Definition of WignerD function?

Question

On Wikipedia, elements of Wigner's D-matrix are defined as

$$D_{m'm}^{j}(\alpha,\beta,\gamma)=\langle jm'|e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z}|jm\rangle=e^{-im'\alpha}d_{m'm}^j (\beta)e^{-im\gamma }$$ where $d_{m'm}^j (\beta)=\langle jm'|e^{-i\beta J_y}|jm\rangle$ is the Wigner (small) d-matrix element. According to the documentation, such Wigner D-functions are implemented in Mathematica via the WignerD command: "WignerD[{j, m1, m2}, ψ, θ, ϕ] gives the Wigner D-function $D_{m_1m_2}^j(\psi,\theta,\phi)$."

However, something seems off. For instance, according to Wikipedia one has

$$D_{1/2,-1/2}(0,\beta,0)=d_{1/2,-1/2}(\beta)=-\sin(\beta/2).$$

whereas Mathematica's WignerD command yields

 In[1]:= WignerD[{1/2, 1/2, -1/2}, 0, β, 0]
 Out[1]= Sin[β/2].

 In[2]:= WignerD[{1/2, -1/2, 1/2}, 0, β, 0]
 Out[2]= -Sin[β/2].

So Mathematica seems to disagree with Wikipedia as to the overall sign (or, equivalently, the order of $m',m$). This presumably reflects some convention for how the Wigner D-functions are defined in Mathematica, but I haven't seen this documented anywhere. Can anyone clear up the confusion?

Answer 1

In physics and on Wikipedia, Wigner's sign convention is used; Mathematica uses a different sign convention. To define a Wigner D matrix that adheres to the Wigner/Wikipedia sign convention, you can use

myWignerD[{j_, m1_, m2_}, α_, β_, ɣ_] = WignerD[{j, -m1, -m2}, α, β, ɣ];

As far as I can tell, this definition confirms the formulas given at https://en.wikipedia.org/wiki/Wigner_D-matrix#List_of_d-matrix_elements for $\alpha=\gamma=0$ (the $d$-matrix elements) as well as the $\alpha$- and $\gamma$-dependences.