Here’s a quick sketch of how it works.

**SO(3) story.** As a representation of SO(3) we have

where is the spin-j representation of SO(3). This is the usual decomposition of functions on the sphere into spherical harmonics, expressed in a less grungy manner: spherical harmonics give explicit bases for the irreps

The space of hydrogen atom bound states at the th energy level is isomorphic to

if we call the lowest energy level the 0th energy level, which is not standard (people usually call it the first). Thus, a basis of bound states is indexed by another number called saying which irrep we’re in, and a third number called indexing a basis of this irrep. These numbers are the usual **quantum numbers** for the electron in the quantum number, except that my is one less than the usual convention.

These energy eigenstates are functions on but we can restrict them to functions on the sphere, giving

**SO(4) story.** On the other hand, the space of all bound states of the hydrogen atom is naturally a representation not only of but also , and thus of the double cover . Since the Peter–Weyl theorem says

as representations of The states in the th energy level form the subspace where The from the previous story is covered by the diagonal subgroup of

To check the consistency of the two stories, note that perfect squares are sums of odd numbers:

where and So, the energy eigenspace

from the first story at least has the same dimension as the energy eigenspace

from the second story. But in fact they’re isomorphic as representations of And indeed, this energy eigenspace is an irrep of So, we’re seeing how an irrep of decomposes as irreps of

It’s nice that this math gives a pretty good approximate description of the most common element in the Universe. When asked if there were anything that could be concluded about God from the study of natural history, Haldane said “He has an inordinate fondness for beetles”. But there are a lot more hydrogen atoms.

]]>The book seems to have appeared after the paper you mention, so it’s not as if the paper corrects a mistake in the book.

]]>Conformal regularization of the Kepler problem,

Commun. Math. Phys. 103 (1986), 403-413.

the 6D phase space to an 8D extended phase space (which also contains energy and time as conjugate variables), and then restricts the action [before (1.5) there] to an orbit, which is 7D. ]]>

I see now that what I really need is not just this stabilizer, but an intuitive understanding of the connection between the group SO(2,4) and the Kepler manifold.

For people just listening in: the **Kepler manifold** is the cotangent bundle of the 3-sphere with the zero covectors removed. Points in here correspond to negative-energy states of a particle in an attractive inverse square force.

Cordani explains a way that SO(2,4) acts on this space. He does it using conformally compactified Minkowski spacetime

which is something I like. But I need to to get more of an intuition of what’s going on! When I can picture things in my head, I can usually make rapid progress. I’m not at that stage yet.

Right this moment I’m stuck on something embarrassingly simple. I’ll explain it publicly just to keep the conversation going, though I should probably figure it out myself. Theorem 6.9 of Cordani’s book seems to say that is diffeomorphic (even symplectomorphic) to the manifold of future-pointing null covectors in . But is 6-dimensional and the space of future-pointing null covectors in is 7-dimensional!

Did Cordani mean to take the manifold of future-pointing null covectors in and *fiberwise projectivize* it, reducing its dimension by one? Am I misreading something? If I can’t get this right, I can’t trust myself… and if he can’t get it right I can’t trust him.

Quantization of the Kepler Manifold,

Commun. Math. Phys. 113 (1988), 649–657.

describes the point stabilizer on p. 651, so that is the Kepler manifold.

]]>If a differential equation have N free parameters, then it is possible to obtain for each transformation in the group a constraint on the parameters, that could be solved using a minimization with gradient descent (or other optimization algorithm) whatever the complexity of the constraint; if the differential equation is the Schrodinger equation, with a polynomial potential with N free parameters, then it is possible to obtain a class of wave function with the symmetries of the group, in the three dimensional space and only for some groups. ]]>

Just to put the problem into my brain, so I can start pondering it in my spare moments, could you tell me what subgroup H ⊂ SO(2,4) we should use, for SO(2,4)/H to be this symmetric space?

When I get a bit more time I’ll email you… or you can email me whenever you want.

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