.
Wigner 6-j symbols
Wigner's 6-j symbols were introduced by Eugene Paul Wigner in 1940, and published in 1965. They are defined by a sum over products of four 3jm symbols,
\( \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = \sum_{m_i} (-1)^S \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & -m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_5 & j_6\\ -m_1 & m_5 & m_6 \end{pmatrix} \begin{pmatrix} j_4 & j_5 & j_3\\ m_4 & -m_5 & m_3 \end{pmatrix} \begin{pmatrix} j_4 & j_2 & j_6\\ -m_4 & -m_2 & -m_6 \end{pmatrix} . \)
with phase \( S=\sum_{k=1}^6 (j_k-m_k) \). The summation is over all six mi, effectively confined by the selection rules of the 3jm symbols. They are related to Racah's W-coefficients by
\( \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = (-1)^{j_1+j_2+j_4+j_5}W(j_1j_2j_5j_4;j_3j_6). \)
They have higher symmetry than Racah's W-coefficients.
Symmetry relations
The 6-j symbol is invariant under the permutation of any two columns:
\( \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_2 & j_1 & j_3\\ j_5 & j_4 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_1 & j_3 & j_2\\ j_4 & j_6 & j_5 \end{Bmatrix} = \begin{Bmatrix} j_3 & j_2 & j_1\\ j_6 & j_5 & j_4 \end{Bmatrix}. \)
The 6-j symbol is also invariant if upper and lower arguments are interchanged in any two columns:
\( \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_4 & j_5 & j_3\\ j_1 & j_2 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_1 & j_5 & j_6\\ j_4 & j_2 & j_3 \end{Bmatrix} = \begin{Bmatrix} j_4 & j_2 & j_6\\ j_1 & j_5 & j_3 \end{Bmatrix}. \)
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
\( \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} \)
is zero unless j1, j2, and j3 satisfy triangle conditions, i.e.,
\( j_1 = |j_2-j_3|, \ldots, j_2+j_3 \)
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.
Special case
When j6 = 0 the expression for the 6-j symbol is:
\( \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & 0 \end{Bmatrix} = \frac{\delta_{j_2,j_4}\delta_{j_1,j_5}}{\sqrt{(2j_1+1)(2j_2+1)}} (-1)^{j_1+j_2+j_3}\{j_1,j_2,j_3\}. \)
The function {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.
Orthogonality relation
The 6-j symbols satisfy this orthogonality relation:
\( \sum_{j_3} (2j_3+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6' \end{Bmatrix} = \frac{\delta_{j_6^{}j_6'}}{2j_6+1} \{j_1,j_5,j_6\} \{j_4,j_2,j_6\}. \)
Asymptotics
A remarkable formula for the asymptotic behavior of the 6-j symbol was first conjectured by Ponzano and Regge[1] and later proven by Roberts.[2] 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,
\( \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} \sim \frac{1}{\sqrt{12 \pi |V|}} \cos{\left( \sum_{i=1}^{6} J_i \theta_i +\frac{\pi}{4}\right)}. \)
The notation is as follows: Each θi is the external dihedral angle about the edge Ji of the associated tetrahedron and the amplitude factor is expressed in terms of the volume, V, of this tetrahedron.
Mathematical interpretation
In representation theory, 6j-symbols are matrix coefficients of the associator isomorphism in a tensor category.[3] For example, if we are given three representations Vi, Vj, Vk of a group (or quantum group), one has a natural isomorphism
\( (V_i \otimes V_j) \otimes V_k \to V_i \otimes (V_j \otimes V_k) \)
of tensor product representations, induced by coassociativity of the corresponding bialgebra. One of the axioms defining a monoidal category is that associators satisfy a pentagon identity, which is equivalent to the Biedenharn-Elliot identity for 6j-symbols.
When a monoidal category is semisimple, we can restrict our attention to irreducible objects, and define multiplicity spaces
\( H_{i,j}^\ell = \operatorname{Hom}(V_{\ell}, V_i \otimes V_j) \)
so that tensor products are decomposed as:
\( V_i \otimes V_j = \bigoplus_\ell H_{i,j}^\ell V_\ell \)
where the sum is over all isomorphism classes of irreducible objects. Then:
\( (V_i \otimes V_j) \otimes V_k \cong \bigoplus_{\ell,m} H_{i,j}^\ell \otimes H_{\ell,k}^m \otimes V_m \qquad \text{while} \qquad V_i \otimes (V_j \otimes V_k) \cong \bigoplus_{m,n} H_{i,n}^m \otimes H_{j,k}^n \otimes V_m \)
The associativity isomorphism induces a vector space isomorphism
\( \Phi_{i,j}^{k,m}: \bigoplus_{\ell} H_{i,j}^\ell \otimes H_{\ell,k}^m \to \bigoplus_n H_{i,n}^m \otimes H_{j,k}^n \)
and the 6j symbols are defined as the component maps:
\( \begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix} = (\Phi_{i,j}^{k,m})_{\ell,n} \)
When the multiplicity spaces have canonical basis elements and dimension at most one (as in the case of SU(2) in the traditional setting), these component maps can be interpreted as numbers, and the 6j-symbols become ordinary matrix coefficients.
In abstract terms, the 6j-symbols are precisely the information that is lost when passing from a monoidal category to its Grothendieck group, since one can reconstruct a monoidal structure using the associator. For the case of representations of a finite group, the character table, together with its 6j-symbols, uniquely determines the group up to isomorphism, while the character table alone does not.
See also
Clebsch–Gordan coefficients
3-jm symbol
Racah W-coefficient
9-j symbol
Notes
Ponzano G and Regge T (1968). "Semiclassical Limit of Racah Coefficients". in Spectroscopy and Group Theoretical Methods in Physics: Amsterdam. pp. 1–58.
Roberts J (1998). "Classical 6j-symbols and the tetrahedron". Geometry and Topology 3: 21–66. doi:10.2140/gt.1999.3.21.
Etingof, P.; Gelaki S.; Nikshych D.; Ostrik V. (2009). Tensor Categories (PDF).
References
Biedenharn, L. C.; van Dam, H. (1965). Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers. New York: Academic Press. ISBN 0-12-096056-7.
Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 0-691-07912-9.
Condon, Edward U.; Shortley, G. H. (1970). "Chapter 3". The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 0-521-09209-4.
Maximon, Leonard C. (2010), "3j,6j,9j Symbols", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
Messiah, Albert (1981). Quantum Mechanics (Volume II) (12th ed.). New York: North Holland Publishing. ISBN 0-7204-0045-7.
Brink, D. M.; Satchler, G. R. (1993). "Chapter 2". Angular Momentum (3rd ed.). Oxford: Clarendon Press. ISBN 0-19-851759-9.
Zare, Richard N. (1988). "Chapter 2". Angular Momentum. New York: John Wiley. ISBN 0-471-85892-7.
Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading, Massachusetts: Addison-Wesley. ISBN 0-201-13507-8.
External links
Regge, T. (1959). "Simmetry Properties of Racah's Coefficients". Nuovo Cimento 11 (1): 116–117. doi:10.1007/BF02724914.
Stone, Anthony. "Wigner coefficient calculator". (Gives exact answer)
Simons, Frederik J. "Matlab software archive, the code SIXJ.M".
Volya, A. "Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator".
Plasma Laboratory of Weizmann Institute of Science. "369j-symbol calculator".
GNU scientific library. "Coupling coefficients".
Mathar, Richard J. "(Python implementation)". Java implementation
Johansson, H.T.; Forssén, C. "(WIGXJPF)". (accurate; C, fortran, python)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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