Wigner-D functions

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

% the Wigner-d function of degree 1
beta = 0.5;
d = Wigner_D(1,beta)

% the Wigner-D function of degree 1
R = rotation.rand;
D = sqrt(3) * Wigner_D(1,R)

d =
    0.9388   -0.3390   -0.0612
    0.3390    0.8776    0.3390
   -0.0612   -0.3390    0.9388
D =
  -0.7268 + 0.0871i   0.9546 + 0.7435i   0.1277 + 0.9919i
   0.8593 - 0.8519i  -0.2681 + 0.0000i   0.8593 + 0.8519i
   0.1277 - 0.9919i   0.9546 - 0.7435i  -0.7268 - 0.0871i

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

D = SO3FunHarmonic([0;0;0;1])
D.eval(R)

D = SO3FunHarmonic (xyz → xyz)
  antipodal: true
  isReal: false
  bandwidth: 1
  weight: 0
 
ans =
   0.1277 - 0.9919i

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 \(n,k,l\).

norm(D)

ans =
     1

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

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

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

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