The n-Category Café

A bit is just a binary alternative: 1 or 0, true or false. That’s how it works in classical logic. We could also have a ‘trit’, meaning 3 alternatives.

In quantum physics we instead have qubits and qutrits.

Qubits and qutrits are usually described using complex numbers. The algebra of observables of a qubit is the Jordan algebra $\mathfrak{h}_2(\mathbb{C})$, consisting of $2 \times 2$ self-adjoint complex matrices. Similarly, the algebra of observables of an qutrit is the Jordan algebra $\mathfrak{h}_3(\mathbb{C})$, consisting of $3 \times 3$ self-adjoint complex matrices.

We can also study systems with more than 3 alternative ways to be. They work the same way, using the Jordan algebras $\mathfrak{h}_n(\mathbb{C})$ with $n \gt 3.$

But we can also do quantum mechanics using other number systems! The options have been mapped out, and the largest allowed number system for this purpose is the algebra of octonions.

A weird thing is that Jordan algebras built using octonions can describe qutrits, but not quantum systems with more than 3 alternative ways to be. The algebra of observables of an octonionic qutrit is the so-called ‘exceptional’ Jordan algebra $\mathfrak{h}_3(\mathbb{O})$, consisting of $3 \times 3$ self-adjoint octonion matrices. What makes it exceptional is that $\mathfrak{h}_n(\mathbb{O})$ is not a Jordan algebra when $n$ is bigger than 3.

So, there’s something special about octonionic qutrits — and it turns out that every symmetry in the gauge group of the Standard Model is a symmetry of an octonionic qutrit!

Not every symmetry of an octonionic qutrit is a symmetry of the Standard Model. But those that do have a simple description. They are those that restrict to give symmetries of an ordinary qutrit sitting inside the octonionic qutrit… and an ordinary qubit sitting inside that!

That sounds exciting, but also vague, so let me make it precise.

While lots of people say the gauge group of the Standard Model of particle physics is $\text{U}(1) \times \text{SU}(2) \times \text{SU}(3)$, in fact a certain subgroup of this acts trivially on all known particles. If we mod out by that, we’re left with

$$\begin{array}{ccl} \text{S}(\text{U}(2) \times \text{U}(3)) &= & \Big\{ x \in \text{SU}(5) : x = \left( \begin{array}{c c c c c} \ast & \ast & 0 & 0 & 0 \\ \ast & \ast & 0 & 0 & 0 \\ 0 & 0 & \ast & \ast & \ast \\ 0 & 0 & \ast & \ast & \ast \\ 0 & 0 & \ast & \ast & \ast \end{array} \right) \; \Big\}. \end{array}$$

and this is the group I’m talking about.

We proved two theorems describing this group in terms of the symmetries of an octonionic qutrit. The group of automorphisms of the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$ is a 52-dimensional Lie group known affectionately as $\text{F}_4$ — so that’s what I mean by the symmetries of an octonionic qutrit.

Here’s our main result:

Theorem 1. Suppose $X,B$ are Jordan subalgebras of $\mathfrak{h}_3(\mathbb{O})$ such that

$$X \cong \mathfrak{h}_2(\mathbb{C}), \;\; B \cong \mathfrak{h}_3(\mathbb{C}), \;\; X \subset B.$$

Then

$$\text{Stab}(X) \cap \text{Stab}(B)_0 \cong \text{S}(\text{U}(2) \times \text{U}(3)).$$

Here $\text{Stab}(X)$ is the stabilizer of $X$ — that is, the subgroup of $\text{F}_4$ consisting of elements that map $X$ to itself — while $\text{Stab}(B)_0$ is the identity component of the stabilizer of $B$.

This ‘identity component’ business is rather sneaky, but it turns out that guys in $\text{Stab}(B)_0$ are symmetries of an ordinary qutrit that can be described as unitary operators on $\mathbb{C}$, while $\text{Stab}(B)$ also contains those symmetries that are described by antiunitary operators. The CPT symmetry of the Standard Model is antiunitary, for example.

Theorem 1 emerged from a related result, which grew out of the work of Todorov and Dubois-Violette:

Theorem 2. Suppose $A,B$ are Jordan subalgebras of $\mathfrak{h}_3(\mathbb{O})$ such that

$$A \cong \mathfrak{h}_2(\mathbb{O}), \;\; B \cong \mathfrak{h}_3(\mathbb{C}), \;\; A \cap B \cong \mathfrak{h}_2(\mathbb{C}).$$

Then

$$\text{Stab}(A) \cap \text{Stab}(B)_0 \cong \text{S}(\text{U}(2) \times \text{U}(3)).$$

Todorov and Dubois–Violette proved this for a certain standard choice of subalgebras $A$ and $B$. Thus, the challenge in proving Theorem 2 was to show that every other choice can be mapped to this standard choice using the action of $\text{F}_4$. This shows that the theorem is not an artifact of a specific choice, but rather a general fact.

How do we prove these results?

We start by constructing the octonion product from $\text{SU}(3)$-invariant operations on $\mathbb{C}$ and $\mathbb{C}^3$. We then use this description to reprove Todorov and Dubois–Violette’s special case of Theorem 2. Then we show that $\text{F}_4$ acts transitively on the set of subalgebras of $\mathfrak{h}_3(\mathbb{O})$ that are isomorphic to $\mathfrak{h}_3(\mathbb{C})$. We also show every Jordan subalgebra of $\mathfrak{h}_3(\mathbb{O})$ isomorphic to $\mathfrak{h}_2(\mathbb{C})$ is contained in a unique Jordan subalgebra isomorphic to $\mathfrak{h}_2(\mathbb{O})$. This lets us prove that $\text{F}_4$ acts transitively on the set of pairs of Jordan subalgebra $A, B \subset \mathfrak{h}_3(\mathbb{O})$ with $A \cong \mathfrak{h}_2(\mathbb{O})$, $B \cong \mathfrak{h}_3(\mathbb{C})$ and $A \cap B \cong \mathfrak{h}_3(\mathbb{C})$. Theorem 2 then follows from Todorov and Dubois-Violette’s special case. We conclude by using these results to prove Theorem 1.

However, if you want to get into the details of the physics, the interesting part is how the strong force gauge group $\text{SU}(3)$ and the electroweak $\text{S}(\text{U}(1) \times \text{U}(2))$ show up from the relation between octonionic qutrits, complex qutrits and complex qubits. You’ll see that in the proof of Lemma 4.

And if you want to get into the details of the math, the main interesting thing here is the use of Jordan algebra technology like ‘Peirce decompositions’ and ‘Jordan frames’ to figure out what it must be like when you have a Jordan algebra $\mathfrak{h}_2(\mathbb{L})$ or $\mathfrak{h}_3(\mathbb{L})$ sitting inside $\mathfrak{h}_3(\mathbb{K})$, where $\mathbb{L}$ is some normed division algebra contained in a bigger normed division algebra $\mathbb{K}$.

What it all ‘really means’, if anything, is a question for later. It could be just a coincidence. Of course I hope not.