The n-Category Café

In 1932, Pascual Jordan tried to isolate some axioms that an ‘algebra of observables’ should satisfy in quantum mechanics The unadorned phrase ‘algebra’ usually signals an associative algebra, but this not the kind of algebra Jordan was led to. The reason is that in traditional quantum mechanics, self-adjoint $n \times n$ complex matrices count as observables. These aren’t closed under matrix multiplication. Instead, they’re closed under linear combinations and the commutative operation

$$a \circ b = \frac{1}{2}( a b + b a)$$

This operation is not associative! It is, however, power-associative: any way of parenthesizing a product of $n$ copies of the same element $a$ gives the same result, which we can call $a^n$.

This led Jordan to define what is now called a formally real Jordan algebra: a real vector space with a bilinear, commutative and power-associative product satisfying

$$a_1^2 + \cdots + a_n^2 = 0 \quad \implies \quad a_1 = \cdots = a_n = 0$$

for all $n$. The last condition, called formal reality, gives a partial ordering: if we write $a \le b$ when the element $b - a$ is a sum of squares, this condition says

$$a \le b \; and \; b \le a \; \quad \implies \quad a = b .$$

In 1934, Jordan published a paper with von Neumann and Wigner classifying all finite-dimensional formally real Jordan algebras. They began by proving that any such algebra is a direct sum of ‘simple’ ones. A formally real Jordan algebra $A$ is simple when its only ideals are $\{0\}$ and $A$ itself, where an ideal is the usual sort of thing: a vector subspace $B \subseteq A$ such that $b \in B$ implies $a \circ b \in B$ for all $a \in A$.

Then they proved that every simple finite-dimensional formally real Jordan algebra is isomorphic to one on this list:

• The algebra $\mathfrak{h}_n(\mathbb{R})$ of $n \times n$ self-adjoint real matrices with the product $a \circ b = \frac{1}{2}(a b + b a)$.
• The algebra $\mathfrak{h}_n(\mathbb{C})$ of $n \times n$ self-adjoint complex matrices with the product $a \circ b = \frac{1}{2}(a b + b a)$.
• The algebra $\mathfrak{h}_n(\mathbb{H})$ of $n \times n$ self-adjoint quaternionic matrices with the product $a \circ b = \frac{1}{2}(a b + b a)$.
• The algebra $\mathfrak{h}_3(\mathbb{O})$ of $n \times n$ self-adjoint octonionic matrices where $n \le 3$, with the product $a \circ b = \frac{1}{2}(a b + b a )$.
• The spin factors: the algebras $\mathbb{R}^n \oplus \mathbb{R}$ with the product $(x,t) \circ (x', t') = (t x' + t' x, x \cdot x' + t t')$

Here we say a square matrix $T$ is self-adjoint if $T_{j i} = (T_{i j})^\ast$, where we use the conjugation that we have on $\mathbb{R}, \mathbb{C}, \mathbb{H}$ and $\mathbb{O}$.

What does all this mean for physics? Self-adjoint matrices with entries in $\mathbb{R},\mathbb{C}$ and $\mathbb{H}$ give observables in three forms of quantum mechanics. Complex quantum mechanics is the most important one, but real and quaternionic quantum mechanics exist too, and I’ve argued that they’re lurking behind the scenes in physics:

• John Baez, Division algebras and quantum theory, Found. Phys. 42 (2012), 819–855.

The spin factors are weirder, but they have an intriguing relation to special relativity. For any formally real Jordan algebra we get a cone

$$\{a : \; a \ge 0\}$$

For the spin factor $\mathbb{R}^n \oplus \mathbb{R}$ this can be identified with the ‘future cone’ in $(n+1)$-dimensional Minkowski spacetime: the set of points you can get to if you start at the origin and move no faster than light. This gives a whole different outlook on formally real Jordan algebras: we can think of them as spacetimes having a concept of causality given by a future cone.

In fact there’s a theorem to justify this: the Koecher–Vinberg theorem! This says that any formally real Jordan algebra gives a specially nice sort of cone called a ‘symmetric cone’ (actually the interior of the cone mentioned above) — and conversely, any symmetric cone gives a formally real Jordan algebra. You can see details here:

• Wikipedia, Symmetric cone.
• nLab, Self-dual homogeneous convex cones.

So, there’s something funny going on: we can think of formally real Jordan algebras either as algebras of observables or as spacetimes with future cones. For example, $\mathfrak{h}_2(\mathbb{C})$ is the Jordan algebra of observables of a qubit, but we can also see it as 4d Minkowski spacetime! This generalizes to qubits defined using other normed division algebras:

• The Jordan algebra $\mathfrak{h}_2(\mathbb{R})$ is isomorphic to the spin factor $\mathbb{R}^2 \oplus \mathbb{R}$.
• The Jordan algebra $\mathfrak{h}_2(\mathbb{C})$ is isomorphic to the spin factor $\mathbb{R}^3 \oplus \mathbb{R}$.
• The Jordan algebra $\mathfrak{h}_2(\mathbb{H})$ is isomorphic to the spin factor $\mathbb{R}^5 \oplus \mathbb{R}$.
• The Jordan algebra $\mathfrak{h}_2(\mathbb{O})$ is isomorphic to the spin factor $\mathbb{R}^9 \oplus \mathbb{R}$.

We get a relation between the real numbers, complex numbers, quaternions and octonions and the Minkowski spacetimes of dimensions 3,4,6 and 10. These are precisely the dimensions where a classical superstring Lagrangian can be written down! And that’s no coincidence. But that’s not where I’m going now.

It’s the self-adjoint octonionic matrices that I want to focus on now. Due to the nonassociativity of the octonions, we can’t get Jordan algebras from self-adjoint octonionic matrices bigger than $3 \times 3$. This is one of those things you need to think about for a while to understand: it’s kind of sad but kind of wonderful.

The $1 \times 1$ self-adjoint octonionic matrices are just a fancy way of talking about $\mathbb{R}$. The $2 \times 2$ ones, as we’ve just seen, are just a way of talking about 10-dimensional Minkowski spacetime. But the $3 \times 3$ self-adjoint octonionic matrices — the only formally real Jordan algebra I haven’t said much about yet — are called the exceptional Jordan algebra.

Here’s what I’d like to explain in future posts. The automorphisms of the exceptional Jordan algebra form a Lie group called F₄. And Dubois-Violette and Todorov found the Standard Model gauge group sitting inside $\mathrm{F}_4$ in a beautiful way! They wrote three papers about this. If you want to read just one, it should probably be this:

• Michel Dubois-Violette and Ivan Todorov, Exceptional quantum geometry and particle physics II, Nucl. Phys. B 938 (2019), 751–761.

They state two main results. For these you need to remember some stuff I’ve already explained. Namely, if you pick a square root of $-1$ in the octonions and call it $i$, this makes $\mathbb{O}$ into a 4-dimensional complex vector space, and the group of automorphisms of $\mathbb{O}$ preserving this extra structure is isomorphic to $\mathrm{SU}(3)$, and $\mathbb{O}$ splits into 1d and a 3d representation of this group:

$$\mathbb{O} \cong \mathbb{C} \oplus V$$

If you’re willing to pick an arbitrary basis of $V$ you can say

$$\mathbb{O} \cong \mathbb{C} \oplus \mathbb{C}^3$$

In other words: an octonion is a complex scalar and a complex vector.

You also need a mental picture of the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$. Any element looks like this:

$$\left( \begin{array}{ccc} a & Z & Y^\ast \\ Z^\ast & b & X \\ Y & X^\ast & c \end{array} \right)$$

where $a,b,c \in \R$ and $X,Y,Z \in \mathbb{O}$. So, it consists of 3 real numbers and 3 octonions. Thus, the dimension of $\mathfrak{h}_3(\mathbb{O})$ is

$$27 = 3 + 3 \times 8$$

Using our choice of $i \in \mathbb{O}$ we can go further and split each octonion into a complex scalar and a complex vector. Then we get this:

Theorem 5. The subgroup of $\mathrm{F}_4$ that preserves the splitting of each off-diagonal octonion into complex scalar and complex vector parts is isomorphic to $(\mathrm{SU}(3) \times \mathrm{SU}(3))/\mathbb{Z}_3$.

Of course I should say how $\mathbb{Z}_3$ sits inside $\mathrm{SU}(3) \times \mathrm{SU}(3)$. I’ll do this later; for now I’ll just say it does so in the best possible, most symmetrical way.

So we’re getting two copies of $\mathrm{SU}(3)$, with a bit of overlap. One of them corresponds to the $\mathrm{SU}(3)$ that acts on the three colors of quark, as explained last time. What about the other? You might naively hope it acts on the three generations of quarks and leptons: that is, the three columns here:

Note the 3 octonions in an element of $\mathfrak{h}_3(\mathbb{O})$ have enough room to describe 3 quarks and 3 leptons. We’d have to double this somehow to describe all 6 quarks and 6 leptons.

These are some vague thoughts — but let’s shelve them for now. The next theorem points us in a different direction.

We can choose a copy of $\mathfrak{h}_2(\mathbb{O})$ sitting inside $\mathfrak{h}_3(\mathbb{O})$. Here when I say a ‘copy’, I mean a Jordan subalgebra isomorphic to $\mathfrak{h}_2(\mathbb{O})$ — for example, the one consisting of these matrices:

$$\left( \begin{array}{ccc} a & Z & 0 \\ Z^\ast & b & 0 \\ 0 & 0 & 0 \end{array} \right)$$

Theorem 6. The subgroup of $\mathrm{F}_4$ that preserves the splitting of each off-diagonal octonion into complex scalar and complex vector parts and preserves a copy of $\mathfrak{h}_2(\mathbb{O})$ in $\mathfrak{h}_3(\mathbb{O})$ is isomorphic to $(\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1))/\mathbb{Z}_6$.

Of course I should say how $\mathbb{Z}_6$ sits inside $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$. I’ll do this later; for now I’ll just say it does so in exactly the way that makes $(\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1))/\mathbb{Z}_6$ into the the true gauge group of the Standard Model!

You see, people often say the gauge group of the Standard Model is $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$, but a certain $\mathbb{Z}_6$ subgroup acts trivially on all known particles, so we can argue that the ‘true’ gauge group is the quotient by this (normal) subgroup. And this is indeed the attitude one takes in many grand unified theories: one embeds not $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ but this quotient $(\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1))/\mathbb{Z}_6$ into a larger group.

So, we’re getting the Standard Model gauge group from the exceptional Jordan algebra together with two further ideas:

• fixing a square root of $-1$ inside the octonions;
• fixing a copy of $\mathfrak{h}_2(\mathbb{O})$ inside $\mathfrak{h}_3(\mathbb{O})$.

What might this mean? Before thinking too much about that, it’s probably good to see why Theorem 5 and Theorem 6 are true. I’ll stop here for now, and try to tackle these theorems in future parts.

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• Part 1. How to define octonion multiplication using complex scalars and vectors, much as quaternion multiplication can be defined using real scalars and vectors. This description requires singling out a specific unit imaginary octonion, and it shows that octonion multiplication is invariant under $\mathrm{SU}(3)$.
• Part 2. A more polished way to think about octonion multiplication in terms of complex scalars and vectors, and a similar-looking way to describe it using the cross product in 7 dimensions.
• Part 3. How a lepton and a quark fit together into an octonion — at least if we only consider them as representations of $\mathrm{SU}(3)$, the gauge group of the strong force. Proof that the symmetries of the octonions fixing an imaginary octonion form precisely the group $\mathrm{SU}(3)$.
• Part 4. Introducing the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$: the $3 \times 3$ self-adjoint octonionic matrices. A result of Dubois-Violette and Todorov: the symmetries of the exceptional Jordan algebra preserving their splitting into complex scalar and vector parts and preserving a copy of the $2 \times 2$ adjoint octonionic matrices form precisely the Standard Model gauge group.
• Part 5. How to think of $2 \times 2$ self-adjoint octonionic matrices as vectors in 10d Minkowski spacetime, and pairs of octonions as left- or right-handed spinors.
• Part 6. The linear transformations of the exceptional Jordan algebra that preserve the determinant form the exceptional Lie group $\mathrm{E}_6$. How to compute this determinant in terms of 10-dimensional spacetime geometry: that is, scalars, vectors and left-handed spinors in 10d Minkowski spacetime.
• Part 7. How to describe the Lie group $\mathrm{E}_6$ using 10-dimensional spacetime geometry.
• Part 8. A geometrical way to see how $\mathrm{E}_6$ is connected to 10d spacetime, based on the octonionic projective plane.
• Part 9. Duality in projective plane geometry, and how it lets us break the Lie group $\mathrm{E}_6$ into the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.