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August 15, 2011

The Strangest Numbers in String Theory

Posted by John Baez

Here’s a really easy introduction to normed division algebras, particularly the octonions, and their role in string theory. You basically just need to have gone to high school:

Our contract with Scientific American let us put this version on our websites 3 months after publication.

If you want to see how Scientific American transformed our writing with top-notch editing but also added more of a pro-string-theory slant, you can compare the original.

For the details, see:

Posted at August 15, 2011 2:52 PM UTC

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Re: The Strangest Numbers in String Theory

The article was intriguing, but John Huerta’s thesis with all the details looked a bit scary, or at least the kind of thing I’d need to spend six months on to really understand.

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?

I’ve recently been looking at four-dimensional Riemannian versions of the Proca and Dirac equations, but the interaction term between the Dirac spinors and the vector potential that I’ve been using is extremely similar to the Lorentzian one; it’s simply eA μψ γ μψe A_\mu \psi^\dagger \gamma^\mu \psi, as opposed to the Lorentzian eA μψ γ 0γ μψe A_\mu \psi^\dagger \gamma^0 \gamma^\mu \psi (and the gamma matrices in the Riemannian version satisfy {γ μ,γ ν}=2δ μν\{\gamma^\mu, \gamma^\nu\}=2\delta^{\mu\nu} rather than {γ μ,γ ν}=2η μν\{\gamma^\mu, \gamma^\nu\}=2\eta^{\mu\nu}). One nice thing about the Riemmanian case is that ψ ψ\psi^\dagger \psi is a scalar, because the representation of SO(4) on the Dirac spinors is unitary.

Anyway, if there’s a secret trick for combining a Dirac spinor and a vector as quaternions and getting a scalar, that would be fascinating to learn. But if I need to understand Lie-n-superalgebras first, I’d better put it off until I have a lot more free time!

Posted by: Greg Egan on August 16, 2011 12:56 PM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

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

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

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

where the inverse is needed to ensure

(g,h)(g,h)v=(gg,hh)v (g,h) (g',h') v = (g g' , h h') v

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

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

(g,h)ϕ=gϕ (g,h) \phi = g \phi

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

(g,h)ψ=hψ (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:

vψvψ v \otimes \psi \mapsto v \psi

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

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

((g,h)v)((g,h)ψ)=(gvh 1)(hψ)=g(vψ) ((g,h) v) \; ((g,h)\psi) = (g v h^{-1}) \; (h \psi) = g (v \psi)

Good! The result vψv \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 Ψ¯A μγ μΨ\overline{\Psi} A_\mu \gamma^\mu \Psi, where now however Ψ\Psi is a pair (ψ,ϕ)(\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)Spin(4) are each isomorphic to their dual, via the standard inner product on \mathbb{H}, namely

x,y=Re(x *y) \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.

Posted by: John Baez on August 16, 2011 3:18 PM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

Nice. Do that with the octonions and you’ll get F4, or with the split-octonions to get F4(4). Right now I’m playing with what you get from doing this with the split-octo-octonions, and symmetry breaking.

Posted by: Garrett on August 17, 2011 6:27 AM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

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 2\mathbb{C}^2; I never knew before that you could simply embed them in \mathbb{H} that way!

Posted by: Greg Egan on August 16, 2011 11:41 PM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

Hi Greg,

You can also think of \mathbb{H} as the `Pauli spinors’ for Spin(3)=SU(2)\mathrm{Spin}(3) = \mathrm{SU}(2). This one space plays lots of roles! For a taste of how this works, see the conclusion of my talk Introducing the Quaternions.

Posted by: John Huerta on August 18, 2011 8:46 AM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

Yes, it may seem like a useless coincidence that the vectors V= 4V = \mathbb{R}^4, right-handed spinors S += 2S^+ = \mathbb{C}^2 and left-handed spinors S = 2S^- = \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!

Posted by: John Baez on August 17, 2011 5:12 AM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

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

int=q(ψ L *Aψ R+ψ R *A *ψ L)\mathcal{L}_int = -q (\psi_L^* A \psi_R + \psi_R^* A^* \psi_L)

Here * is quaternionic conjugation, qq is the particle’s charge, AA is the electromagnetic four-potential and (ψ L,ψ R)(\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.

Posted by: Greg Egan on October 1, 2011 11:33 AM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

Neat, Greg! I haven’t seen this before.

It looks like what you’re using here is triality, which in dimension 4 basically says that vectors, left-handed spinors, and right-handed spinors all live in the quaterions:

A,ψ L,ψ R. A \in \mathbb{H}, \quad \psi_L \in \mathbb{H}, \quad \psi_R \in \mathbb{H} .

But each of these spaces has a different action of SO(4)SO(4), or rather its double cover, Spin(4)Spin(4). To express these actions, one can use the following exceptional isomorphism, that says Spin(4)Spin(4) is the group of pairs of unit quaterions:

Spin(4){(l,r) 2:|l|=|r|=1}. Spin(4) \cong \{ (l,r) \in \mathbb{H}^2 : |l| = |r| = 1 \} .

(One could probably even think of the quaternions as the reason this isomorphism exists!)

Then experience tells me that (l,r)Spin(4)(l,r) \in Spin(4) ought to act on vectors and spinors as follows:

lAr *,lψ L,rψ R. l A r^*, \quad l \psi_L, \quad r \psi_R .

(The vector action is how one constructs that exceptional isomorphism. The spinor actions are used in the Pati–Salam model, a grand unified theory that unifies left- and right-handed particles).

Spin(4)Spin(4) invariance is then a quick calculation, and in fact, both terms are separately invariant! I bet the whole thing is invariant under O(4)O(4), or rather its double cover, Pin(4)Pin(4). It looks like it ought to be, since O(4)O(4) is generated by reflections, and this Lagrangian is symmetric in left and right.

Posted by: John Huerta on October 1, 2011 2:47 PM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

Oops! I just noticed that John Baez explained this in an earlier comment. It’s been awhile since I read this thread.

Posted by: John Huerta on October 1, 2011 2:54 PM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

John Huerta wrote:

I bet the whole thing is invariant under O(4)O(4), or rather its double cover, Pin(4)Pin(4).

That’s right. It’s invariant under the parity operator, which acts on vector quaternions as conjugation and exchanges left and right-handed spinors.

Posted by: Greg Egan on October 1, 2011 4:11 PM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

I had a minute to expand the list of references at super Poincare Lie algebra and division algebra and supersymmetry .

While doing so I noticed a reference that had previously escaped my attention:

Jerzy Lukierski, Francesco Toppan, Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory (2002) (pdf)

They directly construct ordinary (meaning: Lie 1-algebraic) extensions of the super-Poincaré Lie algebra by using the normed division algebras. These “polyvector” extensions should be the automorphism Lie 1-algebras of the corresponding Lie nn-algebra extensions (for the supergravity Lie 3-algebra extension and the “M-Lie algebra” Lie 1-algebra extension this is spelled out here).

I guess. Haven’t checked. Would be interesting to make this a general argument. But I don’t have time for it.

Posted by: Urs Schreiber on August 16, 2011 1:04 PM | Permalink | Reply to this

Re: The Strangest Numbers in String Theory

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?

Posted by: John Baez on August 17, 2011 5:33 AM | Permalink | Reply to this

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