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{{Short description|Sums in quantum mathematics}}

[[File:Jucys diagram for Wigner 6-j symbol.svg|thumb|[[Jucys diagram]] for the Wigner 6-''j'' symbol. The plus sign on the nodes indicates an anticlockwise reading of its surrounding lines. Due to its symmetries, there are many ways in which the diagram can be drawn. An equivalent configuration can be created by taking its mirror image and thus changing the pluses to minuses.]]

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,
:<math>
\begin{align}
\begin{Bmatrix}
j_1 & j_2 & j_3\\
j_4 & j_5 & j_6
\end{Bmatrix}
&= \sum_{m_1, \dots, m_6} (-1)^{\sum_{k = 1}^6 (j_k - m_k)}
\begin{pmatrix}
j_1 & j_2 & j_3\\
-m_1 & -m_2 & -m_3
\end{pmatrix}\times\\
&\times
\begin{pmatrix}
j_1 & j_5 & j_6\\
m_1 & -m_5 & m_6
\end{pmatrix}
\begin{pmatrix}
j_4 & j_2 & j_6\\
m_4 & m_2 & -m_6
\end{pmatrix}
\begin{pmatrix}
j_4 & j_5 & j_3\\
-m_4 & m_5 & m_3
\end{pmatrix}
.
\end{align}
</math>
The summation is over all six {{math|''m''''i''}} allowed by the selection rules of the 3-''j'' symbols.

They are closely related to the [[Racah W-coefficient]]s, 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.<ref>{{Cite journal |first1=J. |last1=Rasch |first2=A. C. H. |last2=Yu |title=Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients |journal=SIAM J. Sci. Comput. |volume=25 |issue=4 |year=2003 |pages=1416–1428 |doi=10.1137/s1064827503422932}}</ref> Their relationship is given by:
:<math>
\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_1 j_2 j_5 j_4; j_3 j_6).
</math>

==Symmetry relations==
The 6-''j'' symbol is invariant under any permutation of the columns:
:<math>
\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}
= \cdots
</math>
The 6-''j'' symbol is also invariant if upper and lower arguments
are interchanged in any two columns:
:<math>
\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}.
</math>
These equations reflect the 24 symmetry operations of the [[Graph automorphism|automorphism group]] that leave the associated [[Table of simple cubic graphs#4 nodes|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
:<math>
\begin{Bmatrix}
j_1 & j_2 & j_3\\
j_4 & j_5 & j_6
\end{Bmatrix}
</math>
is zero unless ''j''1, ''j''2, and ''j''3 satisfy triangle conditions,
i.e.,
:<math>
j_1 = |j_2-j_3|, \ldots, j_2+j_3
</math>
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 the elements of each triad must be an integer. Therefore, the members of each triad are either all integers or contain one integer and two half-integers.

==Special cases==
When ''j''6 = 0 the expression for the 6-''j'' symbol is:
:<math>
\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} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix}.
</math>
The ''triangular delta'' {{math|{''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.

Values for ''j''6 = e = 0, 1/2, 1, 3/2 & 2 can be straightforwardly obtained from the Racah W-coefficients (Brink & Satchler 1994, Table 4, ''j''6 = e = 0, 1/2, & 1; Biedenharn, Blatt, & Rose, 1952, ''j''6 = e = 0, 1/2, 1, 3/2 & 2. Values for ''j''6 = 1/2 and 1 are given below. The formulae for the recouplings for other values of ''j''6 can be easily inferred from these by using the symmetry of the 6''j''-symbols and appropriate substitution.
:<math>
\begin{Bmatrix}
j_1 & j_2 & j_3\\
j_2+\frac{1}{2} & j_1+\frac{1}{2} & \frac{1}{2}
\end{Bmatrix}
= (-1)^{j_1+j_2+j_3+1} \left[ \frac{(j_1 + j_2 + j_3+2)(j_1 + j_2 - j_3+1)}{(2j_1+1)(2j_1+2)(2j_2+1)(2j_2+2)} \right]^{1/2}
</math>

:<math>
\begin{Bmatrix}
j_1 & j_2 & j_3\\
j_2+\frac{1}{2} & j_1-\frac{1}{2} & \frac{1}{2}
\end{Bmatrix}
= (-1)^{j_1+j_2+j_3} \left[ \frac{(j_3 + j_1 - j_2)(j_2 + j_3 - j_1+1)}{(2j_1)(2j_1+1)(2j_2+1)(2j_2+2)} \right]^{1/2}
</math>

:<math>
\begin{Bmatrix}
j_1 & j_2 & j_3\\
j_2-1 & j_1-1 & 1
\end{Bmatrix}
= (-1)^{j_1+j_2+j_3} \left[ \frac{(j_1 + j_2 + j_3)(j_1 + j_2 +j_3+1)(j_1+j_2-j_3)(j_1+j_2-j_3-1)}{(2j_1-1)(2j_1)(2j_1+1)(2j_2-1)(2j_2)(2j_2+1)} \right]^{1/2}
</math>

:<math>
\begin{Bmatrix}
j_1 & j_2 & j_3\\
j_2-1 & j_1 & 1
\end{Bmatrix}
= (-1)^{j_1+j_2+j_3} \left[ \frac{2(j_1 + j_2 + j_3+1)(j_1 + j_2-j_3)(j_2+j_3-j_1)(j_1-j_2+j_3+1)}{(2j_1)(2j_1+1)(2j_1+2)(2j_2-1)(2j_2)(2j_2+1)} \right]^{1/2}
</math>

:<math>
\begin{Bmatrix}
j_1 & j_2 & j_3\\
j_2+1 & j_1-1 & 1
\end{Bmatrix}
= (-1)^{j_1+j_2+j_3} \left[ \frac{(j_1-j_2+j_3-1)(j_1-j_2+j_3)(j_2+j_3-j_1+1)(j_2+j_3-j_1+2)}{(2j_1-1)(2j_1)(2j_1+1)(2j_2+1)(2j_2+2)(2j_2+3)} \right]^{1/2}
</math>

:<math>
\begin{Bmatrix}
j_1 & j_2 & j_3\\
j_2 & j_1 & 1
\end{Bmatrix}
= (-1)^{j_1+j_2+j_3+1} \frac{j_1(j_1+1) + j_2(j_2+1) - j_3(j_3+1)}{[(2j_1)(2j_1+1)(2j_1+2)(j_2)(j_2+1)(2j_2+1)]^{1/2}}
</math>

In practice, rather than using tables of 6j-symbols, one would make use of the available calculators and computer codes listed under External Links below, or, for specific arguments, look them up in a set of tables (Varshalovic, Moskalev, & Khersonskii 1988, Chapter 9).

==Orthogonality relation==
The 6-''j'' symbols satisfy this orthogonality relation:
:<math>
\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} \begin{Bmatrix} j_1 & j_5 & j_6 \end{Bmatrix} \begin{Bmatrix} j_4 & j_2 & j_6 \end{Bmatrix}.
</math>

==Asymptotics==
A remarkable formula for the asymptotic behavior of the 6-''j'' symbol was first conjectured by Ponzano and Regge<ref>{{cite book |last1=Ponzano |first1=G. |last2=Regge |first2=T.|chapter=Semiclassical Limit of Racah Coefficients|year=1968|pages=1–58 |title=Spectroscopy and Group Theoretical Methods in Physics |isbn=978-0-444-10147-1 |publisher=Elsevier}}</ref> and later proven by Roberts.<ref>{{cite journal|last=Roberts J|title=Classical 6j-symbols and the tetrahedron|year=1999|journal=Geometry and Topology|pages=21–66|volume=3|doi=10.2140/gt.1999.3.21|arxiv=math-ph/9812013|s2cid=9678271}}</ref> 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,
:<math>
\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)}.
</math>
The notation is as follows: Each θi is the external [[Dihedral angle|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==

In [[Representation theory|representation theory]], 6-''j'' symbols are matrix coefficients of the associator isomorphism in a tensor category.<ref>{{cite book |last1=Etingof |first1=P. |last2=Gelaki |first2=S.|last3=Nikshych |first3=D.|last4=Ostrik |first4=V. |title=Tensor Categories. Lecture notes for MIT 18.769 |url=http://www-math.mit.edu/~etingof/tenscat1.pdf |year=2009 }}</ref> For example, if we are given three representations ''V''i, ''V''j, ''V''k of a group (or [[Quantum group|quantum group]]), one has a natural isomorphism
:<math>(V_i \otimes V_j) \otimes V_k \to V_i \otimes (V_j \otimes V_k)</math>
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
:<math>H_{i,j}^\ell = \operatorname{Hom}(V_{\ell}, V_i \otimes V_j)</math>
so that tensor products are decomposed as:
:<math>V_i \otimes V_j = \bigoplus_\ell H_{i,j}^\ell \otimes V_\ell</math>
where the sum is over all isomorphism classes of irreducible objects. Then:
:<math>(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</math>
The associativity isomorphism induces a [[Vector space|vector space]] isomorphism
:<math>\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</math>
and the 6j symbols are defined as the component maps:
:<math>
\begin{Bmatrix}
i & j & \ell\\
k & m & n
\end{Bmatrix}
= (\Phi_{i,j}^{k,m})_{\ell,n}</math>
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 group|Grothendieck ring]], since one can reconstruct a monoidal structure using the associator. For the case of representations of a [[Finite group|finite group]], it is well known that the [[Character table|character table]] alone (which determines the underlying [[Abelian category|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.<ref>{{cite journal |last1=Etingof |first1=P. |last2=Gelaki |first2=S. |title=Isocategorical Groups |journal=International Mathematics Research Notices |volume=2001 |issue=2 |pages=59–76 |date=2001 |doi=10.1155/S1073792801000046 |citeseerx=10.1.1.239.6293 |arxiv=math/0007196 |doi-access= }}</ref>

==See also==

* [[Physics:Clebsch–Gordan coefficients|Clebsch–Gordan coefficients]]
* [[Physics:3-j symbol|3-j symbol]]
* [[Racah W-coefficient]]
* [[Physics:9-j symbol|9-j symbol]]
* [[Representations of classical Lie groups]]

==Notes==
{{Reflist}}

==References==

* {{cite journal
|journal=Reviews of Modern Physics
|first1=L. C.
|last1=Biedenharn
|first2=J. M.
|last2=Blatt
|first3=M. E.
|last3=Rose
|title=Some Properties of the Racah and Associated Coefficients
|volume=24
|issue=4
|year=1952
|doi=10.1103/RevModPhys.24.249
|pages=249–257
|bibcode=1952RvMP...24..249B
}}
* {{cite book |last1= Biedenharn |first1= L. C. |authorlink=Lawrence Biedenharn |last2=van Dam |first2=H.
|title= Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers
|year= 1965 |publisher= [[Company:Academic Press|Academic Press]] |isbn= 0-12-096056-7}}
* {{cite book |last= Biedenharn |first= L. C. |author2= Louck, J. D. |title= Angular Momentum in Quantum Physics
|year= 1981 |publisher= Addison-Wesley |isbn= 0-201-13507-8 }}
* {{cite book |last= Brink |first= D. M. |author2= Satchler, G. R. |title= Angular Momentum |year= 1993 |edition= 3rd |publisher= Clarendon Press |isbn= 0-19-851759-9 |chapter= 2. Representations of the Rotation Group |chapter-url-access= registration |chapter-url= https://archive.org/details/angularmomentum0000brin |url-access= registration |url= https://archive.org/details/angularmomentum0000brin }}
* {{cite book |last= Brink |first= D. M. |author2= Satchler, G. R. |title= Angular Momentum |year= 1994 |edition= 3rd |publisher= Clarendon Press |isbn= 0-19-851759-9 |chapter= 3. Coupling Angular Momentum Vectors and Transformation Theory }}
* {{cite book |last1= Condon |first1= Edward U. |last2= Shortley |first2=G. H. |title= The Theory of Atomic Spectra |chapter-url= https://archive.org/details/in.ernet.dli.2015.212979 |year= 1970
|publisher= Cambridge University Press |isbn= 0-521-09209-4 |chapter= 3. Angular Momentum}}
* {{cite book |last= Edmonds |first= A. R. |title= Angular Momentum in Quantum Mechanics |date=January 8, 1996 |orig-year= 1957 |publisher= [[Princeton University Press]] |isbn= 9780691025896 |url-access= registration |url= https://archive.org/details/angularmomentumi0000edmo }}
*{{dlmf|id=34 |title=3j,6j,9j Symbols|first=Leonard C.|last= Maximon}}
* {{cite book |last= Messiah |first= Albert |authorlink=Albert Messiah |title= Quantum Mechanics |volume=II |year= 1981 | edition= 12th
|publisher= [[Company:Elsevier|North Holland Publishing]] |isbn= 0-7204-0045-7}}
* {{cite book
|first1=Dmitrii Aleksandrovich
|last1=Varshalovic
|first2=Anatoli Nikolaevitch
|last2=Moskalev
|first3=Valerij Kel'manovic
|last3=Khersonskii
|title=Quantum Theory of Angular Momentum |year= 1988 |publisher= [[Company:World Scientific|World Scientific]] |isbn= 9-971-50996-2 }}
* {{cite book |last= Zare |first= Richard N. |title= Angular Momentum |year=1988
|publisher= [[Company:John Wiley & Sons|Wiley]] |isbn= 0-471-85892-7 |chapter= 2. Coupling of two Angular Momentum Vectors }}

==External links==

* {{cite web
|first1=Joey
|last1=Dumont
|title=wignerSymbols
|website=[[GitHub]]
|url=https://github.com/joeydumont/wignerSymbols
}} (accurate; C++)
* {{ cite web
|url=https://www.gnu.org/software/gsl/doc/html/specfunc.html#coupling-coefficients
|title=Coupling coefficients
|author=GNU scientific library
}}
* {{cite web
|first1=Richard
|last1=Holt
|title=Wigner 6j Angular Momentum Coupling Coefficient
|url=https://www.mathworks.com/matlabcentral/fileexchange/24258-wigner-6j-angular-momentum-coupling-coefficient
}} (accurate; Matlab)
* {{cite web
|first1=H.T.
|last1=Johansson
|first2=C.
|last2=Forssén
|title=(WIGXJPF)
|url=http://fy.chalmers.se/subatom/wigxjpf/
}} (accurate; C, fortran, python)
* {{cite web
|first1=H.T.
|last1=Johansson
|title=(FASTWIGXJ)
|url=http://fy.chalmers.se/subatom/fastwigxj/
}} (fast lookup, accurate; C, fortran)
* {{cite web
|first1=R. J.
|last1=Mathar
|title=Table of 6j Symbols
|website=[[GitHub]]
|url= https://github.com/rmathar/Clebsch-Gordan/blob/main/6jSymb
}} (accurate; PARI/GP)
* {{cite web
|url=http://plasma-gate.weizmann.ac.il/369j.html
|title=369j-symbol calculator
|author=Plasma Laboratory of Weizmann Institute of Science
}}
* {{cite journal
|journal=Nuovo Cimento
|first1=T.
|last1=Regge
|title=Simmetry Properties of Racah's Coefficients
|volume=11
|issue=1
|year=1959
|doi=10.1007/BF02724914
|pages=116–7
|bibcode=1959NCim...11..116R
|s2cid=121333785
}}
* {{cite web |url=http://geoweb.princeton.edu/people/simons/software.html |first1=Frederik J.|last1=Simons|title=Matlab software archive, the code SIXJ.M}}
* {{cite web
|first1=Anthony
|last1=Stone
|url=http://www-stone.ch.cam.ac.uk/wigner.shtml
|title=Wigner coefficient calculator
}} (Gives exact answer)
* {{cite web
|first1 = A
|last1 = Volya
|url = http://www.volya.net/vc
|archive-url = https://archive.today/20121220081850/http://www.volya.net/vc
|url-status = dead
|archive-date = 2012-12-20
|title = Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator
}}
* {{ cite web
|url=https://www.sympy.org/en/index.html
|title=A Python Library for Symbolic Mathematics
|author=SymPy
}} (accurate; python)
* {{ cite web
|url=https://www.wolframalpha.com/input?i=wigner+6j
|title=WolframAlpha Wigner 6j Calculator
|author=WolframAlpha
}} (accurate)
??
[[Category:Rotational symmetry]]
[[Category:Representation theory of Lie groups]]
[[Category:Quantum mechanics]]

{{Sourceattribution|6-j symbol}}

Physics:6-j symbol - Revision history

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