Torres del Castillo (2003)

Torres Del Castillo [52] is a monograph on 3-D spinors and spin-weighted functions, and one of the few book-length treatments of the spin-weighted spherical harmonics.

Rotations and Wigner's $D$ matrix

He starts by defining a rotation $ℛ$ as transforming a point $x_i$ into another point with coordinates $x_i' = a_{ij}x_j$. Under that rotation, any scalar function $f$ transforms into another function $f' = ℛ f$ defined by [Eq. (2.43)]

\[f'\big(x_i\big) = f\big( a^{-1}_{ij} x_j \big).\]

In particular, $f'(x'_i) = f(x_i)$. This is exactly our rotation operator $U(𝐑)$, which rotates the field and hence the argument of the function inversely. He then defines Wigner's D-matrix to satisfy [Eq. (2.45)]

\[ℛ Y_{l,m} = \sum_{m'} D^l_{m',m}(ℛ) Y_{l,m'}.\]

Including the arguments to the spherical harmonics, this becomes

\[Y_{l,m}\big(ℛ^{-1} R_{θ, ϕ}\big) = \sum_{m'} D^l_{m',m}(ℛ) Y_{l,m'}\big(R_{θ, ϕ}\big),\]

which is our rotation law. In this form, we have [Eq. (2.46)]

\[D^l_{m'',m}(ℛ_1 ℛ_2) = \sum_{m'} D^l_{m'',m'}(ℛ_1) D^l_{m',m}(ℛ_2).\]

He computes [Eq. (2.53)]

\[D^l_{m',m}(ϕ, θ, \chi) = e^{-i m' ϕ} d^l_{m',m}(θ) e^{-i m \chi},\]

where the $d$ matrix is given in the second equation below Eq. (2.53) by

\[d^l_{m',m}(θ) = \sqrt{(l+m)!(l-m)!(l+m')!(l-m')!} \sum_{k} \frac{ (-1)^k (\sin \tfrac{1}{2} θ)^{m-m'+2k} (\cos \tfrac{1}{2} θ)^{2l-m+m'-2k} } { k!(l+m'-k)!(l-m-k)!(m-m'+k)! },\]

and the spin-weighted spherical harmonic is related to $D$ in the equation following Eq. (2.53) by

\[{}_{s}Y_{j,m}(θ, ϕ) = (-1)^m \sqrt{\frac{2j+1}{4π}} d^j_{-m,s}(θ) e^{i m ϕ}.\]

The phases in $D$ are those of our definition, and the last expression is the integer-index form of our definition of ${}_sY_{ℓ,m}$, so we expect complete agreement.

Implementing formulas

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

module TorresDelCastillo

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

import ..ConventionsUtilities: 𝒾, ❗

The $d$ matrix. The sum runs over all $k$ for which the factorials have non-negative arguments, $\max(0, m'-m) \leq k \leq \min(l+m', l-m)$.

function d(l, m′, m, θ::T) where {T<:Real}    √T((l+m)❗ * (l-m)❗ * (l+m′)❗ * (l-m′)❗) *    sum(        (-1)^k * sin(θ/2)^(m-m′+2k) * cos(θ/2)^(2l-m+m′-2k)        / T((k)❗ * (l+m′-k)❗ * (l-m-k)❗ * (m-m′+k)❗)        for k ∈ max(0, m′-m):min(l+m′, l-m);        init=zero(T)    )end

Equation (2.53):

function D(l, m′, m, ϕ, θ, χ)    exp(-𝒾 * m′ * ϕ) * d(l, m′, m, θ) * exp(-𝒾 * m * χ)end

The spin-weighted spherical harmonics, from the equation following Eq. (2.53):

function Y(s, j, m, θ::T, ϕ::T) where {T<:Real}    (-1)^m * √((2j+1) / (4T(π))) * d(j, -m, s, θ) * exp(𝒾 * m * ϕ)end

end  # module TorresDelCastillo

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

and

sₘₐₓ = 2

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)

The $d$ and $D$ matrices agree with ours:

for β ∈ βrange(rng)    for (l, dˡ) ∈ SphericalFunctions.dCalculator(β, ℓₘₐₓ), m′ ∈ -l:l, m ∈ -l:l        @test TorresDelCastillo.d(l, m′, m, β) ≈ dˡ[m′, m] atol=ϵₐ rtol=ϵᵣ    endendfor (α, β, γ) ∈ αβγs    for (l, 𝔇ˡ) ∈ SphericalFunctions.DCalculator(α, β, γ, ℓₘₐₓ), m′ ∈ -l:l, m ∈ -l:l        @test TorresDelCastillo.D(l, m′, m, α, β, γ) ≈ 𝔇ˡ[m′, m] atol=ϵₐ rtol=ϵᵣ    endend

And the spin-weighted spherical harmonics agree with ours:

for (θ, ϕ) ∈ θϕrange(rng)    for (ℓ, Yˡ) ∈ SphericalFunctions.sYlmCalculator(θ, ϕ, ℓₘₐₓ, -sₘₐₓ:sₘₐₓ)        for s ∈ -min(ℓ, sₘₐₓ):min(ℓ, sₘₐₓ), m ∈ -ℓ:ℓ            @test TorresDelCastillo.Y(s, ℓ, m, θ, ϕ) ≈ Yˡ[s, m] atol=ϵₐ rtol=ϵᵣ        end    endend

These successful tests show that Torres del Castillo's $d$ and $D$ matrices and spin-weighted spherical harmonics agree with the corresponding functions defined by the SphericalFunctions package.

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