3}+j_{4}+j_{7}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{5}&j_{4}&j_{7}\end{Bmatrix}}.}
Orthogonality relation The 9-j symbols satisfy this orthogonality relation:
∑ j 7 j 8 ( 2 j 7 + 1 ) ( 2 j 8 + 1 ) { j 1 j 2 j 3 j 4 j 5 j 6