Phase convention for half-integer spins

[syp2001]

Currently, the spherical expansion method and Leaver's method differ by a factor of $\pm i$ for half-integer spins.

In = SpinWeightedSpheroidalHarmonicS[1/2, 1/2, 1/2, 1.5`30,Method->"Leaver"][\[Pi]/2,0]
Out = 0.182527795061294349026 

In = SpinWeightedSpheroidalHarmonicS[1/2, 1/2, 1/2, 1.5`30,Method->"SphericalExpansion"][\[Pi]/2,0]
Out = 0.18252779959664164891544117166 I

This seems to be coming from the phase factor of (-1)^(m+l-s) in SpinWeightedSphericalHarmonicY. Would it make sense to use a different phase convention for half-integer spins to avoid returning imaginary values when phi = 0?

[barrywardell]

Half-integer spins were enabled in issue #22. There were some checks included there, but I don't see anything that covered a comparison of the two methods for the computation of the harmonic in that case. Since this is a choice of phase convention, the only question is what choice we should make. One possibility is to use the relation to the WignerD matrices, i.e.

SpinWeightedSphericalHarmonicY[s, l, m, \[Theta], \[Phi]] ==
  1/(-1)^m Sqrt[(2 l + 1)/(4 \[Pi])] WignerD[{l, m, -s}, \[Phi], \[Theta], \[Psi]] E^(I s \[Psi])

The existing SphericalExpansion method adheres to this for integer s and gives a complex value in your case for half-integer s, but with the opposite sign.

[barrywardell]

@srd24: do you know what the standard phase convention is for the spin-half spherical harmonics?

[srd24]

Hi Barry,

Please see Eq. (17) and Section 3 in my paper with Jonathan Gair. Let me know if this is not what you need.

Best wishes,

Sam

On Fri, 8 Dec 2023 at 16:44, Barry Wardell @.***> wrote:

@srd24 https://github.com/srd24: do you know what the standard phase convention is for the spin-half spherical harmonics?

— Reply to this email directly, view it on GitHub https://github.com/BlackHolePerturbationToolkit/SpinWeightedSpheroidalHarmonics/issues/32#issuecomment-1847507157, or unsubscribe https://github.com/notifications/unsubscribe-auth/AJS6DRHIXDUSGJBKF5T2FZLYIM7WZAVCNFSM6AAAAABAKB4IRCVHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMYTQNBXGUYDOMJVG4 . You are receiving this because you were mentioned.Message ID: <BlackHolePerturbationToolkit/SpinWeightedSpheroidalHarmonics/issues/32/1847507157 @github.com>

[barrywardell]

Thanks @srd24, that clarifies things somewhat. In that paper you reference Eq. (3.1) of Goldberg et al. The spin-weighted spherical harmonics in this package use an expression equivalent to: $$(-1)^m \left[\frac{2l+1}{4\pi}\frac{(l+m)!(l-m)!}{(l+s)!(l-s)!}\right]^{1/2} (\sin \theta/2)^{2l} \sum_{n} \binom{l-s}{n} \binom{l+s}{n+s-m} (-1)^{l-n-s} e^{i m \phi} (\cot \theta/2)^{2n+s-m}$$ which is the same as Goldberg's but with an extra factor of $(-1)^m$. This extra factor is interpreted as being a more standard convention for the Condon-Shortley phase than Goldberg used and is required for consistency with Mathematica's SphericalHarmonicY. In fact, the formula used is slightly different, with $(\sin \theta/2)^{2l} \sum_n \cdots$ replaced by:

$$ \begin{array}{cc} { & \begin{array}{cc} (-1)^{l-s} \binom{l+s}{s-m} \cos ^{s-m}\left(\frac{\theta }{2}\right) _2F_1\left(-l-m,s-l;-m+s+1;-\cot ^2\left(\frac{\theta }{2}\right)\right) \sin ^{2 l+m-s}\left(\frac{\theta }{2}\right) & m\leq s \ (-1)^{l-m} \binom{l-s}{m-s} \cos ^{m-s}\left(\frac{\theta }{2}\right) _2F_1\left(m-l,-l-s;m-s+1;-\cot ^2\left(\frac{\theta }{2}\right)\right) \sin ^{2 l-m+s}\left(\frac{\theta }{2}\right) & \text{otherwise} \ \end{array} \ \end{array}$$

Mathematica's Simplify shows the two expressions to be equal.