Wigner-D functions edit page

## Theorie

The Wigner-D functions are special functions on the rotation group $$SO(3)$$.

In terms of Matthies (ZYZ-convention) Euler angles $${\bf R} = {\bf R}(\alpha,\beta,\gamma)$$ the $$L_2$$-normalized Wigner-D function of degree $$n$$ and orders $$k,l \in \{-n,\dots,n\}$$ is defined by

$D_n^{k,l}({\bf R}) = \sqrt{2n+1} \, \mathrm e^{-\mathrm i k\gamma} \mathrm d_n^{k,l}(\cos\beta) \,e^{-\mathrm i l\alpha}$

where $$d_n^{k,l}$$, denote the real valued Wigner-d functions, which are defined in terms of Jacobi polynomial $$P_s^{a,b}$$ by

$d_n^{k,l}(x) = (-1)^{\nu} \binom{2n-s}{s+a}^{\frac12} \binom{s+b}{b}^{-\frac12} \left(\frac{1-x}{2}\right)^{\frac{a}{2}} \left(\frac{1+x}{2}\right)^{\frac{b}2} P_s^{a,b}(x)$

using the constants $$a =|k-l|$$, $$b =|k+l|$$, $$s = n - \max\{|k|,|l|\}$$ and $$\nu = \min\{0,k\}+\min\{0,l\}$$ if $$l \geq k$$; $$\nu = \min\{0,k\}+\min\{0,l\} + k+l$$ otherwise.

This definition is slightly different to other well known definitions, because the Wigner-D functions are defined compatible to the spherical harmonics which form an orthonormal basis on the 2-sphere.

In MTEX the Wigner-D and Wigner-d functions are available through the command Wigner_D

Here the orders $$k$$, $$l$$ work as row and column indices.

## Series Expansion

The Wigner-D functions form an orthonormal basis in $$L_2(SO(3))$$. Hence, we can describe functions on the rotation group $$SO(3)$$ by there harmonic representation using the class SO3FunHarmonic.

Hence we define the Wigner-D function $$D_1^{1,-1}$$ by

Various normalizations for the Wigner-D functions are common in the literature.

Here we define the $$L_2$$-norm by

$\| f \|_2 = \left(\frac1{8\pi^2}\,\int_{SO(3)} \lvert f( {\bf R}) \rvert^2 \,\mathrm d {\bf R} \right)^{1/2}$

such that the norm of the constant function $$f=1$$ is $$1$$. Take a look on the section Integration of SO3Fun's.

Using that definition the Wigner-D functions in MTEX are normalized, i.e. $$\| D_n^{k,l} \|_2 = 1$$ for all $$n,k,l$$.

## Some important formulas for Wigner-D functions

The Wigner-D functions are the matrix elements of the representations $$D_n \colon SO(3) \to \mathbb C^{(2n+1)\times(2n+1)}$$ on $$SO(3)$$. Since representations are group homomorphisms, we have $$D_n( {\bf R} \, {\bf Q} ) = \frac1{\sqrt{2n+1}} \, D_n( {\bf Q} ) \, D_n( {\bf R} ).$$ Hence we get

$D_n^{k,l}( {\bf R} \, {\bf Q} ) = \frac1{2n+1} \sum_{j=-n}^n D_n^{k,j}( {\bf Q} )\,D_n^{j,l}( {\bf R} ).$

Some symmetry properties of Wigner-D functions yields

$D_n^{k,l}( {\bf R} ) = \overline{D_n^{l,k}( {\bf R}^{-1} )}.$

## Symmetry properties of Wigner-d functions

The Wigner-d functions by construction fulfill a lot of symmetry properties. Some importants are

$d_n^{k,l}(x) = d_n^{-k,-l}(x) = (-1)^{k+l}\, d_n^{l,k}(x) = (-1)^{k+l}\, d_n^{-l,-k}(x)$

$d_n^{k,l}(x) = (-1)^{n+k+l}\,d_n^{-k,l}(-x) = (-1)^{n+k+l}\,d_n^{k,-l}(-x)$

$d_n^{k,l}(\cos\beta) = (-1)^{k+l}\,d_n^{k,l}(\cos(-\beta))$