The Kepler Problem (Part 9)

Azimuth 2025-08-03

Today I want to make a little digression into the quaternions. We won’t need this for anything later—it’s just for fun. But it’s quite beautiful.

We saw in Part 8 that if we take the spin of the electron into account, we can think of bound states of the hydrogen atom as spinor-valued functions on the 3-sphere. Here a ‘spinor’ is a pair of complex numbers.

But we can also think of a spinor as a quaternion! And we can think of the 3-sphere as the unit sphere in the quaternions! So bound states of hydrogen have a nice quaternionic description.

We can go further using quaternionic analysis.

It took a long time for people to figure out the best generalization of complex analysis to the quaternions. Complex analytic functions are incredibly nice, and important in physics. But when you try to generalize them to ‘quaternion analytic functions’, your first few guesses are unlikely to work well. A guy named Rudolf Fueter figured out the right definition:

• Rudolf Fueter, Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Commentarii Mathematici Helvetici 8 (1936), 371–378.

More recently, some very good mathematical physicists have been further developing this subject:

• Anthony Sudbery, Quaternionic analysis, Mathematical Proceedings of the Cambridge Philosophical Society 85 (1979), 199–225.

• Igor Frenkel and Matvei Libine, Quaternionic analysis, representation theory and physics, Advances in Mathematics 217 (2008), 1806–1877.

Using this, we can describe a lot of hydrogen atom bound states as quaternion analytic functions! And even better, the Dirac operator on spinor-valued functions on the 3-sphere, which I described in Part 8, has a nice description in these terms.

To be a bit more precise: we start by describing a bound state of hydrogen as a function

$\psi \colon S^3 \to \mathbb{H}$

obeying

$\int_{S^3} |\psi(q)|^2 < \infty$

Here $\mathbb{H}$ is the quaternions and $S^3$ is the sphere of quaternions with length 1, which forms a group isomorphic to $\text{SU}(2).$ But we’ll show that a dense subspace of functions of this sort extend to functions $\psi \colon \mathbb{H} - \{0\} \to \mathbb{H}$ that obey a quaternionic analogue of the Cauchy–Riemann equations. Remember, those are the equations obeyed by complex analytic functions.

Okay, let’s get started!

Here is the quaternionic Cauchy–Riemann equation:

$\frac{\partial \psi}{\partial q_0} + i \frac{\partial \psi}{\partial q_1} + j \frac{\partial \psi}{\partial q_2} + k \frac{\partial \psi }{\partial q_3} = 0$

Here $\psi$ is some quaternion-valued function defined on some open subset of the quaternions, and $q_0, \dots, q_3$ are the usual real coordinates on $\mathbb{H}$ for which any quaternion $q$ is of the form

$q = q_0 + q_1 i + q_2 j + q_3 k$

For any open set $O \subseteq \mathbb{H},$ people say a function $\psi \colon O \to \mathbb{H}$ is regular if it’s differentiable in the usual real sense and the quaternionic Cauchy–Riemann equation holds. In Theorem 1 of his paper, Sudbery shows that any regular function is infinitely differentiable in the usual real sense, in fact real-analytic.

Let $U_k$ be the space of regular functions on $\mathbb{H} - \{0\}$ that are homogeneous of degree $k \in \mathbb{Z},$ meaning that

$\psi(\alpha q) = \alpha^k f(q) \qquad \qquad \forall q \in \mathbb{H} - \{0\}, \alpha \in \mathbb{R} - \{0\}$

Clearly any function $\psi \in U_k$ is determined by its restriction to the unit sphere $S^3 \subset \mathbb{H}.$ But in the proof of his Theorem 7, Sudbery shows something less obvious: the restriction is an eigenfunction of the Dirac-like operator $D$ that I mentioned in Part 8!

To prove this, the trick is to write the quaternionic Cauchy–Riemann operator

$\overline{\partial} = \frac{\partial}{\partial q_0} + i \frac{\partial}{\partial q_1} + j \frac{\partial}{\partial q_2} + k \frac{\partial}{\partial q_3}$

in something like polar coordinates, involving a radial derivative but also the operator $D$ that I introduced in Part 8. The radial derivative of a homogeneous function $\psi$ is easy to work out, and then using $\overline{\partial} \psi = 0$ we can show

$D(\psi|_{S^3}) = k \psi|_{S^3}$

So, Sudbery shows that

$\psi \in U_k \implies D(\psi|_{S^3}) = k \psi|_{S^3}$

(although he uses different notation).

We saw last time that the Dirac operator on the 3-sphere is

$\partial\!\!\!/ = D + \tfrac{3}{2}$

So, we get

$\psi \in U_k \implies \partial\!\!\!/(\psi|_{S^3}) = (k + \tfrac{3}{2}) \psi$

With more work (see my paper) we can show the converse: any eigenfunction of the Dirac operator with eigenvalue $k + \tfrac{3}{2}$ is the restriction of a function in $U_k.$

Thus, each eigenspace of the Dirac operator on the 3-sphere can be seen as the space of all regular functions $\psi \colon \mathbb{H} - \{0\} \to \mathbb{H}$ that are homogeneous of some particular degree.

So, we can think of hydrogen atom bound states, or at least those that are finite linear combinations of energy eigenstates, as regular functions

$\psi \colon \mathbb{H} - \{0\} \to \mathbb{H}$

And these finite linear combinations are dense in the space of all hydrogen atom bound states!

To summarize in a sensationalistic way: hydrogen is quaternionic!

Nitty-gritty details

I’ve skimmed over some details. Please stop here unless you really love the quaternions. But to get everything from Part 8 to mesh nicely with what we’re doing now, we need to think of spinors as quaternions in a good way. We need to choose an isomorphism of real vector spaces

$\alpha \colon \mathbb{C}^2 \xrightarrow{\sim} \mathbb{H}$

in such a way that

• multiplication by $-i\sigma_1, -i\sigma_2$ and $-i\sigma_3$ on $\mathbb{C}^2$ correspond to left multiplication by the quaternions $i,j$ and $k$ on $\mathbb{H},$ and

• multiplication by $i$ on $\mathbb{C}^2$ corresponds to right multiplication by the quaternion $i.$

In case you know some algebra and are wondering what’s really going on here, the idea is that $\mathbb{H}$ is both a left and a right module of itself in the usual way. We can make it into a 2-dimensional complex vector space in a unique way such that multiplication by $i$ is right multiplication by the quaternion $i.$ Since left and right multiplication commute, this makes $\mathbb{H}$ into a 2-dimensional complex vector space on which $\mathbb{H}$ acts complex-linearly by left multiplication.

But $\mathbb{C}^2$ is also a 2-dimensional complex vector space on which $\mathbb{H}$ acts complex-linearly, with $i, j, k$ acting as matrix multiplication by $-i \sigma_1, -i \sigma_2, -i \sigma_3.$

All this suggests that with these structures chosen, $\mathbb{H}$ and $\mathbb{C}^2$ are isomorphic as complex vector spaces on which $\mathbb{H}$ acts complex-linearly!

But how do we find such an isomorphism

$\alpha \colon \mathbb{C}^2 \xrightarrow{\sim} \mathbb{H}$ ?

I got confused for a while, but here’s a systematic approach. Suppose we have such an isomorphism. We must have

$\alpha(1) = (x,y)$

for some numbers $x,y \in \mathbb{C}.$ We want

$\alpha(i) = \alpha(1 i) = \alpha(1) i = (ix,iy)$

but we also want

$\alpha(i) = \alpha(i 1) = -i \sigma_3 \alpha(1) = (-iy,-ix)$

(I’m going to skip lots of computational steps and focus on explaining the strategy.) So, we must have

$(ix,iy) = (-iy,-ix)$

or in other words

$y = -x$

Because we’re assuming $\alpha$ is complex-linear (where we multiply quaternions on the right by $a + bi$ ), we can assume without loss of generality that