How many truly different multipolar charge distributions are there?

Question

Dipolar charge distributions are essentially all the same: regardless of how one adds up a combination of the form $$ \sigma(\theta,\varphi) = \operatorname{Re}\left[\sum_{m=-1}^1 a_m Y_{1m}(\theta,\varphi)\right], $$ you only ever get a rotated version of the canonical $Y_{10}(\theta,\varphi) \propto \cos(\theta)$, multiplied by some global constant.

Quadrupolar charge distributions, on the other hand, are more interesting, because you can have 'double peanut' distributions of the form $$ \sigma(\theta,\varphi) = \sin(\theta)\cos(\varphi)\cos(\theta) = \frac{xz}{r^2}, $$ but you can also have the usual two-balls-and-a-ring distribution of the form $$ \sigma(\theta,\varphi) = 3\cos^2(\theta)-1 = \frac{2z^2-x^2-y^2}{r^2}, $$ and those two are not rotations of each other. With that in mind, then:

• If you consider two charge distributions to be equivalent if they differ only by a rigid rotation or by a global constant, how many real parameters do you need to describe quadrupolar charge distributions on the unit sphere of the form $$\sigma(\theta,\varphi) = \operatorname{Re}\left[\sum_{m=-2}^2 a_m Y_{2m}(\theta,\varphi)\right]?$$ What is the dimensionality of this manifold, and what is its topology? Can this reduction be done in a systematic way? And, if so, how? What is the physical significance of those parameters, and do they have set names in the literature?

• Similarly, how do those questions look like for octupoles?

• What about for general multipoles?

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Edit: I guess, upon further reflection, what I'm asking for is the essential topology of the orbit space of the action of $\mathrm{SO}(3)$ on its irreducible representations: is the quotient space a manifold? if so, what are its dimensions and its topology? if not, why not? I would like to see this for arbitrary $\ell$ but with a specific emphasis on both $\ell=2$ and $\ell=3$, which strike me as the first two nontrivial examples. (I don't anticipate $\ell=4$ to be much more complicated than the octupole representation, but I do think that the octupole brings in nontrivial wrinkles compared to the quadrupole.) If people can comment on what happens for half-integral $j$ that would be awesome as well.

I am aware, in particular, of the reduction of the quadrupole representation by diagonalizing its coefficient matrix, but it is not at all clear to me how one would generalize this to the octupole layer's rank-3 tensors and beyond, and I would definitely like answers to address this generalization.

On the other hand, I do want the discussion to also address the particulars of these orbit spaces: what do the different points represent, and how do those charge distributions actually look like, at least at the 'extremal' points (like $Y_{20}$ and $\operatorname{Re}(Y_{22})$). The dimension-counting argument in Logan's answer is interesting, but it indicates that there are $2\ell-2$ non-equivalent distributions at $\ell\geq 2$, and this means that several of those dimensions are not encompassed by the usual spherical harmonics: since $\operatorname{Re}(Y_{\ell ,\pm m})$ and $\operatorname{Im}(Y_{\ell ,\pm m})$ are rotation-equivalent, when taken neat the $Y_{\ell m}$ can only produce up to $\ell+1$ distinct distributions (with one knocked out at $\ell=2$ by rotational equivalence), this means that from $\ell=4$ onwards there need to be at least $\ell-3$ independent linear combinations of the $Y_{\ell m}$ that are not rotationally equivalent to any of them. What do these combinations look like? I would like explicit examples for the first nontrivial cases, as well as systematic methods to get them for arbitrary $\ell$.

Now, I realize that this whole package is a big ask, but I do think it's interesting and worth exploring. I'll probably add some fake-internet-points sweetener in a few days, but I do want more in-depth answers than the current ones.

Answer 1

The answer for quadrupoles is $2$. The best way to think of a quadrupole is to consider the elements $Y_{2m}$ as linear combinations of entries in a symmetric traceless matrix: $$ {\cal Q}=\left(\begin{array}{ccc} r^2-x^2&xy&xz\\ xy&r^2-y^2&yz\\ xz&yz&r^2-z^2\end{array}\right) $$ Since this matrix is symmetric, it can be brought to diagonal form with a rotation, so up to rotations there remains two eigenvalues to parametrize your quadrupole. These are usually related to $Y_{20}(\theta,\phi)$ and some linear combinations of $Y_{2\pm 2}(\theta,\phi)$ components so that, in the usual notation $$ {\cal Q}_0\sim 2z^2-x^2-y^2 \, ,\qquad {\cal Q}_2\sim x^2-y^2\, . $$ When ${\cal Q}_2=0$ the figure is axially symmetric, either oblate or prolate depending on the sign of ${\cal Q}_0$. When ${\cal Q}_2\ne 0$ the figure has no symmetry axis (basically, a non-symmetric top.)

The principal octupole moment usually discussed in the literature is proportional to: $$ Y_{30}(\theta,\phi)\sim 5\cos(\theta)^3-3\cos(\theta)\, , $$ but there ought to be at least one more ($\sim Y_{33}(\theta,\phi)-Y_{3,-3}(\theta,\phi)$?) to probe the distribution in the $\phi$ direction.

The place to look for this is the nuclear physics literature since higher multipoles reveal information about nuclear shapes, but the standard textbooks I have handy (Ring and Schuck, Krane) do not discuss the $m\ne 0$ components of the higher multipoles. Octupole moments are used probe the "pear-shapedness" of the figure, and the pear being axially symmetric it may be that the $m\ne 0$ components are too small to bother.

(See also this post for additional comments).

Answer 2

There is an analogous situation in quantum mechanics. It is of course well known that the finite-dimensional irreducible complex projective representations of $SO(3)$ are parametrized by a non-negative half-integer $s=0,1/2,1,3/2,\ldots$ with dimension $2s+1$. However, on these representations the projective action of $SO(3)$ is only transitive if $s=0$ or $s=1/2$. Understanding the orbits of $SO(3)$ on these representations is thus a question of interest.

A simple solution was found by Majorana, commonly known as the Majorana stellar representation. A general nonzero vector in the spin $s$ representation can be written as the symmetrized tensor product of $2s$ nonzero vectors in the spin-$1/2$ representation. This is unique up to scaling the vectors by factors which multiply to $1$ and by reordering the vectors. Passing to the projective spaces, a general state is represented by a collection of $2n$ unlabeled (not necessarily distinct) points on the Riemann sphere. The action of $SO(3)$ here is just rotations of the sphere.

Thus a dense subset of the projective space $P^{2s}$ is mapped to the nondegenerate configurations of $2s$ unlabelled points on a sphere, which is a well-studied example of a configuration space. The topology of configuration spaces is well-understood by mathematicians. For example, its fundamental group is the quotient of the braid group on $2s$ strands by a single relator (see this paper for a proof). That configuration space inherits the $SO(3)$ action and the orbit space is the quotient of that action. There are also degenerate orbits when $2$ or more of the points coincide, and these prevent the orbit space from being a legitimate manifold (it is, however, an orbifold).

To relate this back to multipole distributions, actually we don't really need to do much work at all. The $Y_l^m$ spherical harmonics for $m=-l,\ldots,l$ span a copy of the $s=l$ representation of $SO(3)$. Here we are looking at a real representation rather than a complex representation. As such we must restrict the space to the appropriate real section spanned by real spherical harmonics. Additionally, in this case it is an ordinary representation, not a projective representation. This means that we get $1$ degree of freedom back from rescaling by real numbers (rescaling by complex numbers is disallowed). Finally, the quotient by the $SO(3)$ action kills 3 real degrees of freedom for sufficiently large $l$ such that the action is faithful (in this case already $l=2$ is sufficient). In particular then, the dimensionality of the orbit space is, for $l=0,1,2,3,4,\ldots$, $$d_l=1,1,2,4,6,\ldots.$$