How to evaluate this sum of coupling coefficients?

Question

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?

This post imported from StackExchange Physics at 2015-06-26 11:15 (UTC), posted by SE-user okj

Comments

Is $n$ any integer?

This post imported from StackExchange Physics at 2015-06-26 11:15 (UTC), posted by SE-user Vibert

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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.

This post imported from StackExchange Physics at 2015-06-26 11:15 (UTC), posted by SE-user Michael Brown

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@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.

This post imported from StackExchange Physics at 2015-06-26 11:15 (UTC), posted by SE-user okj

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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)

This post imported from StackExchange Physics at 2015-06-26 11:15 (UTC), posted by SE-user okj

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.

http://wiki.sagemath.org/interact/Loop%20Quantum%20Gravity

This post imported from StackExchange Physics at 2015-06-26 11:15 (UTC), posted by SE-user David Horgan