§34.5 Basic Properties: $6j$ Symbol

Permalink:: http://dlmf.nist.gov/34.5

§34.5(i) Special Cases
§34.5(ii) Symmetry
§34.5(iii) Recursion Relations
§34.5(iv) Orthogonality
§34.5(v) Generating Functions
§34.5(vi) Sums

§34.5(i) Special Cases

Notes:: See Edmonds (1974, pp. 98, 130–132), de-Shalit and Talmi (1963, p. 520).
Keywords:: $6j$ symbols
Permalink:: http://dlmf.nist.gov/34.5.i

In the following equations it is assumed that the triangle inequalities are satisfied and that $J$ is again defined by (34.3.4).

If any lower argument in a $6j$ symbol is 0, $\tfrac{1}{2}$ , or 1, then the $6j$ symbol has a simple algebraic form. Examples are provided by:

34.5.1 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 0&j_{3}&j_{2}\end{Bmatrix}=\frac{(-1)^{J}}{\left((2j_{2}+1)(2j_{3}+1)\right)^{{\frac{1}{2}}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E1
Encodings:: TeX, pMML, png

34.5.2 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ \frac{1}{2}&j_{{3}}-\frac{1}{2}&j_{{2}}+\frac{1}{2}\end{Bmatrix}=(-1)^{{J}}\left(\frac{(j_{1}+j_{3}-j_{2})(j_{1}+j_{2}-j_{3}+1)}{(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E2
Encodings:: TeX, pMML, png

34.5.3 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ \frac{1}{2}&j_{{3}}-\frac{1}{2}&j_{{2}}-\frac{1}{2}\end{Bmatrix}=(-1)^{{J}}\left(\frac{(j_{{2}}+j_{{3}}-j_{{1}})(j_{{1}}+j_{{2}}+j_{{3}}+1)}{2j_{{2}}(2j_{{2}}+1)2j_{{3}}(2j_{{3}}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E3
Encodings:: TeX, pMML, png

34.5.4 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}-1\end{Bmatrix}=(-1)^{J}\left(\frac{J(J+1)(J-2j_{1})(J-2j_{1}-1)}{(2j_{2}-1)2j_{2}(2j_{2}+1)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E4
Encodings:: TeX, pMML, png

34.5.5 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ 1&j_{{3}}-1&j_{{2}}\end{Bmatrix}=(-1)^{{J}}\left(\frac{2(J+1)(J-2j_{{1}})(J-2j_{{2}})(J-2j_{{3}}+1)}{2j_{{2}}(2j_{{2}}+1)(2j_{{2}}+2)(2j_{{3}}-1)2j_{{3}}(2j_{{3}}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E5
Encodings:: TeX, pMML, png

34.5.6 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ 1&j_{{3}}-1&j_{{2}}+1\end{Bmatrix}=(-1)^{{J}}\left(\frac{(J-2j_{{2}}-1)(J-2j_{{2}})(J-2j_{{3}}+1)(J-2j_{{3}}+2)}{(2j_{{2}}+1)(2j_{{2}}+2)(2j_{{2}}+3)(2j_{{3}}-1)2j_{{3}}(2j_{{3}}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E6
Encodings:: TeX, pMML, png

34.5.7 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ 1&j_{{3}}&j_{{2}}\end{Bmatrix}=(-1)^{{J+1}}\frac{2(j_{{2}}(j_{{2}}+1)+j_{{3}}(j_{{3}}+1)-j_{{1}}(j_{{1}}+1))}{\left(2j_{{2}}(2j_{{2}}+1)(2j_{{2}}+2)2j_{{3}}(2j_{{3}}+1)(2j_{{3}}+2)\right)^{{\frac{1}{2}}}}.$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E7
Encodings:: TeX, pMML, png

§34.5(ii) Symmetry

Notes:: See Edmonds (1974, pp. 94, 95).
Keywords:: Regge symmetries, $6j$ symbols
Permalink:: http://dlmf.nist.gov/34.5.ii

The $6j$ symbol is invariant under interchange of any two columns and also under interchange of the upper and lower arguments in each of any two columns, for example,

34.5.8 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}=\begin{Bmatrix}j_{{2}}&j_{{1}}&j_{{3}}\\ l_{{2}}&l_{{1}}&l_{{3}}\end{Bmatrix}=\begin{Bmatrix}j_{{1}}&l_{{2}}&l_{{3}}\\ l_{{1}}&j_{{2}}&j_{{3}}\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E8
Encodings:: TeX, pMML, png

Next,

34.5.9 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}=\begin{Bmatrix}j_{{1}}&\frac{1}{2}(j_{{2}}+l_{{2}}+j_{{3}}-l_{{3}})&\frac{1}{2}(j_{{2}}-l_{{2}}+j_{{3}}+l_{{3}})\\ l_{{1}}&\frac{1}{2}(j_{{2}}+l_{{2}}-j_{{3}}+l_{{3}})&\frac{1}{2}(-j_{{2}}+l_{{2}}+j_{{3}}+l_{{3}})\end{Bmatrix},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E9
Encodings:: TeX, pMML, png

34.5.10 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}=\begin{Bmatrix}\frac{1}{2}(j_{{2}}+l_{{2}}+j_{{3}}-l_{{3}})&\frac{1}{2}(j_{{1}}-l_{{1}}+j_{{3}}+l_{{3}})&\frac{1}{2}(j_{{1}}+l_{{1}}+j_{{2}}-l_{{2}})\\ \frac{1}{2}(j_{{2}}+l_{{2}}-j_{{3}}+l_{{3}})&\frac{1}{2}(-j_{{1}}+l_{{1}}+j_{{3}}+l_{{3}})&\frac{1}{2}(j_{{1}}+l_{{1}}-j_{{2}}+l_{{2}})\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E10
Encodings:: TeX, pMML, png

Equations (34.5.9) and (34.5.10) are called Regge symmetries. Additional symmetries are obtained by applying (34.5.8) to (34.5.9) and (34.5.10). See Srinivasa Rao and Rajeswari (1993, pp. 102–103) and references given there.

§34.5(iii) Recursion Relations

Keywords:: $6j$ symbols
Permalink:: http://dlmf.nist.gov/34.5.iii

In the following equation it is assumed that the triangle conditions are satisfied.

34.5.11 ${(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2}-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer, $l,l_{r}$ : nonnegative integer, $J_{r}$ , $L_{r}$ and $E(j)$
Permalink:: http://dlmf.nist.gov/34.5.E11
Encodings:: TeX, pMML, png

where

34.5.12

$J_{r}=j_{r}(j_{r}+1),$

$L_{r}=l_{r}(l_{r}+1),$

Symbols:: $j,j_{r}$ : nonnegative integer, $l,l_{r}$ : nonnegative integer, $r$ : nonnegative integer, $J_{r}$ and $L_{r}$
Permalink:: http://dlmf.nist.gov/34.5.E12
Encodings:: TeX, TeX, pMML, pMML, png, png

34.5.13 $E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l_{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{{\frac{1}{2}}}.$

Symbols:: $j,j_{r}$ : nonnegative integer, $l,l_{r}$ : nonnegative integer and $E(j)$
Permalink:: http://dlmf.nist.gov/34.5.E13
Encodings:: TeX, pMML, png

For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99).

§34.5(iv) Orthogonality

Notes:: See Edmonds (1974, p. 96).
Keywords:: $6j$ symbols
Permalink:: http://dlmf.nist.gov/34.5.iv

34.5.14 $\sum _{{j_{{3}}}}(2j_{{3}}+1)(2l_{{3}}+1)\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l^{{\prime}}_{{3}}\end{Bmatrix}=\delta _{{l_{3},l^{{\prime}}_{{3}}}}.$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E14
Encodings:: TeX, pMML, png

§34.5(v) Generating Functions

Keywords:: $6j$ symbols
Permalink:: http://dlmf.nist.gov/34.5.v

For generating functions for the $6j$ symbol see Biedenharn and van Dam (1965, p. 255, eq. (4.18)).

§34.5(vi) Sums

Notes:: See de-Shalit and Talmi (1963, pp. 517–518), Varshalovich et al. (1988, §9.8), Edmonds (1974, pp. 95–97), Dunlap and Judd (1975).
Keywords:: $6j$ symbols
Permalink:: http://dlmf.nist.gov/34.5.vi

34.5.15 $\sum _{{j}}(-1)^{{j+j^{{\prime}}+j^{{\prime\prime}}}}(2j+1)\begin{Bmatrix}j_{{1}}&j_{{2}}&j\\ j_{{3}}&j_{{4}}&j^{{\prime}}\end{Bmatrix}\begin{Bmatrix}j_{{1}}&j_{{2}}&j\\ j_{{4}}&j_{{3}}&j^{{\prime\prime}}\end{Bmatrix}=\begin{Bmatrix}j_{{1}}&j_{{4}}&j^{{\prime}}\\ j_{{2}}&j_{{3}}&j^{{\prime\prime}}\end{Bmatrix},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.5(vi), §34.9
Permalink:: http://dlmf.nist.gov/34.5.E15
Encodings:: TeX, pMML, png

34.5.16 $(-1)^{{j_{{1}}+j_{{2}}+j_{{3}}+j_{{1}}^{{\prime}}+j_{{2}}^{{\prime}}+l_{1}+l_{2}}}\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}\begin{Bmatrix}j_{{1}}^{{\prime}}&j_{{2}}^{{\prime}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}^{{\prime}}\end{Bmatrix}=\sum _{{j}}(-1)^{{l_{{3}}+l_{{3}}^{{\prime}}+j}}(2j+1)\begin{Bmatrix}j_{{1}}&j_{{1}}^{{\prime}}&j\\ j_{{2}}^{{\prime}}&j_{{2}}&j_{{3}}\end{Bmatrix}\begin{Bmatrix}l_{{3}}&l_{{3}}^{{\prime}}&j\\ j_{{1}}^{{\prime}}&j_{{1}}&l_{{2}}\end{Bmatrix}\begin{Bmatrix}l_{{3}}&l_{{3}}^{{\prime}}&j\\ j_{{2}}^{{\prime}}&j_{{2}}&l_{{1}}\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.5(vi)
Permalink:: http://dlmf.nist.gov/34.5.E16
Encodings:: TeX, pMML, png

Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the $6j$ symbol.

34.5.17 $\sum _{{j}}(2j+1)\begin{Bmatrix}j_{{1}}&j_{{2}}&j\\ j_{{1}}&j_{{2}}&j^{{\prime}}\end{Bmatrix}=(-1)^{{2(j_{{1}}+j_{{2}})}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E17
Encodings:: TeX, pMML, png

34.5.18 $\sum _{{j}}(-1)^{{j_{{1}}+j_{{2}}+j}}(2j+1)\begin{Bmatrix}j_{{1}}&j_{{2}}&j\\ j_{{2}}&j_{{1}}&j^{{\prime}}\end{Bmatrix}=\sqrt{(2j_{{1}}+1)(2j_{{2}}+1)}\,\delta _{{j^{{\prime}},0}},$

Symbols:: $\delta _{{j,k}}$ : Kronecker delta, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E18
Encodings:: TeX, pMML, png

34.5.19 $\sum _{{l}}\begin{Bmatrix}j_{{1}}&j_{{2}}&l\\ j_{{2}}&j_{{1}}&j\end{Bmatrix}=0,$ $2\mu-j$ odd, $\mu=\min(j_{{1}},j_{{2}})$ ,

Symbols:: $(a,b)$ : open interval, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E19
Encodings:: TeX, pMML, png

34.5.20 $\sum _{{l}}(-1)^{{l+j}}\begin{Bmatrix}j_{{1}}&j_{{2}}&l\\ j_{{1}}&j_{{2}}&j\end{Bmatrix}=\frac{(-1)^{{2\mu}}}{2j+1},$ $\mu=\min(j_{{1}},j_{{2}})$ ,

Symbols:: $(a,b)$ : open interval, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E20
Encodings:: TeX, pMML, png

34.5.21 $\sum _{{l}}(-1)^{{l+j+j_{1}+j_{2}}}\begin{Bmatrix}j_{{1}}&j_{{2}}&l\\ j_{{2}}&j_{{1}}&j\end{Bmatrix}=\frac{1}{2j+1}\left(\frac{(2j_{{1}}-j)!(2j_{{2}}+j+1)!}{(2j_{{2}}-j)!(2j_{{1}}+j+1)!}\right)^{{\frac{1}{2}}},$ $j_{{2}}\leq j_{{1}}$ ,

Symbols:: $!$ : $n!$ : factorial, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E21
Encodings:: TeX, pMML, png

34.5.22 $\sum _{{l}}(-1)^{{l+j+j_{1}+j_{2}}}\frac{1}{l(l+1)}\begin{Bmatrix}j_{{1}}&j_{{2}}&l\\ j_{{2}}&j_{{1}}&j\end{Bmatrix}=\frac{1}{j_{{1}}(j_{{1}}+1)-j_{{2}}(j_{{2}}+1)}\left(\frac{(2j_{{1}}-j)!(2j_{{2}}+j+1)!}{(2j_{{2}}-j)!(2j_{{1}}+j+1)!}\right)^{{\frac{1}{2}}},$ $j_{{2}}<j_{{1}}$ .

Symbols:: $!$ : $n!$ : factorial, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E22
Encodings:: TeX, pMML, png

34.5.23 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}=\sum _{{m^{{\prime}}_{{1}}m^{{\prime}}_{{2}}m^{{\prime}}_{{3}}}}(-1)^{{l_{{1}}+l_{{2}}+l_{{3}}+m^{{\prime}}_{{1}}+m^{{\prime}}_{{2}}+m^{{\prime}}_{{3}}}}\begin{pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{{\prime}}_{2}&-m^{{\prime}}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_{3}\\ -m^{{\prime}}_{1}&m_{2}&m^{{\prime}}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_{3}\\ m^{{\prime}}_{1}&-m^{{\prime}}_{2}&m_{3}\end{pmatrix}.$