# Completeness of Wigner D-functions

I am looking up the properties of the Wigner D-functions (or D-matrix).

I come across the following identity, namely "Completeness of Wigner D-functions" mentioned in this paper $$\sum_{j = 0, 1/2, 1, 3/2, \dots}^\infty \sum_{p=-j}^j \sum_{q=-j}^j \frac{2j+1}{16\pi^2} D^{j*}_{pq} (\alpha_1, \beta_1, \gamma_1) D^j_{pq}(\alpha_2, \beta_2, \gamma_2) = \delta(\alpha_1 - \alpha_2) \delta(\cos\beta_1 - \cos\beta_2) \delta(\gamma_1 - \gamma_2).$$

The above sum is over $j = 0, \frac12, 1, \frac32, 2, \dots$. I am wondering what would be if I just sum over integer $j = 0, 1, 2, 3, \dots$

$$\sum_{j = 0, 1, 2, \dots}^\infty \sum_{p=-j}^j \sum_{q=-j}^j \frac{2j+1}{16\pi^2} D^{j*}_{pq} (\alpha_1, \beta_1, \gamma_1) D^j_{pq}(\alpha_2, \beta_2, \gamma_2) = ?$$

• Why would you suggest the half-integer values are not to be included? Commented Feb 23, 2017 at 5:19
• ... or is this just a question for fun? Commented Feb 23, 2017 at 5:29
• The spherical harmonics are related to the Wigner D-matrix with integer indices. That's why I am wondering the difference between the Wigner D-matrix of integer indices and half integer indices. Commented Feb 23, 2017 at 6:55

Actually I consulted the authority on this, the book by Varshalovich et al. on Quantum Theory of Angular momentum. It seems the sum you are interested in actually exists. For integers, it is as you have written except the denominator factor is $8\pi^2$, not $16\pi^2$, i.e. $$\sum_{j=0,1,2\ldots}^\infty \sum_p\sum_q \frac{2j+1}{8\pi^2} D^{j*}_{pq}(\omega_1)D^{j}_{pq}(\omega_2)=\delta(\alpha_1-\alpha_2)\delta(\cos\beta_1-\cos\beta_2)\delta(\gamma_1-\gamma_2)\, .$$ You might want to double check. The complete reference is: