The n-Category Café

Greg wrote:

Here’s a naive question. The article seemed to imply that you might be able to construct an interaction Lagrangian between fermions and bosons in Riemannian 4-dimensional space by treating both as quaternions, and just multiplying them in the quaternionic way. But don’t you need a scalar for the Lagrangian?

Yes you do. In our Scientific American article, when we discuss the interaction, we aren’t talking directly about the Lagrangian that eats a vector and two spinors and spits out a scalar. Rather, we’re reinterpreting this via duality as a linear map that eats a vector and a spinor and spits out a spinor. This is often drawn as a Feynman diagram:

depicting a process where a spinor absorbs a vector and turns into another spinor.

To make this precise we need to distinguish between right- and left-handed spinors. Vectors, right-handed spinors and left-handed spinors are all 4-dimensional representations of $Spin(4)$, the double cover of the rotation group in 4 dimensions. We can identify all these representations with the quaternions, and then the Feynman diagram above corresponds to multiplication of quaternions. However, $Spin(4)$ acts in a different way on vectors, right-handed spinors and left-handed spinors. Let’s see how this goes.

First of all, $Spin(4) \cong SU(2) \times SU(2)$. We’ll think of an element of $SU(2)$ as a ‘unit quaternion’, meaning a quaternion of norm 1. So, we’ll think of an element of $Spin(4)$ as a pair $(g,h)$ of unit quaternions.

In the vector representation of $Spin(4)$ on the quaternions $\mathbb{H}$, the pair $(g,h)$ acts on a vector $v \in \mathbb{H}$ like this:

$$(g,h) v = g v h^{-1}$$

where the inverse is needed to ensure

$$(g,h) (g',h') v = (g g' , h h') v$$

Here it’s easy to guess why $Spin(4)$ is a double cover of the rotation group: both $(1,1)$ and $(-1,-1)$ act in the same way on vectors.

In the left-handed spinor representation of $Spin(4)$ on the quaternions $\mathbb{H}$, the pair $(g,h)$ acts on a left-handed spinor $\phi \in \mathbb{H}$ like this:

$$(g,h) \phi = g \phi$$

Similarly, in the right-handed spinor representation of $Spin(4)$ on the quaternions $\mathbb{H}$, the pair $(g,h)$ acts on a right-handed spinor $\psi \in \mathbb{H}$ like this:

$$(g,h) \psi = h \psi$$

To be precise, the diagram

depicts a process where a right-handed spinor absorbs a vector and turns into a left-handed spinor (or the other way around, but let’s do this case). This map is just quaternion multiplication:

$$v \otimes \psi \mapsto v \psi$$

where I’m treating the vector $v$ and the right-handed spinor $\psi$ as quaternions.

The key thing to check is that this map is an intertwining operator. Let’s transform both $v$ and $\psi$ by an element $(g,h)$ in $Spin(4)$, and then multiply them:

$$((g,h) v) \; ((g,h)\psi) = (g v h^{-1}) \; (h \psi) = g (v \psi)$$

Good! The result $v \psi$ transforms like a left-handed spinor!

With some extra work one can turn this Feynman diagram back into a Lagrangian, which will be a recipe that takes a vector, a left-handed spinor and a right-handed spinor and spits out a number. And modulo various conventions that I’m too tired to worry about, this Lagrangian will be something very much like the famous $\overline{\Psi} A_\mu \gamma^\mu \Psi$, where now however $\Psi$ is a pair $(\psi,\phi)$ consisting of a right-handed and a left-handed spinor.

Sorry to fizzle out before giving you the Lagrangian, but it’s late here, and really the only other ingredient you need is to know that the vector rep, left-handed spinor rep and right-handed spinor of $Spin(4)$ are each isomorphic to their dual, via the standard inner product on $\mathbb{H}$, namely

$$\langle x, y \rangle = Re(x^* y)$$

where the $*$ stands for quaternionic conjugation. Using this fact, you can dualize quaternion multiplication and get an explicit map that takes takes a vector, a left-handed spinor and a right-handed spinor and spits out a number. And this then gives the interaction Lagrangian you’re looking for.

Thanks! That’s incredibly beautiful. I’m used to thinking of the left-handed and right-handed spinors only as two-dimensional, with the elements of SU(2) acting on $\mathbb{C}^2$; I never knew before that you could simply embed them in $\mathbb{H}$ that way!

Yes, it may seem like a useless coincidence that the vectors $V = \mathbb{R}^4$, right-handed spinors $S^+ = \mathbb{C}^2$ and left-handed spinors $S^- = \mathbb{C}^2$ in a 4-dimensional Riemannian universe all form 4-dimensional real vector spaces. People don’t talk about it much. But it’s the tip of an iceberg: it lets us identify vectors and both kinds of spinors with quaternions, and describe all their physical interactions using quaternion multiplication and the quaternion inner product, which is just the usual 4d Euclidean inner product.

Since I knew that you’re writing a trilogy set in a 4-dimensional Riemannian universe, I should have thought more about this. Now I can imagine an other-universe version of Hamilton inventing the quaternions, quickly leading to the thrilling realization that a complex Feynman diagram describing interaction between matter and light is really nothing but an elaborate tapestry woven from quaternion multiplication!

But somehow lately I’ve been trying to treat the 1d, 2d, 4d and 8d Riemannian universes on an equal footing using real numbers, complex numbers, quaternions and octonions — they all work about the same way — and thinking of them as mere warmups to the 3d, 4d, 6d and 10d Lorentzian universes, where superstrings make their appearance.

Now that I think of it, Hamilton would have been a much happier man in a Riemannian universe. He was trying to develop an algebra of ‘triplets’ to describe vectors in 3d space, and was forced by the sheer power of mathematics to switch to 4 dimensions, and the quaternions. When he applied these to physics, every triplet needed to be supplemented by an extra number to give a full-fledged quaternion! He intuited that this 4th number had something to do with time. Indeed he wrote this poem:

THE TETRACTYS

Or high Mathesis, with her charm severe,
Of line and number, was our theme; and we
Sought to behold her unborn progeny,
And thrones reserved in Truth’s celestial sphere:
While views, before attained, became more clear;
And how the One of Time, of Space the Three,
Might, in the Chain of Symbol, girdled be:
And when my eager and reverted ear
Caught some faint echoes of an ancient strain,
Some shadowy outlines of old thoughts sublime,
Gently he smiled to see, revived again,
In later age, and occidental clime,
A dimly traced Pythagorean lore,
A westward floating, mystic dream of FOUR.

The tetractys is this symbol, beloved by the Pythagoreans:

‘Tetra-’ means ‘four’, for the 4 rows, but the figure has 10 dots — and if you stare at it, you can see the Pythagoreans knew spacetime was 10-dimensional, with a 6-dimensional Calabi-Yau manifold as fiber over a 4-dimensional base. Hamilton got the 4 but not the 10.

Seriously, he was onto something… but his insights would have worked a lot better in a Riemannian universe. As it was, Gibbs had to laboriously separate Hamilton’s quaternions into the vector and scalar part to get the setup we learn today. The quaternionists battled the advocates of vectors for decades, lost the fight, and sank into obscurity.

Later, when developing special relativity, Einstein had to re-integrate the vector and scalar into a single 4-vector, but with the key switch from signature ++++ to signature +++-. Then Pauli had to reinvent the quaternions, under the guise of ‘Pauli matrices’, and Dirac had to reinvent Clifford algebras, in the special case of signature +++-, to describe spinors.

It might all have happened a lot more smoothly in your universe!

I figured out the Riemannian interaction Lagrangian between a photon and a Dirac spinor, with everything expressed as quaternions:

$\mathcal{L}_int = -q (\psi_L^* A \psi_R + \psi_R^* A^* \psi_L)$

Here * is quaternionic conjugation, $q$ is the particle’s charge, $A$ is the electromagnetic four-potential and $(\psi_L, \psi_R)$ is the Dirac spinor. This is obviously real-valued, and it’s not hard to see that it’s SO(4)-invariant.

That paper looks fascinating — lots of good stuff. It very much deserves to be reworked and clarified using more of a mathematician’s approach. It would take someone who understands the normed division algebras well. John H?