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

## Β§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 $\displaystyle\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}&j_{2}\end{Bmatrix}$ $\displaystyle=(-1)^{J+1}\frac{2(j_{2}(j_{2}+1)+j_{3}(j_{3}+1)-j_{1}(j_{1}+1))}% {\left(2j_{2}(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)(2j_{3}+2)\right)^{\frac{1}{2% }}}.$

## Β§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. 102β103) and references given there.

## Β§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 $\displaystyle J_{r}$ $\displaystyle=j_{r}(j_{r}+1),$ $\displaystyle L_{r}$ $\displaystyle=l_{r}(l_{r}+1),$ β 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 Β§34.5(iii), Β§34.5 and Ch.34
 34.5.13 $E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l% _{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{\frac{1}{2}}.$ β 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 Β§34.5(iii), Β§34.5 and Ch.34

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

## Β§34.5(iv) Orthogonality

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

## Β§34.5(v) Generating Functions

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

## Β§34.5(vi) Sums

 34.5.15 $\sum_{j}(-1)^{j+j^{\prime}+j^{\prime\prime}}(2j+1)\begin{Bmatrix}j_{1}&j_{2}&j% \\ j_{3}&j_{4}&j^{\prime}\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j\\ j_{4}&j_{3}&j^{\prime\prime}\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{4}&j^{\prime% }\\ j_{2}&j_{3}&j^{\prime\prime}\end{Bmatrix},$ β 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: Β§34.5(vi), Β§34.9 Permalink: http://dlmf.nist.gov/34.5.E15 Encodings: TeX, pMML, png See also: Annotations for Β§34.5(vi), Β§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).