# 6-j symbol

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 *m*_{i} 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.^{[1]} Their relationship is given by:

## Symmetry relations[edit]

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 *j*_{1}, *j*_{2}, and *j*_{3} 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 (*j*_{1}, *j*_{5}, *j*_{6}), (*j*_{4}, *j*_{2}, *j*_{6}), and (*j*_{4}, *j*_{5}, *j*_{3}).
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[edit]

When *j*_{6} = 0 the expression for the 6-*j* symbol is:

The *triangular delta* {*j*_{1} *j*_{2} *j*_{3}} is equal to 1 when the triad (*j*_{1}, *j*_{2}, *j*_{3}) 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[edit]

The 6-*j* symbols satisfy this orthogonality relation:

## Asymptotics[edit]

A remarkable formula for the asymptotic behavior of the 6-*j* symbol was first conjectured by Ponzano and Regge^{[2]} and later proven by Roberts.^{[3]} The asymptotic formula applies when all six quantum numbers *j*_{1}, ..., *j*_{6} 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 *j*_{1}, ..., *j*_{6} the associated tetrahedron has edge lengths *J*_{i} = *j*_{i}+1/2 (i=1,...,6) and the asymptotic formula is given by,

The notation is as follows: Each θ_{i} is the external dihedral angle about the edge *J*_{i} of the associated tetrahedron and the amplitude factor is expressed in terms of the volume, *V*, of this tetrahedron.

## Mathematical interpretation[edit]

In representation theory, 6-*j* symbols are matrix coefficients of the associator isomorphism in a tensor category.^{[4]} For example, if we are given three representations *V*_{i}, *V*_{j}, *V*_{k} of a group (or quantum group), one has a natural isomorphism

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 6-*j* symbols.

When a monoidal category is semisimple, we can restrict our attention to irreducible objects, and define multiplicity spaces

so that tensor products are decomposed as:

where the sum is over all isomorphism classes of irreducible objects. Then:

The associativity isomorphism induces a vector space isomorphism

and the 6j symbols are defined as the component maps:

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 6-*j* symbols become ordinary matrix coefficients.

In abstract terms, the 6-*j* symbols are precisely the information that is lost when passing from a semisimple monoidal category to its Grothendieck ring, since one can reconstruct a monoidal structure using the associator. For the case of representations of a finite group, it is well-known that the character table alone (which determines the underlying abelian category and the Grothendieck ring structure) does not determine a group up to isomorphism, while the symmetric monoidal category structure does, by Tannaka-Krein duality. In particular, the two nonabelian groups of order 8 have equivalent abelian categories of representations and isomorphic Grothdendieck rings, but the 6-*j* symbols of their representation categories are distinct, meaning their representation categories are inequivalent as monoidal categories. Thus, the 6-*j* symbols give an intermediate level of information, that in fact uniquely determines the groups in many cases, such as when the group is odd order or simple.^{[5]}

## See also[edit]

## Notes[edit]

**^**Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients".*SIAM J. Sci. Comput*.**25**(4): 1416–1428. doi:10.1137/s1064827503422932.**^**Ponzano G and Regge T (1968). "Semiclassical Limit of Racah Coefficients". in Spectroscopy and Group Theoretical Methods in Physics: Amsterdam: 1–58. Cite journal requires`|journal=`

(help)**^**Roberts J (1999). "Classical 6j-symbols and the tetrahedron".*Geometry and Topology*.**3**: 21–66. arXiv:math-ph/9812013. doi:10.2140/gt.1999.3.21. S2CID 9678271.**^**Etingof, P.; Gelaki S.; Nikshych D.; Ostrik V. (2009).*Tensor Categories*(PDF).**^**Etingof, P.; Gelaki S. (2000). "Isocategorical Groups". arXiv:math/0007196.

## References[edit]

- 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. (eds.),
*NIST Handbook of Mathematical Functions*, Cambridge University Press, ISBN 978-0-521-19225-5, 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[edit]

- Regge, T. (1959). "Simmetry Properties of Racah's Coefficients".
*Nuovo Cimento*.**11**(1): 116–117. Bibcode:1959NCim...11..116R. doi:10.1007/BF02724914. S2CID 121333785. - 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". Archived from the original on 2012-12-20.
- Plasma Laboratory of Weizmann Institute of Science. "369j-symbol calculator".
- GNU scientific library. "Coupling coefficients".
- Johansson, H.T.; Forssén, C. "(WIGXJPF)". (accurate; C, fortran, python)
- Johansson, H.T. "(FASTWIGXJ)". (fast lookup, accurate; C, fortran)