# §34.5 Basic Properties: $\mathit{6j}$ Symbol

## §34.5(i) Special Cases

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

 34.5.1 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 0&j_{3}&j_{2}\end{Bmatrix}$ $\displaystyle=\frac{(-1)^{J}}{\left((2j_{2}+1)(2j_{3}+1)\right)^{\frac{1}{2}}},$ 34.5.2 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}+\frac{1}{2}\end{Bmatrix}$ $\displaystyle=(-1)^{J}\left(\frac{(j_{1}+j_{3}-j_{2})(j_{1}+j_{2}-j_{3}+1)}{(2% j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},$ 34.5.3 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}-\frac{1}{2}\end{Bmatrix}$ $\displaystyle=(-1)^{J}\left(\frac{(j_{2}+j_{3}-j_{1})(j_{1}+j_{2}+j_{3}+1)}{2j% _{2}(2j_{2}+1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},$ 34.5.4 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}-1\end{Bmatrix}$ $\displaystyle=(-1)^{J}\left(\frac{J(J+1)(J-2j_{1})(J-2j_{1}-1)}{(2j_{2}-1)2j_{% 2}(2j_{2}+1)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},$ 34.5.5 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}\end{Bmatrix}$ $\displaystyle=(-1)^{J}\left(\frac{2(J+1)(J-2j_{1})(J-2j_{2})(J-2j_{3}+1)}{2j_{% 2}(2j_{2}+1)(2j_{2}+2)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},$ 34.5.6 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}+1\end{Bmatrix}$ $\displaystyle=(-1)^{J}\left(\frac{(J-2j_{2}-1)(J-2j_{2})(J-2j_{3}+1)(J-2j_{3}+% 2)}{(2j_{2}+1)(2j_{2}+2)(2j_{2}+3)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}% {2}},$ 34.5.7 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}&j_{2}\end{Bmatrix}$ $\displaystyle=(-1)^{J+1}\frac{2(j_{2}(j_{2}+1)+j_{3}(j_{3}+1)-j_{1}(j_{1}+1))}% {\left(2j_{2}(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)(2j_{3}+2)\right)^{\frac{1}{2% }}}.$

## §34.5(ii) Symmetry

The $\mathit{6j}$ 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 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\ l_{2}&l_{1}&l_{3}\end{Bmatrix}=\begin{Bmatrix}j_{1}&l_{2}&l_{3}\\ l_{1}&j_{2}&j_{3}\end{Bmatrix}.$

Next,

 34.5.9 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}$ $\displaystyle=\begin{Bmatrix}j_{1}&\frac{1}{2}(j_{2}+l_{2}+j_{3}-l_{3})&\frac{% 1}{2}(j_{2}-l_{2}+j_{3}+l_{3})\\ l_{1}&\frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})&\frac{1}{2}(-j_{2}+l_{2}+j_{3}+l_{3% })\end{Bmatrix},$ 34.5.10 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}$ $\displaystyle=\begin{Bmatrix}\frac{1}{2}(j_{2}+l_{2}+j_{3}-l_{3})&\frac{1}{2}(% j_{1}-l_{1}+j_{3}+l_{3})&\frac{1}{2}(j_{1}+l_{1}+j_{2}-l_{2})\\ \frac{1}{2}(j_{2}+l_{2}-j_{3}+l_{3})&\frac{1}{2}(-j_{1}+l_{1}+j_{3}+l_{3})&% \frac{1}{2}(j_{1}+l_{1}-j_{2}+l_{2})\end{Bmatrix}.$

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

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

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

where

 34.5.12 $\displaystyle J_{r}$ $\displaystyle=j_{r}(j_{r}+1),$ $\displaystyle L_{r}$ $\displaystyle=l_{r}(l_{r}+1),$ ⓘ Defines: $J_{r}$ (locally) and $L_{r}$ (locally) Symbols: $j,j_{r}$: non-negative integers or non-negative integers plus one half., $l,l_{r}$: non-negative integers or non-negative integers plus one half. and $r$: nonnegative integer Permalink: http://dlmf.nist.gov/34.5.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §34.5(iii), §34.5 and Ch.34
 34.5.13 $E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l% _{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{\frac{1}{2}}.$ ⓘ Defines: $E(j)$ (locally) Symbols: $j,j_{r}$: non-negative integers or non-negative integers plus one half. and $l,l_{r}$: non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.5.E13 Encodings: TeX, pMML, png See also: Annotations for §34.5(iii), §34.5 and Ch.34

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

## §34.5(iv) Orthogonality

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

## §34.5(v) Generating Functions

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

## §34.5(vi) Sums

 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: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$: $\mathit{6j}$ symbol and $j,j_{r}$: non-negative integers or non-negative integers plus one half. Referenced by: §34.5(vi), §34.9 Permalink: http://dlmf.nist.gov/34.5.E15 Encodings: TeX, pMML, png See also: Annotations for §34.5(vi), §34.5 and Ch.34
 34.5.16 $(-1)^{j_{1}+j_{2}+j_{3}+j_{1}^{\prime}+j_{2}^{\prime}+l_{1}+l_{2}}\begin{% Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}\begin{Bmatrix}j_{1}^{\prime}&j_{2}^{\prime}&j_{% 3}\\ l_{1}&l_{2}&l_{3}^{\prime}\end{Bmatrix}=\sum_{j}(-1)^{l_{3}+l_{3}^{\prime}+j}(% 2j+1)\begin{Bmatrix}j_{1}&j_{1}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&j_{3}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{1}^{\prime}&j_{1}&l_{2}\end{Bmatrix}\begin{Bmatrix}l_{3}&l_{3}^{\prime}&j\\ j_{2}^{\prime}&j_{2}&l_{1}\end{Bmatrix}.$

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

 34.5.17 $\displaystyle\sum_{j}(2j+1)\begin{Bmatrix}j_{1}&j_{2}&j\\ j_{1}&j_{2}&j^{\prime}\end{Bmatrix}$ $\displaystyle=(-1)^{2(j_{1}+j_{2})},$ 34.5.18 $\displaystyle\sum_{j}(-1)^{j_{1}+j_{2}+j}(2j+1)\begin{Bmatrix}j_{1}&j_{2}&j\\ j_{2}&j_{1}&j^{\prime}\end{Bmatrix}$ $\displaystyle=\sqrt{(2j_{1}+1)(2j_{2}+1)}\,\delta_{j^{\prime},0},$ 34.5.19 $\displaystyle\sum_{l}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{2}&j_{1}&j\end{Bmatrix}$ $\displaystyle=0,$ $2\mu-j$ odd, $\mu=\min(j_{1},j_{2})$, 34.5.20 $\displaystyle\sum_{l}(-1)^{l+j}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{1}&j_{2}&j\end{Bmatrix}$ $\displaystyle=\frac{(-1)^{2\mu}}{2j+1},$ $\mu=\min(j_{1},j_{2})$,
 34.5.21 $\displaystyle\sum_{l}(-1)^{l+j+j_{1}+j_{2}}\begin{Bmatrix}j_{1}&j_{2}&l\\ j_{2}&j_{1}&j\end{Bmatrix}$ $\displaystyle=\frac{1}{2j+1}\left(\frac{(2j_{1}-j)!(2j_{2}+j+1)!}{(2j_{2}-j)!(% 2j_{1}+j+1)!}\right)^{\frac{1}{2}},$ $j_{2}\leq j_{1}$, 34.5.22 $\displaystyle\sum_{l}(-1)^{l+j+j_{1}+j_{2}}\frac{1}{l(l+1)}\begin{Bmatrix}j_{1% }&j_{2}&l\\ j_{2}&j_{1}&j\end{Bmatrix}$ $\displaystyle=\frac{1}{j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}\left(\frac{(2j_{1}-j)!(2% j_{2}+j+1)!}{(2j_{2}-j)!(2j_{1}+j+1)!}\right)^{\frac{1}{2}},$ $j_{2}.
 34.5.23 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\sum_{m^{\prime}_{1}m^{\prime}_{2}m^{\prime}_{3% }}(-1)^{l_{1}+l_{2}+l_{3}+m^{\prime}_{1}+m^{\prime}_{2}+m^{\prime}_{3}}\begin{% pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix}.$

Equation (34.5.23) can be regarded as an alternative definition of the $\mathit{6j}$ symbol.

For other sums see Ginocchio (1991).