The n-Category Café

The behavior of spin-1/2 particles depends heavily on the dimension of space, or spacetime, modulo 8. For example, in Minkowski spacetime of any dimension that’s 2 more than a multiple of 8, and only in these dimensions, Majorana–Weyl spinors are possible. These are spin-1/2 particles that have an intrinsic handedness (that’s the ‘Weyl’ part) and are their own antiparticles (that’s the ‘Majorana’ part).

In 10d Minkowski spacetime something special happens: Majorana–Weyl spinors can be described using octonions! Mathematically this originates from the fact that there are two representations of $Spin(9,1)$, called the left-handed and right-handed Majorana–Weyl spinor representations, which can both be identified with $\mathbb{O}^2$, the space of pairs of octonions.

This in turn follows from a simpler fact: 10-dimensional Minkowski spacetime can be identified with $\mathfrak{h}_2(\mathbb{O})$, the space of self-adjoint $2 \times 2$ octonionic matrices:

$$\mathfrak{h}_2(\mathbb{O}) = \left\{ \left( \begin{array}{cc} t+x & y \\ y^\ast & t-x \end{array} \right) : \; t,x \in \mathbb{R}, y \in \mathbb{O} \right\}$$

Here $t$ is time and there are 9 space coordinates: the real number $x$ and the 8 real numbers describing the octonion $y$. Of course Lorentz transformations and rotations let us change these coordinates. The Lorentz group $SO(9,1)$ is precisely the group of transformations of $\mathfrak{h}_2(\mathbb{O})$ that preserve the determinant

$$\mathrm{det} \left( \begin{array}{cc} t+x & y \\ y^\ast & t-x \end{array} \right) = t^2 - x^2 - y y^\ast$$

Even though the octonions are noncommutative, we have $y y^\ast = y^\ast y = |y|^2$ where $|y|$ is the norm of the octonion $y$, so there’s no ambiguity here.

All this is very similar to something that happens in 4 dimensions. 4d Minkowski spacetime can be identified with $\mathfrak{h}_2(\mathbb{C})$, the space of self-adjoint $2 \times 2$ complex matrices, and $Spin(3,1)$ has two representations on $\mathbb{C}^2$, the left-handed and right-handed Weyl spinor representations. All this should be familiar to students of particle physics. The 10d case works essentially the same way: we just replace complex numbers with octonions. Of course the octonions are noncommutative and nonassociative, but it doesn’t really cause a problem here.

In fact, something similar also happens in 3d Minkowski spacetime, which can be identified with $\mathfrak{h}_2(\mathbb{R})$, and 6d Minkowski spacetime, which can be identified with $\mathfrak{h}_2(\mathbb{H})$. The special features of the 10d case take a while to manifest themselves.

So, to get the story started we’ll let $\mathbb{K}$ be any normed division algebra — $\mathbb{R}, \mathbb{C}, \mathbb{H}$ or $\mathbb{O}$. We’ll say its dimension is $n$, so $n = 1,2,4$ or $8$. We’ll show how to identify $(n+2)$-dimensional Minkowski spacetime with $\mathfrak{h}_2(\mathbb{K})$, the space of self-adjoint $2 \times 2$ matrices with entries in $\mathbb{K}$. And we’ll describe two kinds of spinors, both of which can be identified with elements of $\mathbb{K}^2$.

The rest of today’s post will be based on this:

• John C. Baez, John Huerta, Division algebras and supersymmetry I, in Superstrings, Geometry, Topology, and C${}^\ast$-algebras, eds. R. Doran, G. Friedman and J. Rosenberg, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65–80.

Vectors

We begin by writing an element of $\mathfrak{h}_2(\mathbb{K})$ as

$$v = \left( \begin{array}{cc} t + x & y \\ y^\ast & t - x \end{array} \right)$$

with $t,x \in \mathbb{R}$ and $x \in \mathbb{K}$.

It is clear that

$$-det(v) = -t^2 + x^2 + y y^\ast$$

is a quadratic form on $\mathfrak{h}_2(\mathbb{K})$ of signature $(n+1,1)$. This is how $\mathfrak{h}_2(\mathbb{K})$ gets identified with $(n+2)$-dimensional Minkowski spacetime.

But we also want to get our hands on the Minkowski metric — that is, a bilinear form on $\mathfrak{h}_2(\mathbb{K})$. For this the operation of trace reversal is crucial:

$$\tilde{v} = v - tr(v)1$$

Concretely, this looks like:

$$\tilde{v} = \left( \begin{array}{cc} -t + x& y \\ y^\ast & -t - x \end{array} \right)$$

So trace reversal amounts to negating $t$: it’s what physicists call ‘time reversal’. Thanks to this time reversal, if we define the bilinear form

$$g \colon \mathfrak{h}_2(\mathbb{K}) \times \mathfrak{h}_2(\mathbb{K}) \to \mathbb{R}$$

by

$$g(v,w) = \frac{1}{2} Re \, tr(v \tilde{w})$$

the inner product of time-like vectors with themselves will become negative, and we get the Minkowski metric. Indeed, a little calculation shows

$$v \tilde{v} = \tilde{v} v = -det(v) 1$$

so

$$g(v,v) = -det(v)$$

Thus, $g$ is our Minkowski metric.

In what follows we’ll call elements of $\mathfrak{h}_2(\mathbb{K})$ vectors, and write

$$V = \mathfrak{h}_2(\mathbb{K})$$

Spinors

Next we introduce two spaces of spinors, which we’ll call right-handed spinors, $S_+$, and left-handed spinors, $S_-$. As real vector spaces, both will be just $\mathbb{K}^2$. However, the group $Spin(n+1,1)$ will act on them in different ways.

To construct these representations, we begin by defining two ways for vectors to act on spinors. First, a map

$$\gamma \colon V \otimes S_+ \to S_-$$

given by

$$\gamma(v \otimes \psi) = v \psi$$

where in $v \psi$ we’re multiplying a matrix and a column vector. Second, a map

$$\tilde{\gamma} \colon V \otimes S_- \to S_+$$

given by

$$\tilde{\gamma}(v \otimes \psi) = \tilde{v} \psi$$

We can also think of these as maps that send elements of $V$ to linear operators:

$$\begin{array}{cccl} \gamma \colon & V & \to & Hom(S_+, S_-), \\ \tilde{\gamma} \colon & V & \to & Hom(S_-, S_+) \end{array}$$

Here a word of caution is needed: 25% of the time, and every time we really care about it, our normed division algebra $\mathbb{K}$ will be nonassociative! So $2 \times 2$ matrices with entries in $\mathbb{K}$ cannot be identified with linear operators on $\mathbb{K}^2$ in the usual way. They certainly induce linear operators via left multiplication:

$$L_v(\psi) = v \psi$$

Indeed, this is how $\gamma$ and $\tilde{\gamma}$ turn elements of $V$ into linear operators:

$$\begin{array}{ccl} \gamma(v) & = & L_v \\ \tilde{\gamma}(v) & = & L_{\tilde{v}} \end{array}$$

However, because of nonassociativity, composing such linear operators may be different from multiplying the matrices:

$$L_v L_w(\psi) = v( w \psi) \neq (v w) \psi = L_{v w} (\psi)$$

Now, back to business. Since vectors act on elements of $S_+$ to give elements of $S_-$ and vice versa, they map the space $S_+ \oplus S_-$ to itself. This gives rise to an action of the Clifford algebra $Cliff(V)$ on $S_+ \oplus S_-$:

Theorem 7. The vectors $V = \mathfrak{h}_2(\mathbb{K})$ act on the spinors $S_+ \oplus S_- = \mathbb{K}^2 \oplus \mathbb{K}^2$ via the map

$$\Gamma \colon V \to End(S_+ \oplus S_-)$$

given by

$$\Gamma(v)(\psi, \,\phi) = (\widetilde{v} \phi, \, v \psi)$$

Furthermore, $\Gamma(v)$ satisfies the Clifford algebra relation:

$$\Gamma(v)^2 = g(v,v) 1$$

and so extends to a homomorphism $\Gamma \colon Cliff(V) \to End(S_+ \oplus S_-)$, i.e. a representation of the Clifford algebra $Cliff(V)$ on $S_+ \oplus S_-$.

Proof. Suppose $v \in V$ and $\Psi = (\psi, \phi) \in S_+ \oplus S_-$. We need to check that

$$\Gamma(v)^2(\Psi) = -\det(v) \Psi$$

Here we must be mindful of nonassociativity: we have

$$\Gamma(v)^2(\Psi) = ( \tilde{v}(v \psi), \, v(\tilde{v} \phi))$$

But it’s easy to check that the expressions $\tilde{v}(v \psi)$ and $v (\tilde{v} \phi)$ involve multiplying at most two different nonreal elements of $\mathbb{K}$. The associative law holds in this case, even for the octonions, so in fact

$$\Gamma(v)^2(\Psi) = ( (\tilde{v}v) \psi, \, (v \tilde{v}) \phi)$$

To conclude, we use the fact that $v \tilde{v} = \tilde{v} v = -det(v) 1$.   █

The action of a vector swaps $S_+$ and $S_-$, so acting by vectors twice sends $S_+$ to itself and $S_-$ to itself. This means that while $S_+$ and $S_-$ are not modules for the Clifford algebra $Cliff(V)$, they are both modules for the ‘even part’ of the Clifford algebra, generated by products of pairs of vectors. The group $Spin(n+1,1)$ lives in this even part: as is well-known, it is generated by products of pairs of ‘unit vectors’ in $V$: that is, vectors $v$ with $g(v,v) = \pm 1$. As a result, $S_+$ and $S_-$ are both representations of $Spin(n+1, 1)$.

While we will not need this in what follows, one can check that:

• When $\mathbb{K} = \mathbb{R}$, $S_+ \cong S_-$ is the Majorana spinor representation of $Spin(2,1)$.

• When $\mathbb{K} = \mathbb{C}$, $S_+$ and $S_-$ are the Weyl spinor representations of $Spin(3,1)$.

• When $\mathbb{K} = \mathbb{H}$, $S_+$ and $S_-$ are the Weyl spinor representations of $Spin(5,1)$.

• When $\mathbb{K} = \mathbb{O}$, $S_+$ and $S_-$ are the Majorana–Weyl spinor representations of $Spin(9,1)$.

A taste of $\mathfrak{h}_3(\mathbb{K})$

Now that we’ve gotten vectors and spinors set up using $\mathfrak{h}_2(\mathbb{K})$, let me give you a tiny taste of how they’ll shed light on $\mathfrak{h}_3(\mathbb{K})$. A self-adjoint $3 \times 3$ matrix with entries in $\mathbb{K}$ looks like this:

$$a = \left( \begin{array}{ccc} \alpha & z & y^\ast \\ z^\ast & \beta & x \\ y & x^\ast & \gamma \end{array} \right)$$

where $x,y,z \in \mathbb{K}$ and $\alpha, \beta, \gamma \in \mathbb{R}$. But we can think of the lower right $2 \times 2$ block as a vector in $V = \mathfrak{h}_2(\mathbb{K})$, and write

$$a = \left( \begin{array}{cc} \alpha & \psi^\dagger \\ \psi & v \end{array} \right)$$

where $\alpha \in \mathbb{R}$ is a scalar,

$$v = \left( \begin{array}{cc} \beta & x \\ x^\ast & \gamma \end{array} \right) \in V$$

is a vector, and

$$\psi = \left( \begin{array}{cc} z^\ast \\ y \end{array} \right) \in S_-$$

is a left-handed spinor. So, we get an isomorphism of vector spaces

$$\mathfrak{h}_3(\mathbb{K}) \cong \mathbb{R} \oplus V \oplus S_-$$

Yes: we’ve combined a scalar, a vector and a left-handed spinor into a single entity!

You may wonder why we’re treating $\psi$ as a left-handed spinor. I’ll talk about that more next time. We’ll see that right-handed spinors also play an important role.

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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.