I would like to evaluate the following summation of Clebsch-Gordan and Wigner 6-j symbols in closed form:

$$\sum_{l,m} C_{l_2,m_2,l_1,m_1}^{l,m} C_{\lambda_2,\mu_2,\lambda_1,\mu_1}^{l,m} \left\{ \begin{array}{ccc} l & l_2 & l_1 \\ n/2 & n/2 & n/2 \end{array}\right\} \left\{ \begin{array}{ccc} l & \lambda_2 & \lambda_1 \\ n/2 & n/2 & n/2 \end{array}\right\}$$

with $n \in \left[0,\infty\right)$, $l,l_1,l_2,\lambda_1,\lambda_2 \in \left[0,n\right]$, $m \in \left[-l,l\right]$, $m_1 \in \left[-l_1,l_1\right]$, $m_2 \in \left[-l_2,l_2\right]$, $\mu_1 \in \left[-\lambda_1,\lambda_1\right]$ and $\mu_2 \in \left[-\lambda_2,\lambda_2\right]$. All indices are integers and n must be also even.

I have been using Varshalovich's Book, but can't find any identities that have been useful to simplify this. I am hoping that the result is something like $\delta_{l_2,\lambda_2}\delta_{m_2,\mu_2}\delta_{l_1,\lambda_1}\delta_{m_1,\mu_1}$, but I'm not sure that that will be the case. Any ideas of how to evaluate this?

  • $\begingroup$ Is $n$ any integer? $\endgroup$
    – Vibert
    Jun 25, 2013 at 23:08
  • 1
    $\begingroup$ Well Mathematica has ClebschGordan and SixJSymbol functions but I can't get it to simplify your expression. Even evaluating simple cases is taking me a long time. Maybe somebody with more Mathematica and/or combinatorics knowledge than me can find a trick. $\endgroup$
    – Michael
    Jun 26, 2013 at 0:57
  • $\begingroup$ @Vibert: $n \geq 0$ is an even integer, $0 \leq l \leq n$ is an integer, all other indices take integer values and their limits follow from the definition of the CG coefficients and 6j symbols. Sorry about not stating that before. $\endgroup$
    – okj
    Jun 26, 2013 at 1:47
  • $\begingroup$ Specifically: $$n \in \left[0,\infty\right)$$ $$l,l_1,l_2,\lambda_1,\lambda_2 \in \left[0,n\right]$$ $$m \in \left[-l,l\right]$$ $$m_1 \in \left[-l_1,l_1\right]$$ $$m_2 \in \left[-l_2,l_2\right]$$ $$\mu_1 \in \left[-\lambda_1,\lambda_1\right]$$ $$\mu_2 \in \left[-\lambda_2,\lambda_2\right]$$ ($n$ is an even integer, all other indices are integers) $\endgroup$
    – okj
    Jul 2, 2013 at 12:47

1 Answer 1


Well this is pretty similar to the calculations I have done to find the spectra of the quantum geometric volume operator in Loop Quantum Gravity. Given that I don't think that you will be able to find a closed analytical expression for this summation. I would be reasonably straightforward to write a numerical routine to calculate this.

Here is the link to the interactive Sage math routines I wrote to calculate operator spectra. You could probably adapt it to your purpose. If you would like any help with this just let me know.



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