Wigner-D functions

Wigner-D functions play the role of sine and cosine waves for rotations. Weighted sums of them can represent any square-integrable function on the rotation group \(\mathrm{SO}(3)\).

Each function has a non-negative degree \(n\) and two orders \(k\) and \(l\), both between \(-n\) and \(n\). The command WignerD evaluates a complete matrix of orders at once.

Evaluate one degree

A real number passed to WignerD is the second Euler angle \(\beta\). The result below contains all Wigner-d functions of degree one at \(\beta=90\) degrees.

beta = 90 * degree;
d = WignerD(beta,1)

d =
    0.5000   -0.7071   -0.5000
    0.7071   -0.0000    0.7071
   -0.5000   -0.7071    0.5000

The rows and columns correspond to \(k=-1,0,1\) and \(l=-1,0,1\), respectively. In particular, the central value is zero at 90 degrees, while the corner values have magnitude \(0.5\).

Passing a rotation instead evaluates the generally complex Wigner-D functions. The 'normalize' flag selects MTEX's \(L_2\) normalization.

R = rotation.byEuler(20*degree,40*degree,10*degree);
W = WignerD(R,1,'normalize')

W =
   1.3245 + 0.7647i   0.1367 - 0.7753i   0.1995 - 0.0352i
   0.2693 - 0.7398i   1.3268 + 0.0000i   0.2693 + 0.7398i
   0.1995 + 0.0352i   0.1367 + 0.7753i   1.3245 - 0.7647i

Follow the Wigner-d curves

The Wigner-d functions are ordinary functions of the single angle \(\beta\), so they can be drawn directly. Here are three entries from the degree-one matrix.

beta = linspace(0,pi,181);
d = zeros(numel(beta),3);
for j = 1:numel(beta)
  Dj = WignerD(beta(j),1);
  d(j,:) = [Dj(1,1) Dj(2,2) Dj(1,3)];
end

plot(beta./degree,d,'linewidth',2)
xlim([0 180]); xlabel('beta in degree')
legend('d^1_{-1,-1}','d^1_{0,0}','d^1_{-1,1}', ...
  'location','best')

The first curve decreases from 1 to 0. The central curve crosses zero at 90 degrees, while the corner entry decreases from 0 to -1. These smooth shapes arise because the functions are polynomials in \(\cos(\beta/2)\) and \(\sin(\beta/2)\). Series in Wigner-d functions therefore converge quickly for smooth functions.

Build one basis function

The Wigner-D functions form an orthonormal basis of \(L_2(\mathrm{SO}(3))\). MTEX stores a weighted series in this basis as an SO3FunHarmonic object.

Its coefficient vector starts with the degree-zero block. The next nine entries are the degree-one matrix, stored one column at a time. The vector below therefore selects \(D_1^{1,-1}\).

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

seriesValue =
   0.1995 + 0.0352i

The selected coefficient is row 3 and column 1 of the normalized matrix W above. Both evaluations give approximately 0.1995 + 0.0352i.

directValue = W(3,1)
comparisonError = abs(seriesValue-directValue)

directValue =
   0.1995 + 0.0352i
comparisonError =
   1.4913e-13

Normalization

Several normalizations of Wigner-D functions occur in the literature. MTEX defines the \(L_2\) norm by

\[ \|f\|_2 = \left(\frac{1}{8\pi^2}\int_{\mathrm{SO}(3)} \lvert f(\mathbf{R})\rvert^2\,\mathrm{d}\mathbf{R}\right)^{1/2}. \]

With this measure, the constant function \(f=1\) has norm 1. Every normalized Wigner-D function also has \(\|D_n^{k,l}\|_2=1\).

norm(F)

ans =
     1

The section Integration and Norms explains integration and norms for rotational functions in more detail.

The maths behind Wigner-D functions

In the Matthies ZYZ convention, write a rotation as \(\mathbf{R}=\mathbf{R}(\alpha,\beta,\gamma)\). MTEX defines the \(L_2\)-normalized Wigner-D function by

\[ D_n^{k,l}(\mathbf{R})=\sqrt{2n+1}\, \mathrm{e}^{-\mathrm{i}k\gamma}\,d_n^{k,l}(\beta)\, \mathrm{e}^{-\mathrm{i}l\alpha}. \]

The real-valued Wigner-d function \(d_n^{k,l}\colon[0,\pi]\to\mathbb{R}\) is defined through the Jacobi polynomial \(P_s^{a,b}\) by

\[ d_n^{k,l}(\beta)=(-1)^\nu \binom{2n-s}{s+a}^{\frac{1}{2}} \binom{s+b}{b}^{-\frac{1}{2}} \left(\frac{\sin\beta}{2}\right)^a \left(\frac{\cos\beta}{2}\right)^b P_s^{a,b}(\cos\beta), \]

where \(a=|k-l|\), \(b=|k+l|\), and \(s=n-\max\{|k|,|l|\}\). The sign exponent is

\[ \nu=\begin{cases} \min\{0,k\}+\min\{0,l\}, & \text{if }l\geq k,\\ \min\{0,k\}+\min\{0,l\}+k+l, & \text{otherwise}. \end{cases} \]

How the MTEX convention differs

MTEX's definition differs slightly from other common definitions. It is compatible with the spherical harmonics, which form an orthonormal basis on the 2-sphere and are used to build harmonic spherical functions.

Composition and symmetry

For each degree \(n\), the Wigner-D functions are the matrix elements of a representation \(D_n\colon\mathrm{SO}(3)\to \mathbb{C}^{(2n+1)\mathbin{\times}(2n+1)}\). The representation is a group homomorphism. With MTEX's order and normalization conventions,

\[ D_n(\mathbf{R}\mathbf{Q})=\frac{1}{\sqrt{2n+1}} D_n(\mathbf{Q})D_n(\mathbf{R}), \]

and hence

\[ D_n^{k,l}(\mathbf{R}\mathbf{Q})= \frac{1}{\sqrt{2n+1}}\sum_{j=-n}^{n} D_n^{k,j}(\mathbf{Q})D_n^{j,l}(\mathbf{R}). \]

Inversion exchanges the two orders and complex conjugates the value:

\[ D_n^{k,l}(\mathbf{R})= \overline{D_n^{l,k}(\mathbf{R}^{-1})}. \]

The Wigner-d functions also satisfy several useful symmetries:

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

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

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

References