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August 3, 2020

Octonions and the Standard Model (Part 4)

Posted by John Baez

Last time we saw what we can do by choosing a square root of 1-1 in the octonions. They become a 4-dimensional complex vector space, and their automorphisms fixing this square root of 1-1 form the group SU(3)\mathrm{SU}(3). This is the symmetry group of the strong force —and even better, its representation on the octonions matches the one we see for one quark and one lepton in the Standard Model.

What happens if we play the same game for some larger structures built from octonions? For example 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}), the space of 3×33 \times 3 self-adjoint matrices with octonion entries?

Maybe some of you can guess where I’m going with this, but I think I should start at the beginning and go slow, so more people can jump aboard the train!

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×nn \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

ab=12(ab+ba) 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 nn copies of the same element aa gives the same result, which we can call a na^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++a n 2=0a 1==a n=0 a_1^2 + \cdots + a_n^2 = 0 \quad \implies \quad a_1 = \cdots = a_n = 0

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

abandbaa=b. 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 AA is simple when its only ideals are {0}\{0\} and AA itself, where an ideal is the usual sort of thing: a vector subspace BAB \subseteq A such that bBb \in B implies abBa \circ b \in B for all aAa \in A.

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

  • The algebra 𝔥 n()\mathfrak{h}_n(\mathbb{R}) of n×nn \times n self-adjoint real matrices with the product ab=12(ab+ba)a \circ b = \frac{1}{2}(a b + b a).
  • The algebra 𝔥 n()\mathfrak{h}_n(\mathbb{C}) of n×nn \times n self-adjoint complex matrices with the product ab=12(ab+ba)a \circ b = \frac{1}{2}(a b + b a).
  • The algebra 𝔥 n()\mathfrak{h}_n(\mathbb{H}) of n×nn \times n self-adjoint quaternionic matrices with the product ab=12(ab+ba)a \circ b = \frac{1}{2}(a b + b a).
  • The algebra 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) of n×nn \times n self-adjoint octonionic matrices where n3n \le 3, with the product ab=12(ab+ba)a \circ b = \frac{1}{2}(a b + b a ).
  • The spin factors: the algebras n\mathbb{R}^n \oplus \mathbb{R} with the product (x,t)(x,t)=(tx+tx,xx+tt) (x,t) \circ (x', t') = (t x' + t' x, x \cdot x' + t t')

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

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:a0} \{a : \; a \ge 0\}

For the spin factor n\mathbb{R}^n \oplus \mathbb{R} this can be identified with the ‘future cone’ in (n+1)(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:

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, 𝔥 2()\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 𝔥 2()\mathfrak{h}_2(\mathbb{R}) is isomorphic to the spin factor 2\mathbb{R}^2 \oplus \mathbb{R}.
  • The Jordan algebra 𝔥 2()\mathfrak{h}_2(\mathbb{C}) is isomorphic to the spin factor 3\mathbb{R}^3 \oplus \mathbb{R}.
  • The Jordan algebra 𝔥 2()\mathfrak{h}_2(\mathbb{H}) is isomorphic to the spin factor 5\mathbb{R}^5 \oplus \mathbb{R}.
  • The Jordan algebra 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}) is isomorphic to the spin factor 9\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 3×33 \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 1×11 \times 1 self-adjoint octonionic matrices are just a fancy way of talking about \mathbb{R}. The 2×22 \times 2 ones, as we’ve just seen, are just a way of talking about 10-dimensional Minkowski spacetime. But the 3×33 \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 F 4\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:

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 1-1 in the octonions and call it ii, this makes 𝕆\mathbb{O} into a 4-dimensional complex vector space, and the group of automorphisms of 𝕆\mathbb{O} preserving this extra structure is isomorphic to SU(3)\mathrm{SU}(3), and 𝕆\mathbb{O} splits into 1d and a 3d representation of this group:

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

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

𝕆 3\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 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}). Any element looks like this:

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

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

27=3+3×8 27 = 3 + 3 \times 8

Using our choice of i𝕆i \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 F 4\mathrm{F}_4 that preserves the splitting of each off-diagonal octonion into complex scalar and complex vector parts is isomorphic to (SU(3)×SU(3))/ 3(\mathrm{SU}(3) \times \mathrm{SU}(3))/\mathbb{Z}_3.

Of course I should say how 3\mathbb{Z}_3 sits inside SU(3)×SU(3)\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 SU(3)\mathrm{SU}(3), with a bit of overlap. One of them corresponds to the SU(3)\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 𝔥 3(𝕆)\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 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}) sitting inside 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}). Here when I say a ‘copy’, I mean a Jordan subalgebra isomorphic to 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}) — for example, the one consisting of these matrices:

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

Theorem 6. The subgroup of F 4\mathrm{F}_4 that preserves the splitting of each off-diagonal octonion into complex scalar and complex vector parts and preserves a copy of 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}) in 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) is isomorphic to (SU(3)×SU(2)×U(1))/ 6(\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1))/\mathbb{Z}_6.

Of course I should say how 6\mathbb{Z}_6 sits inside SU(3)×SU(2)×U(1)\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 (SU(3)×SU(2)×U(1))/ 6(\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 SU(3)×SU(2)×U(1)\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1), but a certain 6\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 SU(3)×SU(2)×U(1)\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1) but this quotient (SU(3)×SU(2)×U(1))/ 6(\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 1-1 inside the octonions;
  • fixing a copy of 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}) inside 𝔥 3(𝕆)\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.

Posted at August 3, 2020 1:03 PM UTC

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Re: Octonions and the Standard Model (Part 4)

The second, third and fourth entries in your list of simple finite-dimensional formally real Jordan algebras have plain letters that should be Fraktur’ed (𝔥 n()\mathfrak{h}_n(\mathbb{C}), 𝔥 n()\mathfrak{h}_n(\mathbb{H}) and 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O})).

Posted by: Blake Stacey on August 4, 2020 6:07 AM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 4)

Thanks! I fraktured them.

Posted by: John Baez on August 4, 2020 7:12 AM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 4)

There’s one obvious corollary you’re not mentioning. F 4F_4 has only real (self-conjugate) representations. Obviously, any real representation of F 4F_4 decomposes as a real representation of any subgroup of F 4F_4.

So you may have found an (SU(3)×SU(2)×U(1))/ 6(SU(3)\times SU(2)\times U(1))/\mathbb{Z}_6 subgroup of F 4F_4. But, famously, the quarks and leptons of the Standard Model transform as a complex (non-self-conjugate) representation of the SM gauge group, which is not what you can get from any embedding of (SU(3)×SU(2)×U(1))/ 6(SU(3)\times SU(2)\times U(1))/\mathbb{Z}_6 in F 4F_4.

So I’m not sure where this is headed…

Posted by: Jacques Distler on August 4, 2020 7:27 AM | Permalink | PGP Sig | Reply to this

Re: Octonions and the Standard Model (Part 4)

You didn’t specify which real form of F 4F_4 is the aforementioned automorphism group. But my remark is true for both the compact real form and for the noncompact forms F 4(4)F_{4(4)} and F 4(20)F_{4(-20)}.

  • F 4(4)F_{4(4)} has maximal compact subgroup (Sp(3)×SU(2))/ 2(Sp(3)\times SU(2))/\mathbb{Z}_2
  • F 4(20)F_{4(-20)} has maximal compact subgroup Spin(9)Spin(9).

In both cases, the maximal compact has only real (self-conjugate) representations. Even if we’re willing to forget about the U(1)U(1) of the Standard Model, the SU(3)×SU(2)SU(3)\times SU(2) must be a subgroup of the maximal compact. In the Standard Model, the quarks and leptons form a complex (non-self-conjugate) representation of SU(3)×SU(2)SU(3)\times SU(2).

Posted by: Jacques Distler on August 4, 2020 7:58 PM | Permalink | PGP Sig | Reply to this

Re: Octonions and the Standard Model (Part 4)

I mean the compact real form of F 4\mathrm{F}_4. The automorphism group of any formally real Jordan algebra is compact, because on any such algebra you can define an inner product in a canonical way, and the automorphism group preserves that inner product.

Your point is a great one! So, these ideas can’t give rise to a grand unified theory where F 4\mathrm{F}_4 acts on quarks and leptons.

That’s fine. I think if these ideas amount to anything at all for particle physics, they will have to work in some novel way. As I said at the start of this series, I do not have my heart set on them amounting to anything at all for particle physics. I’m mainly trying get better at understanding the octonions and the exceptional Jordan algebra.

Posted by: John Baez on August 4, 2020 11:27 PM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 4)

To be more precise, it’s not whether there’s an action of F 4F_4 on the quarks and leptons. It’s whether we can identify the elements of the exceptional Jordan algebra with quarks and leptons such that the (SU(3)×SU(2)×U(1))/ 6(SU(3)\times SU(2)\times U(1))/\mathbb{Z}_6 subgroup of F 4F_4 acts correctly. That’s what’s impossible.

Posted by: Jacques Distler on August 5, 2020 12:34 AM | Permalink | PGP Sig | Reply to this

Re: Octonions and the Standard Model (Part 4)

Right; for example we can let F 4\mathrm{F}_4 act trivially on quarks and leptons; that’s not ruled out.

For those who are listening in who don’t know much group representation theory, let me expand:

Irreducible unitary group representations on complex Hilbert spaces HH come in 3 kinds:

  • real representations, which are isomorphic to their dual via an invariant symmetric bilinear form g:HHg: H \otimes H \to \mathbb{C}. These are complexifications of representations on real Hilbert spaces.

  • pseudoreal or (better) quaternionic representations, which are isomorphic to their dual via an invariant antisymmetric bilinear form ω:HH\omega : H \otimes H \to \mathbb{C}. These are the underlying complex representations of representations on quaternionic Hilbert spaces.

  • complex representations, which are not isomorphic to their dual.

I expand on this in Quantum theory and division algebras, but it’s well-known stuff.

If every element of a group is conjugate to its inverse, all its irreducible unitary representations are real or quaternionic — so all its unitary representations are self-dual (isomorphic to their dual). F 4\mathrm{F}_4 has this property. The Standard Model gauge group

G SM=(SU(3)×SU(2)×U(1))/ 6 G_{SM} = (\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1))/\mathbb{Z}_6

does not. And indeed, the usual representation of G SMG_{SM} on one generation of quarks and leptons (but not their antiparticles) is not self-dual.

But if we take a self-dual rep of a group and restrict it to a subgroup, we get a self-dual rep of that subgroup. So no matter how we include G SMG_{SM} in F 4\mathrm{F}_4, any representation of F 4\mathrm{F}_4 will restrict to a self-dual rep of G SMG_{SM}.

So, we cannot get the usual representation of G SMG_{SM} on one generation of quarks and leptons by taking some representation of F 4\mathrm{F}_4 and restricting it to G SMG_{SM}.

So, let’s not try to do that.

Posted by: John Baez on August 6, 2020 2:44 AM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 4)

For example, 𝔥 2()\mathfrak{h}_2(\mathbb{C}) is the Jordan algebra of observables of a qubit, but we can also see it as 4d Minkowski spacetime!

To make this concrete, we can use the good old Pauli matrices as a basis. To the point (t,x,y,z)(t,x,y,z), we associate the matrix

M=12(tI+xσ x+yσ y+zσ z)=12(t+z xiy x+iy tz). M = \frac{1}{2}(t I + x \sigma_x + y \sigma_y + z \sigma_z) = \frac{1}{2}\begin{pmatrix} t + z & x - i y \\ x + i y & t - z\end{pmatrix}.

The future cone becomes the cone of positive semidefinite operators on 2\mathbb{C}^2. Where this cone intersects the t=1t = 1 hyperplane, we have the set of quantum states for a qubit. More generally, points in the future cone can serve as effect operators in a POVM. The determinant of MM is

detM=14(t 2x 2y 2z 2), det M = \frac{1}{4}(t^2 - x^2 - y^2 - z^2),

which up to that prefactor is the squared Minkowski interval between our chosen point and the origin. Points that are lightlike with respect to the origin correspond to positive semidefinite matrices with vanishing determinant; where the light rays intersect the t=1t = 1equitemp” corresponds to the pure qubit states — rank-1 projection operators whose determinants vanish because they have only one nonzero eigenvalue.

Wild speculation: Is there a way to view 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) as an exotic spacetime, with the determinant-preserving noncompact real form of E 6\mathrm{E}_6 as its group of Lorentz boosts?

Posted by: Blake Stacey on August 4, 2020 6:30 PM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 4)

Blake wrote:

Wild speculation: Is there a way to view 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) as an exotic spacetime, with the determinant-preserving noncompact real form of E 6\mathrm{E}_6 as its group of Lorentz boosts?

Yes. Greg Egan, John Huerta and I did a bunch of work of this. Sometime maybe I’ll talk about it.

Posted by: John Baez on August 4, 2020 11:33 PM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 4)

Neat!

Posted by: Blake Stacey on August 6, 2020 7:32 AM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 4)

“Maybe some of you can guess where I am going with this, but I think I should start at the beginning and go slow, so more people can jump aboard the train!”

I quit mathphysics research about 10 years ago and I surely don’t want to jump aboard the train again, but this overall standard model discussion (I saw Peter Woit also posted some wild theories) is interesting because for me it clarifies in retrospect a bit of the sociological dynamics of who was interested in what back then.

I knew even back then next to nothing about the standard model. So my question might sound very stupid, but why is the spin factor called spin factor? I mean what is its connection to spin? It seems hardly related to this spin factor.

Posted by: nad on August 18, 2020 8:57 AM | Permalink | Reply to this

Re: Octonions and the Standard Model (Part 4)

“Maybe some of you can guess where I am going with this, but I think I should start at the beginning and go slow, so more people can jump aboard the train!”

I quit mathphysics research about 10 years ago and I surely don’t want to jump aboard the train again, but this overall standard model discussion (I saw Peter Woit also posted some wild theories) is interesting because for me it clarifies in retrospect a bit of the sociological dynamics of who was interested in what back then.

I knew even back then next to nothing about the standard model. So my question might sound very stupid, but why is the spin factor called spin factor? I mean what is its connection to spin? It seems hardly related to this spin factor.

Posted by: nad on August 18, 2020 8:58 AM | Permalink | Reply to this

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