Wigner (1959)

Wigner [20] is the English translation of Wigner's 1931 book, which introduced the representation matrices of the rotation group that now bear his name. As with most early sources, the conventions take some care to extract.

Euler angles

Figure 2 on page 59 shows Wigner's Euler angles $\{α, β, γ\}$. In that figure, the position of the $z'$ axis is independent of $α$, which appears to represent a final rotation about the $z'$ axis; in our convention, this rotation would be described by the final Euler angle, $γ$, so the labels appear to be swapped relative to ours. On the other hand, on page 156, if $𝔇^{(ℓ)}$ obeys the representation-composition property, then $\{α, β, γ\}$ represents the rotation $\{α, 0, 0\} \circ \{0, β, 0\} \circ \{0, 0, γ\}$, which is the same as our convention. Wigner is most explicit about his Euler angles in Appendix A: Eq. (A.2) gives the rotation matrix in terms of the Euler angles, and multiplying it on the left by the column vector $(0, 0, 1)$ shows where the point on the $z$ axis is rotated. It is independent of $γ$ and depends on $α$ (and $β$), which is inconsistent with Fig. 2 but consistent with our convention; we conclude that the labels in Fig. 2 have simply been swapped, and that Wigner's Euler angles are otherwise ours.

Representation matrices

Equation (15.27) gives the explicit expression

\[𝔇^{(j)}(α, β, γ)_{\mu'\mu} = \sum_κ (-1)^κ \frac{\sqrt{(j+\mu)!\,(j-\mu)!\,(j+\mu')!\,(j-\mu')!}} {(j-\mu'-κ)!\,(j+\mu-κ)!\,κ!\,(κ+\mu'-\mu)!}\, e^{i\mu'α} \left(\cos\frac{β}{2}\right)^{2j+\mu-\mu'-2κ} \left(\sin\frac{β}{2}\right)^{2κ+\mu'-\mu} e^{i\muγ},\]

which we implement below. The phases $e^{+i\mu'α}$ and $e^{+i\muγ}$ are the conjugates of ours, and the $β$-dependent factor differs from Wigner's own $d$ formula as quoted by later authors by the sign $(-1)^{\mu'-\mu}$. Using the symmetry $\overline{𝔇_{\mu',\mu}} = (-1)^{\mu'-\mu} 𝔇_{-\mu',-\mu}$ of our matrices, the relation in the summary box follows, and is tested below.

Wigner relates the spherical harmonics to $d$, Eq. (A.11), as

\[Y_{l,m}(ϑ, ϕ) = \mathrm{const}\, (-1)^m\, e^{imϕ}\, \mathbf{d}^{(l)}(ϑ)_{m 0},\]

for a positive normalization constant that does not depend on $m$. This is equivalent to

\[Y_{l,m}(ϑ, ϕ) = \mathrm{const}\, (-1)^m\, e^{imϕ}\, 𝔇^{(l)}(ϕ, ϑ, 0)_{m0}.\]

Note the factor $(-1)^m$, which does not appear in our relation $Y_{ℓ,m} = \sqrt{(2ℓ+1)/4π}\, \overline{𝔇_{m,0}}$. In fact, with the relation between Wigner's $𝔇$ and ours given above, $(-1)^m 𝔇^{(ℓ)}(ϕ, θ, 0)_{m0}|_{\text{Wigner}} = \overline{𝔇_{m,0}(ϕ, θ, 0)}$, so the two relations are the same and Wigner's spherical harmonics are the standard ones. We verify this as well.

Implementing formulas

We begin by writing code that implements the formulas from Wigner. We encapsulate the formulas in a module so that we can test them against the SphericalFunctions package.

module Wigner

We'll use some predefined utilities to make the code look more like the equations.

import ..ConventionsUtilities: 𝒾, ❗

Equation (15.27). The sum runs over all $κ$ for which the factorials have non-negative arguments, $\max(0, \mu-\mu') \leq κ \leq \min(j-\mu', j+\mu)$.

function 𝔇(j, μ′, μ, α::T, β::T, γ::T) where {T<:Real}    sum(        (-1)^κ        * T(√((j+μ)❗ * (j-μ)❗ * (j+μ′)❗ * (j-μ′)❗)            / ((j-μ′-κ)❗ * (j+μ-κ)❗ * (κ)❗ * (κ+μ′-μ)❗))        * exp(𝒾 * μ′ * α) * cos(β/2)^(2j+μ-μ′-2κ) * sin(β/2)^(2κ+μ′-μ) * exp(𝒾 * μ * γ)        for κ ∈ max(0, μ-μ′):min(j-μ′, j+μ);        init=zero(Complex{T})    )end

end  # module Wigner

Tests

We can now test the functions against the equivalent functions from the SphericalFunctions package. We will need to test approximate floating-point equality, so we set absolute and relative tolerances (respectively) in terms of the machine epsilon:

ϵₐ = 100eps()ϵᵣ = 1000eps()

We only test up to

ℓₘₐₓ = 4

because the formulas are slow, and this will be sufficient to sort out any sign or normalization differences, which are the most likely source of error. For the same reason we use a modest grid of Euler angles.

αβγs = αβγrange(rng, Float64, 5)

Wigner's $𝔇$ is $(-1)^{\mu'-\mu}$ times the complex conjugate of ours, or equivalently ours with both indices negated:

for (α, β, γ) ∈ αβγs    for (j, 𝔇ʲ) ∈ SphericalFunctions.DCalculator(α, β, γ, ℓₘₐₓ), μ′ ∈ -j:j, μ ∈ -j:j        @test Wigner.𝔇(j, μ′, μ, α, β, γ) ≈            (-1)^(μ′-μ) * conj(𝔇ʲ[μ′, μ]) atol=ϵₐ rtol=ϵᵣ        @test Wigner.𝔇(j, μ′, μ, α, β, γ) ≈            𝔇ʲ[-μ′, -μ] atol=ϵₐ rtol=ϵᵣ    endend

And Wigner's Eq. (A.11) reproduces the standard spherical harmonics (with $c = \sqrt{(2ℓ+1)/4π}$):

for (θ, ϕ) ∈ θϕrange(rng)    for (ℓ, Yˡ) ∈ SphericalFunctions.YlmCalculator(θ, ϕ, ℓₘₐₓ), m ∈ -ℓ:ℓ        @test √((2ℓ+1) / (4π)) * (-1)^m * Wigner.𝔇(ℓ, m, 0, ϕ, θ, zero(θ)) ≈            Yˡ[m] atol=ϵₐ rtol=ϵᵣ    endend

These successful tests show that Wigner's representation matrices are $(-1)^{\mu'-\mu}$ times the complex conjugates of the $𝔇$ matrices defined by the SphericalFunctions package, and that his spherical harmonics are ours.

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