{{Short description|Special functions}} {{distinguish|spinor spherical harmonics}} In special functions, a topic in mathematics, '''spin-weighted spherical harmonics''' are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are {{math|U(1)}} gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted harmonics are organized by degree {{mvar|l}}, just like ordinary spherical harmonics, but have an additional '''spin weight''' {{mvar|s}} that reflects the additional {{math|U(1)}} symmetry. A special basis of harmonics can be derived from the Laplace spherical harmonics {{math|''Y''<sub>''lm''</sub>}}, and are typically denoted by {{math|<sub>''s''</sub>''Y''<sub>''lm''</sub>}}, where {{mvar|l}} and {{mvar|m}} are the usual parameters familiar from the standard Laplace spherical harmonics. In this special basis, the spin-weighted spherical harmonics appear as actual functions, because the choice of a polar axis fixes the {{math|U(1)}} gauge ambiguity. The spin-weighted spherical harmonics can be obtained from the standard spherical harmonics by application of spin raising and lowering operators. In particular, the spin-weighted spherical harmonics of spin weight {{math|''s'' {{=}} 0}} are simply the standard spherical harmonics: :<math>{}_0Y_{l m} = Y_{l m}\ .</math>

Spaces of spin-weighted spherical harmonics were first identified in connection with the representation theory of the Lorentz group {{harv|Gelfand|Minlos|Shapiro|1958}}. They were subsequently and independently rediscovered by {{harvtxt|Newman|Penrose|1966}} and applied to describe gravitational radiation, and again by {{harvtxt|Wu|Yang|1976}} as so-called "monopole harmonics" in the study of Dirac monopoles.

==Spin-weighted functions== Regard the sphere {{math|''S''<sup>2</sup>}} as embedded into the three-dimensional Euclidean space {{math|'''R'''<sup>3</sup>}}. At a point {{math|'''x'''}} on the sphere, a positively oriented orthonormal basis of tangent vectors at {{math|'''x'''}} is a pair {{math|'''a''', '''b'''}} of vectors such that :<math> \begin{align} \mathbf{x}\cdot\mathbf{a} = \mathbf{x}\cdot\mathbf{b} &= 0\\ \mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b} &= 1\\ \mathbf{a}\cdot\mathbf{b} &= 0\\ \mathbf{x}\cdot (\mathbf{a}\times\mathbf{b}) &> 0, \end{align} </math> where the first pair of equations states that {{math|'''a'''}} and {{math|'''b'''}} are tangent at {{math|'''x'''}}, the second pair states that {{math|'''a'''}} and {{math|'''b'''}} are unit vectors, the penultimate equation that {{math|'''a'''}} and {{math|'''b'''}} are orthogonal, and the final equation that {{math|('''x''', '''a''', '''b''')}} is a right-handed basis of {{math|'''R'''<sup>3</sup>}}.

A spin-weight {{mvar|s}} function {{mvar|f}} is a function accepting as input a point {{math|'''x'''}} of {{math|''S''<sup>2</sup>}} and a positively oriented orthonormal basis of tangent vectors at {{math|'''x'''}}, such that :<math>f\bigl(\mathbf x,(\cos\theta)\mathbf{a}-(\sin\theta)\mathbf{b}, (\sin\theta)\mathbf{a} + (\cos\theta)\mathbf{b}\bigr) = e^{is\theta}f(\mathbf x,\mathbf{a},\mathbf{b})</math> for every rotation angle {{mvar|θ}}.

Following {{harvtxt|Eastwood|Tod|1982}}, denote the collection of all spin-weight {{mvar|s}} functions by {{math|'''B'''(''s'')}}. Concretely, these are understood as functions {{mvar|f}} on {{math|'''C'''<sup>2</sup>\{0}}} satisfying the following homogeneity law under complex scaling :<math>f\left(\lambda z,\overline{\lambda}\bar{z}\right) = \left(\frac{\overline{\lambda}}{\lambda}\right)^s f\left(z,\bar{z}\right).</math> This makes sense provided {{mvar|s}} is a half-integer.

Abstractly, {{math|'''B'''(''s'')}} is isomorphic to the smooth vector bundle underlying the antiholomorphic vector bundle {{math|{{overline|'''O'''(2''s'')}}}} of the Serre twist on the complex projective line {{math|'''CP'''<sup>1</sup>}}. A section of the latter bundle is a function {{mvar|g}} on {{math|'''C'''<sup>2</sup>\{0}}} satisfying :<math>g\left(\lambda z,\overline{\lambda}\bar{z}\right) = \overline{\lambda}^{2s} g\left(z,\bar{z}\right).</math> Given such a {{mvar|g}}, we may produce a spin-weight {{mvar|s}} function by multiplying by a suitable power of the hermitian form :<math>P\left(z,\bar{z}\right) = z\cdot\bar{z}.</math> Specifically, {{math|''f'' {{=}} ''P''<sup>−''s''</sup>''g''}} is a spin-weight {{mvar|s}} function. The association of a spin-weighted function to an ordinary homogeneous function is an isomorphism.

===The operator {{mvar|ð}}=== The spin weight bundles {{math|'''B'''(''s'')}} are equipped with a differential operator {{mvar|ð}} (eth). This operator is essentially the Dolbeault operator, after suitable identifications have been made, :<math>\partial : \overline{\mathbf O(2s)} \to \mathcal{E}^{1,0}\otimes \overline{\mathbf O(2s)} \cong \overline{\mathbf O(2s)}\otimes\mathbf{O}(-2).</math> Thus for {{math|''f'' ∈ '''B'''(''s'')}}, :<math>\eth f \ \stackrel{\text{def}}{=}\ P^{-s+1}\partial \left(P^s f\right) </math> defines a function of spin-weight {{math|''s'' + 1}}.

==Spin-weighted harmonics== Just as conventional spherical harmonics are the eigenfunctions of the Laplace-Beltrami operator on the sphere, the spin-weight {{mvar|s}} harmonics are the eigensections for the Laplace-Beltrami operator acting on the bundles {{math|{{mathcal|E}}(''s'')}} of spin-weight {{mvar|s}} functions. ==Representation as functions== The spin-weighted harmonics can be represented as functions on a sphere once a point on the sphere has been selected to serve as the North pole. By definition, a function {{mvar|η}} with ''spin weight {{mvar|s}}'' transforms under rotation about the pole via :<math>\eta \rightarrow e^{i s \psi}\eta.</math>

Working in standard spherical coordinates, we can define a particular operator {{mvar|ð}} acting on a function {{mvar|η}} as: :<math>\eth\eta = - \left(\sin{\theta}\right)^s \left\{ \frac{\partial}{\partial \theta} + \frac{i}{\sin{\theta}} \frac{\partial} {\partial \phi} \right\} \left[ \left(\sin{\theta}\right)^{-s} \eta \right].</math> This gives us another function of {{mvar|θ}} and {{mvar|φ}}. (The operator {{mvar|ð}} is effectively a covariant derivative operator in the sphere.)

An important property of the new function {{mvar|ðη}} is that if {{mvar|η}} had spin weight {{mvar|s}}, {{mvar|ðη}} has spin weight {{math|''s'' + 1}}. Thus, the operator raises the spin weight of a function by 1. Similarly, we can define an operator {{mvar|{{overline|ð}}}} which will lower the spin weight of a function by 1: :<math>\bar\eth\eta = - \left(\sin{\theta}\right)^{-s} \left\{ \frac{\partial}{\partial \theta} - \frac{i}{\sin{\theta}} \frac{\partial} {\partial \phi} \right\} \left[ \left(\sin{\theta}\right)^{s} \eta \right].</math>

The spin-weighted spherical harmonics are then defined in terms of the usual spherical harmonics as: :<math> {}_sY_{l m} = \begin{cases} \sqrt{\frac{(l-s)!}{(l+s)!}}\ \eth^s Y_{l m},&& 0\leq s \leq l; \\ \sqrt{\frac{(l+s)!}{(l-s)!}}\ \left(-1\right)^s \bar\eth^{-s} Y_{l m},&& -l\leq s \leq 0; \\ 0,&&l < |s|.\end{cases}</math> The functions {{math|<sub>''s''</sub>''Y''<sub>''lm''</sub>}} then have the property of transforming with spin weight {{mvar|s}}.

Other important properties include the following: :<math>\begin{align} \eth\left({}_sY_{l m}\right) &= +\sqrt{(l-s)(l+s+1)}\, {}_{s+1}Y_{l m};\\ \bar\eth\left({}_sY_{l m}\right) &= -\sqrt{(l+s)(l-s+1)}\, {}_{s-1}Y_{l m}; \end{align}</math>

== Orthogonality and completeness == The harmonics are orthogonal over the entire sphere: :<math>\int_{S^2} {}_sY_{l m}\, {}_s\bar{Y}_{l'm'}\, dS = \delta_{ll'} \delta_{mm'}, </math> and satisfy the completeness relation :<math>\sum_{l m} {}_s\bar Y_{l m}\left(\theta',\phi'\right) {}_s Y_{l m}(\theta,\phi) = \delta\left(\phi'-\phi\right)\delta\left(\cos\theta'-\cos\theta\right)</math>

==Calculating== These harmonics can be explicitly calculated by several methods. The obvious recursion relation results from repeatedly applying the raising or lowering operators. Formulae for direct calculation were derived by {{harvtxt|Goldberg|Macfarlane|Newman|Rohrlich|1967}}. Note that their formulae use an old choice for the [http://mathworld.wolfram.com/Condon-ShortleyPhase.html Condon–Shortley phase]. The convention chosen below is in agreement with Mathematica, for instance.

The more useful of the Goldberg, et al., formulae is the following: :<math>{}_sY_{l m} (\theta, \phi) = \left(-1\right)^{l+m-s} \sqrt{ \frac{(l+m)! (l-m)! (2l+1)} {4\pi (l+s)! (l-s)!} } \sin^{2l} \left( \frac{\theta}{2} \right) e^{i m \phi} \times\sum_{r=0}^{l-s} \left(-1\right)^{r} {l-s \choose r} {l+s \choose r+s-m} \cot^{2r+s-m} \left( \frac{\theta} {2} \right)\, .</math>

A Mathematica notebook using this formula to calculate arbitrary spin-weighted spherical harmonics can be found [http://www.black-holes.org/SpinWeightedSphericalHarmonics.nb here].

With the phase convention here: :<math>\begin{align} {}_s\bar Y_{l m} &= \left(-1\right)^{s+m}{}_{-s}Y_{l(-m)}\\ {}_sY_{l m}(\pi-\theta,\phi+\pi) &= \left(-1\right)^l {}_{-s}Y_{l m}(\theta,\phi). \end{align}</math>

== First few spin-weighted spherical harmonics ==

Analytic expressions for the first few orthonormalized spin-weighted spherical harmonics:

=== Spin-weight {{math|''s'' {{=}} 1}}, degree {{math|''l'' {{=}} 1}} ===

:<math>\begin{align} {}_1 Y_{10}(\theta,\phi) &= \sqrt{\frac{3}{8\pi}}\,\sin\theta \\ {}_1 Y_{1\pm 1}(\theta,\phi) &= -\sqrt{\frac{3}{16\pi}}(1 \mp \cos\theta)\,e^{\pm i\phi} \end{align}</math>

== Relation to Wigner rotation matrices ==

:<math>D^l_{-m s}(\phi,\theta,-\psi) =\left(-1\right)^m \sqrt\frac{4\pi}{2l+1} {}_sY_{l m}(\theta,\phi) e^{is\psi}</math>

This relation allows the spin harmonics to be calculated using recursion relations for the {{mvar|D}}-matrices.

== Triple integral ==

The triple integral in the case that {{math|''s''<sub>1</sub> + ''s''<sub>2</sub> + ''s''<sub>3</sub> {{=}} 0}} is given in terms of the 3-{{mvar|j}} symbol: :<math>\int_{S^2} \,{}_{s_1} Y_{j_1 m_1} \,{}_{s_2} Y_{j_2m_2}\, {}_{s_3} Y_{j_3m_3} = \sqrt{\frac{\left(2j_1+1\right)\left(2j_2+1\right)\left(2j_3+1\right)}{4\pi}} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j_3\\ -s_1 & -s_2 & -s_3 \end{pmatrix} </math>

==See also== * Spherical basis

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Category:Fourier analysis Category:Rotational symmetry Category:Special functions