Sums of $6j$ symbols -- reference request

Question

In the book Quantum Theory of Angular Momentum by Varshalovich, Moskalev and Khersonskii, page 305 lists many formulas for sums involving Wigner $6j$ symbols, such as $$\sum\limits_X (-1)^{p+q+X}(2X+1) \left\{\begin{matrix} a & b & X \\ c & d & p\end{matrix}\right\} \left\{\begin{matrix} a & b & X \\ d & c & q\end{matrix}\right\} = \left\{\begin{matrix} a & c & q \\ b & d & p\end{matrix}\right\}.$$ I'm looking for a reference containing a proof of these formulas (or even just the one I singled out above). This book doesn't give a specific reference for these, as far as I can tell.

Answer 1

There's an illuminating discussion of this in

Edmonds, A.R. Angular momentum in quantum mechanics. Princeton University Press, 1957.

The key is to recognize that the $6j$ are, up to a normalization, elements of a unitary matrix because they arise as the overlap of the same state expressed in two different bases: this is the contents of 9.1.5 of Varshalovich: $$ \langle j_1j_2(j_{12})j_3 JM\vert j_1, j_2j_3(j_{23}) J' M'\rangle :=U(j_1j_2 Jj_3;j_{12} j_{23})\delta_{JJ'}\delta_{MM'}\\ =\delta_{JJ'}\delta_{MM'} (-1)^{j_1+j_2+j_3+j}\sqrt{(2j_{12}+1)(2j_{23}+1)} \left\{\begin{array}{ccc} j_1&j_2&j_{12}\\ j_3&J&j_{23} \end{array}\right\}\, . $$ By unitarity of $U$ in the construction: $$ \sum_{j_{12}}U(j_1j_2 Jj_3;j_{12} j_{23}) U(j_1j_2 Jj_3;j_{12} j'_{23})= \delta_{j_{23}j'_{23}}\\ =\sum_{j_{12}} (2j_{12}+1)(2j_{23}+1) \left\{\begin{array}{ccc} j_1&j_2&j_{12}\\ j_3&J&j_{23} \end{array}\right\}\left\{\begin{array}{ccc} j_1&j_2&j_{12}\\ j_3&J&j'_{23} \end{array}\right\}\, . $$ This is (9.8.3) of Varshalovich, and (6.2.9) of Edmonds, with the triangularity conditions arising because $j_1\times j_{23}$ and $j_2\times j_3$ must be triangular with $j_{23}$.

Edmonds also mentions that the equality (6.2.11): $$ \sum_X (-1)^{p+q+X} (2X+1) \left\{\begin{array}{ccc} a&b&X\\ c&d&p \end{array}\right\}\left\{\begin{array}{ccc} a&b&X\\ c&d&q \end{array}\right\} = \left\{\begin{array}{ccc} a&b&q\\ b&d&p \end{array}\right\} $$ arises as a recomposition of recoupling transformations $$ \sum_{j_{23}}\langle j_1 j_2 (j_{12}) j_3 JM\vert j_1 j_2 j_3 (j_{23}) JM\rangle \langle j_1 j_2 j_3(j_{23}) JM\vert j_2 j_3 j_1(j_{31}) JM\rangle= \langle j_1 j_2 (j_{12}) j_3 JM\vert j_2 j_3 j_1 (j_{31}) JM\rangle\, . $$ The sum over $j_{23}$ is a sum over a complete set of states so $$ \sum_{j_{23}}\vert j_1 j_2 j_3 (j_{23}) JM\rangle \langle j_1 j_2 j_3(j_{23}) JM\vert = \mathbf{1} $$ This yields Varshalovich (9.8.4).

Rose has an equivalent discussion but everything is expressed in terms of $W$ coefficients rather than $6j$'s. If this is covered by Rose and by Edmonds, it must be in lots of other texts on angular momentum theory where coupling-recoupling is discussed in some depth.