${}_{s}Y_{β,m}$ and $π^{(β)}_{m', m}$
The previous page introduced the left and right Lie derivative operators acting on functions of a quaternion argument. Here, we show how the spin-weighted spherical harmonics arise naturally as eigenfunctions of these operators.
${}_{s}Y_{β,m}$ as eigenfunctions
The familiar way of arriving at the standard (scalar, $s=0$) spherical harmonics is to consider solutions to the Laplace equation in three-dimensional space, then to separate variables in spherical coordinates, and finally to identify the angular part of the solution as the spherical harmonics. Since scalar spherical harmonics really are just functions on the sphere $πΒ²$, rather than the full group $\mathrm{Spin}(3)$, they can be sensibly written as functions of spherical coordinates $(ΞΈ, Ο)$ alone. We'll suspend for a moment what, exactly, the angular-momentum operators mean for functions on the sphere, and just accept the usual definition of $L$ as the relevant operator in this case. Recall that we can only simultaneously diagonalize operators (find nontrivial functions that are eigenfunctions of both operators at the same time) if they commute. In this case, the only two commuting operators are $L^2 = L_x^2 + L_y^2 + L_z^2$ and some chosen component of $L$ β conventionally chosen to be $L_z$. Specifically, the spherical harmonics are defined to satisfy
\[\begin{aligned} LΒ² \left\{ Y_{β,m} \right\}(ΞΈ, Ο) &= β(β+1) \left\{ Y_{β,m} \right\}(ΞΈ, Ο), \\ L_z \left\{ Y_{β,m} \right\}(ΞΈ, Ο) &= m \left\{ Y_{β,m} \right\}(ΞΈ, Ο). \end{aligned}\]
(As usual, we do not include the factors of $\hbar$ that would normally be included in quantum mechanics texts.) Familiar arguments β either from spectra of operators, or continuity of the functions β tell us that the allowed values of $β$ are non-negative integers, and for each $β$, the allowed values of $m$ are integers satisfying $-β β€ m β€ β$.
But now, we consider functions defined on the full group $\mathrm{Spin}(3)$. In that case, we have both left and right Lie derivative operators available. As before, we can choose eigenfunctions of $LΒ²$ and $L_z$. It turns out that $[L_g, R_h] = 0$, which means that we could also simultaneously diagonalize with respect to some chosen component of $R$ β which again is conventionally chosen to be $R_z$. We might also expect to be able to diagonalize with respect to $RΒ²$, but that turns out to equal $LΒ²$, so while it is true that the spin-weighted spherical harmonics are also eigenfunctions of $RΒ²$, that statement contains no additional information. Thus, we select the spin-weighted spherical harmonics as functions satisfying
\[\begin{aligned} LΒ² \left\{ {}_{s}Y_{β,m} \right\}(π) &= β(β+1) \left\{ {}_{s}Y_{β,m} \right\}(π), \\ L_z \left\{ {}_{s}Y_{β,m} \right\}(π) &= m \left\{ {}_{s}Y_{β,m} \right\}(π), \\ R_z \left\{ {}_{s}Y_{β,m} \right\}(π) &= s \left\{ {}_{s}Y_{β,m} \right\}(π). \end{aligned}\]
In this case, similar arguments show that the allowed values of $β$ are non-negative integers or half-integers, and for each $β$, the allowed values of $m$ and $s$ are correspondingly integers or half-integers satisfying $-β β€ m β€ β$ and $-β β€ s β€ β$. These conditions only determine the functions up to a normalization factor, which we discuss in the next section.
First, though, we take a moment to consider the effect of the raising and lowering operators. We know the commutators from the previous page, and we now know the eigenvalues, so we can compute just like we do in elementary treatments of scalar spherical harmonics. $L_Β±$ operates just the same as usual, modifying the $m$ index, while $R_Β±$ in an exactly analogous way, but modifies the $s$ index. Thus, we can follow the standard derivation to find the standard ladder relations:
\[\begin{aligned} L_Β± \left\{ {}_{s}Y_{β,m} \right\}(π) &= \sqrt{(β β m)(β Β± m + 1)}\, \left\{ {}_{s}Y_{β,mΒ±1} \right\}(π), \\ R_Β± \left\{ {}_{s}Y_{β,m} \right\}(π) &= \sqrt{(β β s)(β Β± s + 1)}\, \left\{ {}_{sΒ±1}Y_{β,m} \right\}(π). \end{aligned}\]
As usual, we have chosen the coefficients to be real and positive. This is the CondonβShortley condition; imposing it on both $L_Β±$ and $R_Β±$ β and requiring agreement with the ordinary spherical harmonics for $s=0$ β fixes the phases of the ${}_{s}Y_{β,m}$ completely. See the conventions summary.
Integration and normalization
It is natural to define an inner product on the space of functions on $\mathrm{Spin}(3)$ using integration over the group itself. That is, for two functions $f$ and $g$, we define[1]
\[β¨f|gβ©_{\mathrm{Spin}(3)} = \int_{\mathrm{Spin}(3)} fΜ(π)\, g(π)\, dπ,\]
where the integration can be implemented using extended Euler angles. This induces a norm on the space of functions on $\mathrm{Spin}(3)$:
\[\left\| f \right\|Β²_{\mathrm{Spin}(3)} =β¨f|fβ©_{\mathrm{Spin}(3)} =\int_{\mathrm{Spin}(3)} |f(π)|Β²\, dπ.\]
One important point is the norm of the constant function $f=1$, which gives the total volume of $\mathrm{Spin}(3)$. We can compute it explicitly using extended Euler angles and find that the result is $2ΟΒ²$. Recalling that $\mathrm{Spin}(3)$ considered as a subset of $ββ΄$ is just the unit three-sphere $πΒ³$, which has volume $2ΟΒ²$, this makes sense.
Unfortunately, these are not actually the conventional inner product and norm used in the literature. The difference is not too difficult to understand or deal with, however. Note that $|f(π)|Β²$ is actually a field of spin weight 0, no matter what the spin weight of $f$ itself is.[2] Therefore, we can push it forward to a nontrivial function on $πΒ²$ as described previously, and then integrate over $πΒ²$ instead of $\mathrm{Spin}(3)$. Thus, we can define a distinct second norm
\[\left\| f \right\|Β²_{πΒ²} = \int_{πΒ²} |f|Β²\, dΞ©.\]
This is actually the conventional norm used in the literature on spin-weighted spherical functions. Again, we can compute the norm of the constant function $f=1$ and find that it is just the area of the unit 2-sphere, $4Ο$. We can relate the two norms with a simple constant factor:
\[\left\| f \right\|Β²_{πΒ²} =\frac{2}{Ο} \left\| f \right\|Β²_{\mathrm{Spin}(3)}.\]
Clearly, we must choose one or the other of these norms when normalizing the spin-weighted spherical harmonics. To agree with standard scalar spherical harmonics, the conventional choice is to use the $πΒ²$ norm:[3]
\[\left\| {}_{s}Y_{β,m} \right\|Β²_{πΒ²} = 1.\]
Therefore, the spin-weighted spherical harmonics satisfy
\[\left\| {}_{s}Y_{β,m} \right\|Β²_{\mathrm{Spin}(3)} = \frac{Ο}{2}.\]
In fact, while this says that the SWSHs are not orthonormal over $\mathrm{Spin}(3)$, they are still orthogonal:
\[\left\langle {}_{s'}Y_{β',m'} \middle| {}_{s}Y_{β,m} \right\rangle_{\mathrm{Spin}(3)} = \frac{Ο}{2} Ξ΄_{β',β} Ξ΄_{m',m} Ξ΄_{s',s}.\]
However, it is important to remember that the SWSHs with different spin weights are not orthogonal under the $πΒ²$ inner product; all we can say is that if the spin weights are the same, we have this weaker orthonormality over $πΒ²$:
\[\left\langle {}_{s}Y_{β',m'} \middle| {}_{s}Y_{β,m} \right\rangle_{πΒ²} = Ξ΄_{β',β} Ξ΄_{m',m}.\]
For more details about integration and normalization, see the discussion on invariant measure. For explicit calculations, see the page on metrics and integration.
SWSHs with different spin weights are not orthogonal under the $πΒ²$ inner product. This is a common misconception, which is even found in at least one textbook. Specifically,
\[\left\langle {}_{s'}Y_{β',m'} \middle| {}_{s}Y_{β,m} \right\rangle_{πΒ²} β Ξ΄_{β',β}\, Ξ΄_{m',m}\, Ξ΄_{s',s}.\]
Note the inequality sign! The actual integral is a complicated function that should never be used because it depends on the coordinate system.
Defining $π^{(β)}_{m', m}$
Finally, we can consider the Wigner $π$ matrices. These are usually defined as
\[π^{(β)}_{m', m}(Ξ±, Ξ², Ξ³) = \big\langle β, m' \big| e^{-iL_z Ξ±} e^{-iL_y Ξ²} e^{-iL_z Ξ³} \big| β, m \big\rangle.\]
Note that the bra-ket notation usually represents integration over $πΒ²$. We can extend that to obtain the corresponding norm, while integrating over $\mathrm{Spin}(3)$ instead. Thus, we can rewrite this as
\[π^{(β)}_{m', m}(π) = \frac{2}{Ο} \int_{\mathrm{Spin}(3)} \bar{Y}_{β,m'}(π)\, Y_{β,m}\left(πβ»ΒΉ π\right)\, dπ.\]
We can evaluate the action of the differential operators on this function pretty easily, noting that the derivative $d/dΟ΅$ can pass through the integral sign by the Leibniz integral rule. Take the left derivative with respect to $π³$ as an example. Replacing $π$ by $e^{-Ο΅π³/2}\,π$ in the definition puts a factor of $e^{Ο΅π³/2}$ into the argument of $Y_{β,m}$; invariance of the measure then lets us substitute $π β e^{-Ο΅π³/2}π$, which cancels that factor and moves the entire $Ο΅$ dependence onto the conjugated factor:
\[π^{(β)}_{m', m}\left(e^{-Ο΅π³/2}\,π\right) = \frac{2}{Ο} \int_{\mathrm{Spin}(3)} \overline{Y_{β,m'}\left(e^{-Ο΅π³/2}\,π\right)}\, Y_{β,m}\left(πβ»ΒΉ π\right)\, dπ.\]
Because that factor is conjugated, the $i$ in the definition of $L_π³$ effectively becomes $-i$, so the eigenvalue $+m'$ of $Y_{β,m'}$ comes back with the opposite sign. The result is that $π$ satisfies
\[\begin{aligned} LΒ² \left\{ π^{(β)}_{m', m} \right\}(π) &= β(β+1) \left\{ π^{(β)}_{m', m} \right\}(π), \\ L_z \left\{ π^{(β)}_{m', m} \right\}(π) &= -m' \left\{ π^{(β)}_{m', m} \right\}(π), \\ R_z \left\{ π^{(β)}_{m', m} \right\}(π) &= m \left\{ π^{(β)}_{m', m} \right\}(π). \end{aligned}\]
So it is the complex conjugate $\overline{π^{(β)}_{m', m}}$ that behaves like a wavefunction, with $L_z$ eigenvalue $+m'$; see the conventions summary, where these relations are collected, and the details page for the derivation.
That is, Wigner's $π$ matrices are proportional to the spin-weighted spherical harmonics. We can get the proportionality factor by applying the definition above with $π=π$, in which case the integral simplifies to the orthonormality condition for the spin-weighted spherical harmonics:
\[π^{(β)}_{m', m}(π) = \delta_{m', m}.\]
Comparing eigenvalues fixes which index of $π$ is which: a function of spin weight $s$ has $R_z$ eigenvalue $s$, and the conjugate of $π^{(β)}_{m, -s}$ has $L_z$ eigenvalue $m$ and $R_z$ eigenvalue $s$, exactly like ${}_{s}Y_{β,m}$. Sharing all three eigenvalues, the two can differ only by a constant:
\[{}_{s}Y_{β,m}(π) = (-1)^s \sqrt{\frac{2β+1}{4Ο}}\, \overline{π^{(β)}_{m, -s}(π)}.\]
This is the definition this package uses; see the conventions summary and the detailed derivation.
The magnitude $\sqrt{(2β+1)/4Ο}$ is the ratio of the two normalizations. The harmonics are normalized on $πΒ²$, which as we saw above makes $\left\| {}_{s}Y_{β,m} \right\|Β²_{\mathrm{Spin}(3)} = Ο/2$, whereas the $π$ matrices satisfy
\[\int_{\mathrm{Spin}(3)} \left| π^{(β)}_{m', m}(π) \right|Β²\, dπ = \frac{2ΟΒ²}{2β+1},\]
and indeed $\frac{2β+1}{4Ο} \cdot \frac{2ΟΒ²}{2β+1} = \frac{Ο}{2}$.
The remaining factor $(-1)^s$ is not an arbitrary convention: it is the CondonβShortley condition applied to $R_+$ β the same demand that fixed the ladder coefficients above to be real and positive β and it is exactly the factor that appears in the LALSuite and NINJA expressions. For half-integer $s$ its branch matters, and the one settled in the conventions is $(-1)^s β‘ e^{iΟs} = i^{2s}$.
Equivalently, the constant as a whole is pinned by the value at the identity: because $π^{(β)}_{m,-s}(π) = Ξ΄_{m,-s}$, the relation above says
\[{}_{s}Y_{β,-s}(π) = i^{2s} \sqrt{\frac{2β+1}{4Ο}},\]
which is how the summary established the convention.
- 1Note that we are using the physicists' bra-ket notation here [6]. Specifically, the inner product is conjugate-linear in its first argument and linear in its second argument: for complex numbers $a$ and $b$, we have $β¨af|bgβ©=aΜβ¨f|gβ©b$. This convention is common in physics, engineering, and computer science. But note that mathematicians more commonly write $β¨f, gβ©$ (with a comma instead of a vertical bar) and define the inner product with opposite linearity: linear in its first argument and conjugate-linear in its second argument, so that $β¨af|bgβ©=aβ¨f|gβ©bΜ$.
- 2It is possible for a function to have no definite spin weight β that is, to be a sum of components with different spin weights, and therefore to simply not be an eigenfunction of $R_z$. In that case, $|f(π)|Β²$ would also have indefinite spin weight. Nonetheless, its integral over $\mathrm{Spin}(3)$ would only pick up nonzero contributions from the part of that function with spin weight 0.
- 3Wikipedia claims that there are different normalization conventions for spherical harmonics, so that there is a factor of $4Ο$ or $4Ο/(2β+1)$ on the right-hand side of the orthonormality condition. The cited text does not quite say this, though; it's talking about associated Legendre functions.