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July 24, 2020

Octonions and the Standard Model (Part 3)

Posted by John Baez

Now I’ll finally explain how a quark and a lepton fit together into an octonion — in the very simplified picture where we treat these particles merely as representations of SU(3)\mathrm{SU}(3), the symmetry group of the strong force. I’ll say just enough about physics for mathematicians to get a sense of what this means. (The most substantial part of this post will be a quick intro to ‘basic triples’, a powerful technique for working with octonions.)

One of curious features of the Standard Model is that fermions — the particles that make up ‘matter’, as opposed to particles that carry ‘forces’ — come in two kinds: leptons and quarks.

Quarks are affected by the strong nuclear force, while leptons are not. Each kind of quark comes in 3 ‘colors’ — three states conventionally called red, green and blue — while leptons do not. These two statements say almost the same thing, since strong force is governed by the group SU(3)\mathrm{SU}(3). Mathematically, the point is that quarks transform in the usual representation of SU(3)\mathrm{SU}(3) on 3\mathbb{C}^3, while leptons transform in the trivial representation of this group on \mathbb{C}.

An interesting fact is that for each kind of quark there’s a corresponding lepton. For example, ordinary matter is mainly made of two kinds of quarks, called up and down quarks, and two kinds of leptons, called electrons and electron neutrinos. These four form the ‘first generation’ of quarks and leptons, and there are two more.

There are physical reasons to expect this ‘one quark per lepton’ pattern: the GIM mechanism and the need to avoid anomalies. However, it is still tempting to seek a theory that unifies quarks and leptons.

Many such theories are known. For example, in the Pati–Salam model, the group SU(3)\mathrm{SU}(3) is replaced with SU(4)\mathrm{SU}(4) and each quark together with its partner lepton are considered at a fundamental level to be a single particle transforming in the usual representation of SU(4)\mathrm{SU}(4) on 4\mathbb{C}^4. Extending the color metaphor, the three colors of quark join with the ‘white’ lepton to form a basis of 4\mathbb{C}^4.

The SU(5) grand unified theory has a different take on this issue: here red, green and blue are joined by two other states called ‘up’ and ‘down’. And the Pati–Salam model and SU(5) theory are themselves reconciled in the larger SO(10) theory.

A different approach, less explored and less successful so far, is to use octonions. Here are two references:

  • Murat Günaydin and Feza Gürsey, Quark structure and octonions, Journal of Mathematical Physics 14 (1973), 1651–1667.

  • Geoffrey Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics, Springer, Berlin, 2013.

though I could probably list dozens more if I worked at it. I don’t know the whole history of this approach. Now I’m getting curious.

The basic idea is this: if we fix an octonion i𝕆i \in \mathbb{O} with i 2=1i^2 = -1, the group of automorphisms of the octonions that fix this element is SU(3)\mathrm{SU}(3). This element ii also determines a copy of \mathbb{C} in 𝕆\mathbb{O}. This makes 𝕆\mathbb{O} into a complex vector space, and SU(3)\mathrm{SU}(3) acts on this vector space. As a representation of SU(3)\mathrm{SU}(3), this vector space is isomorphic to

3 \mathbb{C} \oplus \mathbb{C}^3

where SU(3)\mathrm{SU}(3) acts trivially on \mathbb{C} and in the usual way on 3\mathbb{C}^3. This is just the representation of SU(3)\mathrm{SU}(3) on one kind of lepton and its corresponding quark!

So, we can imagine trying to come up with a theory where a quark and a lepton are unified into a single octonion, and then somehow the choice of a square root of 1-1 in 𝕆\mathbb{O} breaks the symmetry, splitting the octonions into two separate representations of SU(3)\mathrm{SU}(3): the quark and the lepton. This might be part of a larger story in which the complex numbers, so important in quantum physics, are secretly just one special copy of \mathbb{C} sitting inside 𝕆\mathbb{O}.

I have no good idea for how this would work. Since our usual paradigm for spontaneous symmetry breaking involves the Higgs mechanism, one could try to use that. There is a 6-sphere of square roots of 1-1 sitting in the octonions, each giving a copy of \mathbb{C}, so one might imagine some sort of Higgs potential, for example a quartic polynomial, that has minima on this 6-sphere. Someone must have tried this. But I don’t find this idea especially inspiring unless it arises naturally somehow, or explains several things at once.

I’ve already proved all the mathematical claims I just made about octonions and representations of SU(3)\mathrm{SU}(3) — except for one: that the group SU(3)\mathrm{SU}(3) contains all the automorphisms of the octonions preserving ii. For that it’s handy to use a very powerful tool for working with octonions: ‘basic triples’.

For those who prefer their math free of physics, let me state this in the form of a theorem, and then prove it:

Theorem 4. Let i𝕆i \in \mathbb{O} be any element with i 2=1i^2 = -1. Then ii generates a copy of \mathbb{C} in 𝕆\mathbb{O}. Left multiplication by this copy of \mathbb{C} makes 𝕆\mathbb{O} into a complex vector space. The group of automorphisms of the octonions that fix the element ii is isomorphic to SU(3)\mathrm{SU}(3), and this group acts in a complex-linear way on 𝕆\mathbb{O}. This representation is isomorphic to the direct sum of the trivial representation on 𝕆\mathbb{C} \subseteq \mathbb{O} and the usual representation of SU(3)\mathrm{SU}(3) on 3\mathbb{C}^3 by matrix multiplication.

Proof. In Part 1 we saw a description of the octonions as 3\mathbb{C} \oplus \mathbb{C}^3 where \mathbb{C} is the subalgebra generated by a chosen square root of 1-1. One can see from this description that left multiplication by \mathbb{C} makes the octonions into a complex vector space with 3\mathbb{C}^3 as a complex subspace. In this description, octonion multiplication is given by operations that are manifestly invariant under SU(3)\mathrm{SU}(3), which acts trivially on \mathbb{C} and by matrix multiplication on 3\mathbb{C}^3.

Thus, the only thing missing is a proof that no bigger group than SU(3)\mathrm{SU}(3) acts as automorphisms of 𝕆\mathbb{O} fixing the chosen element ii. To do this, let’s show the group of such automorphisms is an 8-dimensional connected Lie group. Since its subgroup SU(3)\mathrm{SU}(3) is also an 8-dimensional and connected, it must be the whole thing.

For this I find it enjoyable to use the concept of ‘basic triple’.

In a normed division algebra the norm always comes from an inner product. Any unit vector orthogonal to 11 is a square root of 1-1, and vice versa. Further, orthogonal square roots of 1-1 anticommute.

So, if our normed division algebra isn’t the real numbers, we can find a square root of 1-1 in it. Choose one and call it ii. Then 11 and ii span a copy of \mathbb{C}.

If this isn’t our whole normed division algebra, we can find another square root of 1-1 that is orthogonal to this copy of \mathbb{C}. Choose one and call it jj. Then you can show 1,i,j1, i, j, and k=ijk = i j span a copy of \mathbb{H}.

If this isn’t our whole normed division algebra yet, we can find another square root of 1-1 that is orthogonal to this whole copy of \mathbb{H}. Choose one and call it \ell. Then you can show 1,i,j,k,,i,j1, i, j, k, \ell, \ell i, \ell j and k\ell k span a copy of 𝕆\mathbb{O}.

We say three elements i,j,𝕆i, j, \ell \in \mathbb{O} are a basic triple if they are orthogonal square roots of 1-1 and \ell is also orthogonal to k=ijk = i j. The multiplication table of octonions looks the same for any basic triple: you can work out what it must be.

Fix a basic triple i,j,𝕆i, j, \ell \in \mathbb{O}. From what I’ve said, any automorphism of 𝕆\mathbb{O} maps it to some other basic triple — and conversely, given any other basic triple, there is a unique automorphism mapping i,j,i, j, \ell to that one. So, the automorphism group of 𝕆\mathbb{O} acts freely and transitively on the set of basic triples. We say the basic triples form a torsor for Aut(𝕆)\mathrm{Aut}(\mathbb{O}).

The important consequence is that Aut(𝕆)\mathrm{Aut}(\mathbb{O}) is homeomorphic to the space of basic triples. This lets us understand the topology of Aut(𝕆)\mathrm{Aut}(\mathbb{O}).

Since unit octonions form a 7-sphere with 11 at the ‘north pole’, the square roots of 1-1 form the ‘equator’, which is a 6-sphere. If we choose one of these, say ii, then the square roots of 1-1 that are orthogonal to it form a 5-sphere. And if we choose one of these, say jj, then the square roots of 1-1 that are orthogonal to i,ji, j and k=ijk = i j form a 3-sphere.

So, the group Aut(𝕆)\mathrm{Aut}(\mathbb{O}) is a bundle of 3-spheres over a bundle of 5-spheres over the 6-sphere. As a spinoff, this implies it is a connected Lie group, and its dimension is 6+5+3=14.6 + 5 + 3 = 14.

But we’re actually interested in automorphisms of the octonions that fix a choice of ii. These correspond to basic triples with this fixed choice of ii. This is just a bundle of 3-spheres over a 5-sphere. So, this is a connected Lie group with dimension 5+3=85 + 3 = 8. And that’s what we needed to show!   █

As another spinoff of all this, it follows that SU(3)\mathrm{SU}(3) is a bundle of 3-spheres over a 5-sphere. But that’s easy to see directly by letting SU(3)\mathrm{SU}(3) act on the unit sphere of 3\mathbb{C}^3. This is a 5-sphere, and each point is stabilized by an SU(2)\mathrm{SU}(2) subgroup, which is a 3-sphere.

Posted at July 24, 2020 12:17 PM UTC

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Re: Octonions and the Standard Model (Part 3)

John wrote,

Mathematically, the point is that quarks transform in the tautologous representation of SU(3)\mathrm{SU}(3) on 3\mathbb{C}^3, while leptons transform in the trivial representation of this group on \mathbb{C}.

The fact that trivial and tautologous mean two different things here disquiets me more than I’d like to admit.

Posted by: Blake Stacey on July 24, 2020 11:48 PM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 3)

Hmm, I thought I edited that out.

I used to speak of the “fundamental” representation of groups like SU(n). Then I learned too much Lie theory and discovered that a simple Lie group has a bunch of so-called “fundamental” representations, one for each dot in its Dynkin diagram. The fundamental representation of SU(n) is still fundamental in this sense — it corresponds to the dot at the left end of the diagram — but now it’s just one of the “fundamental representations”, so saying “the” fundamental representation sounds silly.

James Dolan indoctrinated me into the idea that when a group is defined to be a group of linear transformations of something, it has a “tautologous” representation on that something.

I think I’ve heard other people say this, but I’m not sure if it’s standard enough to fling it around and expect it to be understood. It sounds sort of fancy, when it’s supposed to sound as boring as possible.

So after some thought I decided in these posts to try out something else: the “usual” representation of SU(n) on n\mathbb{C}^n. But I left in one “tautologous”.

I’ll get rid of it.

Posted by: John Baez on July 25, 2020 12:18 AM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 3)

I think I prefer “standard” or “defining” representation. Both words make clear (I feel) that we’re using something arbitrary, historical, rather than mathematically fundamental. (How could one argue mathematically that the standard representation of SU(n)SU(n) is more natural than its dual?)

Posted by: Allen Knutson on July 25, 2020 12:22 PM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 3)

I think I’ve seen defining representation more often than the other suggestions, but it’s possible that standard representation is bland enough that I just read over it and don’t remember the encounter. When speaking, I’d probably say, “the, you know [flails arms], SU(n)SU(n) representation of SU(n)SU(n)”.

Posted by: Blake Stacey on July 25, 2020 5:50 PM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 3)

I’m happy with “defining” or “standard”. Presumably anyone with any inkling of what’s going on will correctly guess what these mean — that’s what I want.

I think “defining” may be a bit more standard than “standard”, here. I remembered it as soon as I posted my last comment!

Posted by: John Baez on July 25, 2020 10:05 PM | Permalink | Reply to this

William

I suppose, as in the old joke about the spherical cow, that a cow is tautologous to a sphere, in your sense of tautology.

Posted by: William Herrera on July 25, 2020 1:15 AM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 3)

I guess instead of the Higgs mechanism, here one may need a 2-Higgs mechanism!

Posted by: Fen Zuo on July 25, 2020 2:51 AM | Permalink | Reply to this

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