Differential operators

On the previous page, we saw that the domain on which our most important functions — spherical harmonics and Wigner 𝔇 matrices — are defined is most naturally regarded as the group of unit quaternions, $\mathrm{Spin}(3) \cong \mathrm{SU}(2)$. As a result, we can define a variety of differential operators acting on these functions, relating to infinitesimal motions in this group, acting either from the left or the right on their arguments. Right or left matters because this group is non-commutative. The left derivative operator we define here turns out to be exactly the angular-momentum operator familiar from physics. In the coming pages, we will use these operators to define the spin-weighted spherical harmonics and Wigner's 𝔇 matrices as eigenfunctions.

Definitions and properties

In general, the left Lie derivative of a function $f(𝐐)$ over the unit quaternions with respect to a generator of rotation $𝐠$ is defined as

\[L_𝐠(f)\{𝐐\} := i \left. \frac{df\left(e^{-ϵ\,𝐠/2}\, 𝐐\right)}{d ϵ} \right|_{ϵ=0}.\]

Note that the exponential multiplies $𝐐$ on the left — hence the name. The half-angle in the exponent is the usual quaternionic one: $e^{-ϵ𝐠/2}$ is the rotation through $-ϵ$ about $𝐠$. This is precisely the definition used throughout the conventions summary; the prefactor $i$ is not an arbitrary choice, but is derived — up to its sign — from the mere requirement that raising and lowering operators exist, with the sign then fixed by the standard convention $L_\pm = L_x \pm i L_y$. We will see below that this agrees with the usual definition of the angular momentum from physics, except that in quantum physics a factor of $\hbar$ is usually included.

So, for example, a rotation about the $𝐳$ axis has the quaternion $𝐤$ as its generator of rotation, and $L_𝐤$ defined in this way agrees with the usual angular-momentum operator $L_z$ familiar from spherical-harmonic theory, and reduces to it when the function has spin weight 0, but also applies to functions of general spin weight. Similarly, we can compute $L_x$ and $L_y$, and take appropriate combinations to find the usual raising and lowering (ladder) operators $L_+$ and $L_-$.

In just the same way, we can define the right Lie derivative of a function $f(𝐐)$ over the unit quaternions with respect to a generator of rotation $𝐠$ as

\[R_𝐠(f)\{𝐐\} := -i \left. \frac{df\left(𝐐\, e^{-ϵ\,𝐠/2}\right)}{d ϵ} \right|_{ϵ=0}.\]

Note that the exponential multiplies $𝐐$ on the right — hence the name — and that the prefactor is $-i$ here, in contrast to the $+i$ for $L$. That relative sign is again derived rather than assumed: it is what makes $R_\pm = R_x \pm i R_y$ the raising and lowering operators for $R_z$, and it is what gives $R$ the same commutation relations as $L$ (below) and the eigenvalue $+s$ — rather than $-s$ — on a function of spin weight $s$.

Also observe that

\[𝐐\, e^{-ϵ\,𝐠/2} = 𝐐\, e^{-ϵ\,𝐠/2}\, 𝐐⁻¹\, 𝐐 = e^{-ϵ\,𝐐\, 𝐠\, 𝐐⁻¹/2}\, 𝐐\]

and

\[e^{-ϵ\,𝐠/2}\, 𝐐 = 𝐐\, 𝐐⁻¹\, e^{-ϵ\,𝐠/2}\, 𝐐 = 𝐐\, e^{-ϵ\,𝐐⁻¹\, 𝐠\, 𝐐/2},\]

which mean that

\[R_𝐠(f)\{𝐐\} = -L_{𝐐\, 𝐠\, 𝐐⁻¹}(f)\{𝐐\} \qquad \text{and} \qquad L_𝐠(f)\{𝐐\} = -R_{𝐐⁻¹\, 𝐠\, 𝐐}(f)\{𝐐\}.\]

That is, the right Lie derivative with respect to $𝐠$ is minus the left Lie derivative with respect to the "rotated" version of the algebra element $𝐠$, and vice versa — which is the relation $R_𝐠 = -L_{𝐐\, 𝐠\, 𝐐^{-1}}$ quoted in the summary, identifying $R$ as (minus) the angular-momentum operator expressed in the body-fixed frame. But the arguments depend on the value $𝐐$, which means, for example, that commutator expressions can't be evaluated by simple static substitution.

This $R_𝐠$ operator is less common in physics, because it represents the dependence of the function on the choice of frame (or coordinate system), which is not usually interesting. Multiplication on the left represents a rotation of the physical system, while rotation on the right represents a rotation of the coordinate system. However, this dependence on coordinate system is precisely what defines the spin weight of a function, so this class of operators is relevant in discussions of spin-weighted spherical functions. In particular, $R_z$ is the spin-weight operator — meaning that when it acts on a spin-weighted spherical harmonic of spin weight $s$, it returns $s$ times that same function. Moreover, the operators $R_\pm$ are the spin-raising and -lowering operators $\eth$ and $\bar{\eth}$ originally introduced by Newman and Penrose [4]; with the conventions settled here the identification is

\[\eth = R_+ \qquad \text{and} \qquad \bar{\eth} = -R_-,\]

the minus sign being peculiar to Newman and Penrose.

Note that these definitions are extremely general, in that they can be used for any Lie group, and for any complex-valued function on that group. And in full generality, we have the useful properties of linearity:

\[L_{s𝐚} = sL_{𝐚} \qquad \text{and} \qquad R_{s𝐚} = sR_{𝐚},\]

and

\[L_{𝐚+𝐛} = L_{𝐚} + L_{𝐛} \qquad \text{and} \qquad R_{𝐚+𝐛} = R_{𝐚} + R_{𝐛},\]

for any real scalar $s$ and any elements of the Lie algebra $𝐚$ and $𝐛$. In particular, if the Lie algebra has a basis $𝐞_{(j)}$, we use the shorthand $L_j$ and $R_j$ for $L_{𝐞_{(j)}}$ and $R_{𝐞_{(j)}}$, respectively, and we can expand any operator in terms of these basis operators:

\[L_{𝐚} = \sum_{j} a_j L_j \qquad \text{and} \qquad R_{𝐚} = \sum_{j} a_j R_j.\]

Commutators and angular momentum

In general, for generators $𝐚$ and $𝐛$, we have the commutator relations

\[\left[ L_𝐚, L_𝐛 \right] = \frac{i}{2} L_{[𝐚,𝐛]} \qquad \left[ R_𝐚, R_𝐛 \right] = \frac{i}{2} R_{[𝐚,𝐛]} \qquad \left[ L_𝐚, R_𝐛 \right] = 0,\]

where $[𝐚,𝐛]$ is the commutator of the two generators, which can be obtained directly as the commutator of the corresponding quaternions. Note that these two equations have the same signs, and that the factors of $i/2$ are inherited directly from the definitions of $L_𝐠$ and $R_𝐠$ given above. This sameness is not an accident of notation: multiplying the quaternions in opposite orders flips the sign of the bracket, and the opposite sign of the prefactor in the two definitions ($+i$ for $L$, $-i$ for $R$) flips it back. Had we used the same prefactor for both, $R$ would obey "anomalous" commutation relations with an extra minus sign. The remaining freedom is an overall sign in each definition, which is fixed — as derived in detail — by demanding $L_\pm = L_x \pm i L_y$ and $R_\pm = R_x \pm i R_y$, thereby reproducing the standard $L$ operator and giving $R$ matching commutators.^[1]

Again, these results are valid for general (finite-dimensional) Lie groups, but a particularly interesting case is in application to the three-dimensional rotation group. In the following, we will apply our results to this group.

The commutator relations for $L$ are consistent — except for the differing use of $\hbar$ — with the usual relations from quantum mechanics:

\[\left[ L_j, L_k \right] = i \hbar \sum_{l=1}^{3} ε_{jkl} L_l.\]

Here, $j$, $k$, and $l$ are indices that run from 1 to 3, and index the set of basis vectors $(\hat{x}, \hat{y}, \hat{z})$. If we represent an arbitrary basis vector as $\hat{e}_j$, then the quaternion commutator $[𝐚,𝐛]$ in the expression for $[L_𝐚, L_𝐛]$ becomes

\[[\hat{e}_j, \hat{e}_k] = 2 \sum_{l=1}^{3} ε_{jkl} \hat{e}_l.\]

That is, the commutator $[𝐚,𝐛]$ is essentially twice the cross product of the corresponding vectors. Plugging this into the general expression $[L_𝐚, L_𝐛] = \frac{i}{2} L_{[𝐚,𝐛]}$, we obtain (except for that factor of $\hbar$) the version used in quantum physics.

The raising and lowering operators relative to $L_z$ and $R_z$ satisfy — by definition of raising and lowering operators — the relations

\[[L_z, L_\pm] = \pm L_\pm \qquad [R_z, R_\pm] = \pm R_\pm.\]

As shown in detail elsewhere, these allow us to solve for those operators up to overall signs in terms of the basic generators:

\[L_\pm = L_x \pm i L_y \qquad R_\pm = R_x \pm i R_y.\]

(Interestingly, the solution process also shows that raising and lowering operators can only exist if the factor in front of the derivatives in the definitions of $L_g$ and $R_g$ are pure imaginary numbers.) In particular, this results in the commutator relations

\[[L_+, L_-] = 2L_z \qquad [R_+, R_-] = 2R_z.\]

In the functions this package provides, these operators are returned as matrices acting on vectors of mode weights. As such, we can actually evaluate these commutators as given to cross-validate the expressions and those functions.