3.3 $\OP^1$ and Lorentzian Geometry

In Section 3.1 we sketched a systematic approach to projective lines over the normed division algebras. The most famous example is the Riemann sphere, $\CP^1$. As emphasized by Penrose [72], this space has a fascinating connection to Lorentzian geometry -- or in other words, special relativity. All conformal transformations of the Riemann sphere come from fractional linear transformations

$$z \mapsto {az + b\over cz + d}, \qquad \qquad a,b,c,d \in \C.$$

It is easy to see that the group of such transformations is isomorphic to $\PSL (2,\C)$: $2 \times 2$ complex matrices with determinant 1, modulo scalar multiples of the identity. Less obviously, it is also isomorphic to the Lorentz group $\SO _0(3,1)$: the identity component of the group of linear transformations of $\R^4$ that preserve the Minkowski metric

$$x \cdot y = x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4 .$$

This fact has a nice explanation in terms of the 'heavenly sphere'. Mathematically, this is the 2-sphere consisting of all lines of the form $\{\alpha x\}$ where $x \in \R^4$ has $x \cdot x = 0$. In special relativity such lines represent light rays, so the heavenly sphere is the sphere on which the stars appear to lie when you look at the night sky. This sphere inherits a conformal structure from the Minkowski metric on $\R^4$. This allows us to identify the heavenly sphere with $\CP^1$, and it implies that the Lorentz group acts as conformal transformations of $\CP^1$. In concrete terms, what this means is that if you shoot past the earth at nearly the speed of light, the constellations in the sky will appear distorted, but all angles will be preserved.

In fact, these results are not special to the complex case: the same ideas work for the other normed division algebras as well! The algebras $\R,\C,\H$ and $\O$ are related to Lorentzian geometry in 3, 4, 6, and 10 dimensions, respectively [63,64,65,78,86]. Even better, a full explanation of this fact brings out new relationships between the normed division algebras and spinors. In what follows we explain how this works for all 4 normed division algebras, with special attention to the peculiarities of the octonionic case.

To set the stage, we first recall the most mysterious of the four infinite series of Jordan algebras listed at the beginning of Section 3: the spin factors. We described these quite concretely, but a more abstract approach brings out their kinship to Clifford algebras. Given an $n$-dimensional real inner product space $V$, let the spin factor $\J (V)$ be the Jordan algebra freely generated by $V$ modulo relations

$$% latex2html id marker 1604 v^2 = \Vert v\Vert^2 .$$

Polarizing and applying the commutative law, we obtain

$$v\circ w = \langle v, w \rangle,$$

so $\J (V)$ is isomorphic to $V \oplus \R$ with the product

$$(v,\alpha) \circ (w, \beta) = (\alpha w + \beta v, \langle v,w\rangle + \alpha \beta).$$

Though Jordan algebras were invented to study quantum mechanics, the spin factors are also deeply related to special relativity. We can think of $\J (V) \iso V \oplus \R$ as Minkowksi spacetime, with $V$ as space and $\R$ as time. The reason is that $\J (V)$ is naturally equipped with a symmetric bilinear form of signature $(n,1)$, the Minkowski metric:

$$(v,\alpha)\cdot (w,\beta) = \langle v,w\rangle - \alpha \beta.$$

The group of linear transformations preserving the Minkowski metric is called $\OO (n,1)$, and the identity component of this is called the Lorentz group, $\SO _0(n,1)$. We define the lightcone ${\rm C}(V)$ to consist of all nonzero $x \in \J (V)$ with $x \cdot x = 0$. A 1-dimensional subspace of $\J (V)$ spanned by an element of the lightcone is called a light ray, and the space of all light rays is called the heavenly sphere ${\rm S}(V)$. We can identify the heavenly sphere with the unit sphere in $V$, since every light ray is spanned by an element of the form $(v,1)$ where $v \in V$ has norm one. Here is a picture of the lightcone and the heavenly sphere when $V$ is 2-dimensional:

When $V$ is at least 2-dimensional, we can build a projective space from the Jordan algebra $\J (V)$. The result is none other than the heavenly sphere! To see this, note that aside from the elements 0 and 1, all projections in $\J (V)$ are of the form $p = \textstyle{1\over 2}(v, 1)$ where $v \in V$ has norm one. These are the points of our projective space, but as we have seen, they also correspond to points of the heavenly sphere. Our projective space has just one line, corresponding to the projection $1 \in \J (V)$. We can visualize this line as the heavenly sphere itself.

What does all this have to do with normed division algebras? To answer this, let $\K$ be a normed division algebra of dimension $n$. Then the Jordan algebra $\h _2(\K)$ is secretly a spin factor! There is an isomorphism

$$\phi \maps \h _2(\K) \to J(\K \oplus \R) \iso \K \oplus \R \oplus \R$$

given by

$$% latex2html id marker 1609 \phi \left( \begin{array}{cc} \... ..., \alpha) , \qquad \qquad x \in \K, \; \alpha, \beta \in \R .$$

Furthermore, the determinant of matrices in $\h _2(\K)$ is well-defined even when $\K$ is noncommutative or nonassociative:

$$% latex2html id marker 1610 \det \left( \begin{array}{cc} \... ...\\ \end{array} \right) = \alpha^2 - \beta^2 - \Vert x\Vert^2 ,$$

and clearly we have

$$\det(a) = -\phi(a) \cdot \phi(a)$$

for all $a \in \h _2(\K)$.

These facts have a number of nice consequences. First of all, since the Jordan algebras $\J (\K \oplus \R)$ and $\h _2(\K)$ are isomorphic, so are their associated projective spaces. We have seen that the former space is the heavenly sphere ${\rm S}(\K \oplus \R)$, and that the latter is $\KP^1$. It follows that

$$% latex2html id marker 1612 \KP^1 \iso {\rm S}(\K \oplus \R) .$$

This gives another proof of something we already saw in Section 3.1: $\KP^1$ is an $n$-sphere. But it shows more. The Lorentz group $\SO _0(n+1,1)$ has an obvious action on the heavenly sphere, and the usual conformal structure on the sphere is invariant under this action. Using the above isomorphism we can transfer this group action and invariant conformal structure to $\KP^1$ in a natural way.

Secondly, it follows that the determinant-preserving linear transformations of $\h _2(\K)$ form a group isomorphic to $\OO (n+1,1)$. How can we find some transformations of this sort? If $\K = \R$, this is easy: when $g \in \SL (2,\R)$ and $x \in \h _2(\R)$, we again have $gxg^* \in \h _2(\R)$, and

$$\det(gxg^*) = \det(x).$$

This gives a homomorphism from $\SL (2,\R)$ to ${\rm O}(2,1)$. This homomorphism is two-to-one, since both $g = 1$ and $g = -1$ act trivially, and it maps $\SL (2,\R)$ onto the identity component of ${\rm O}(2,1)$. It follows that $\SL (2,\R)$ is a double cover of $\SO _0(2,1)$. The exact same construction works for $\K = \C$, so $\SL (2,\C)$ is a double cover of $\SO _0(3,1)$.

For the other two normed division algebras the above calculation involving determinants breaks down, and it even becomes tricky to define the group $\SL (2,\K)$, so we start by working at the Lie algebra level. We say a $m \times m$ matrix with entries in the normed division algebra $\K$ is traceless if the sum of its diagonal entries is zero. Any such traceless matrix acts as a real-linear operator on $\K^m$. When $\K$ is commutative and associative, the space of operators coming from $m \times m$ traceless matrices with entries in $\K$ is closed under commutators, but otherwise it is not, so we define $\Sl (m,\K)$ to be the Lie algebra of operators on $\K^m$ generated by operators of this form. This Lie algebra in turn generates a Lie group of real-linear operators on $\K^m$, which we call $\SL (m,\K)$. Note that multiplication in this group is given by composition of real-linear operators, which is associative even for $\K = \O$.

The Lie algebra $\Sl (m,\K)$ comes born with a representation: its fundamental representation as real-linear operators on $\K^m$, given by

$$a \maps x \mapsto ax , \qquad \qquad x \in \K^m$$

whenever $a \in \Sl (m,\K)$ actually corresponds to a traceless $m \times m$ matrix with entries in $\K$. Tensoring the fundamental representation with its dual, we get a representation of $\Sl (m,\K)$ on the space of matrices $\K[m]$, given by

$$a \maps x \mapsto ax + xa^*, \qquad \qquad x \in \K[m]$$

whenever $a$ is a traceless matrix with entries in $\K$. Since $ax + xa^*$ is hermitian whenever $x$ is, this representation restricts to a representation of $\Sl (m,\K)$ on $\h _m(\K)$. This in turn can be exponentiated to obtain a representation of the group $\SL (m,\K)$ on $\h _m(\K)$.

Now let us return to the case $m = 2$. One can prove that the representation of $\SL (2,\K)$ on $\h _2(\K)$ is determinant-preserving simply by checking that

$$% latex2html id marker 1616 {d \over dt} \det(x + t(ax + xa^*))\, \Bigr\vert _{t = 0} = 0$$

when $x$ lies in $\h _2(\K)$ and $a \in \K[2]$ is traceless. Here the crucial thing is to make sure that the calculation is not spoiled by noncommutativity or nonassociativity. It follows that we have a homomorphism

$$\alpha_\K \maps \SL (2,\K) \to \SO _0(n+1,1)$$

One can check that this is onto, and that its kernel consists of the matrices $\pm 1$. Thus if we define

$$% latex2html id marker 1618 \PSL (2,\K) = \SL (2,\K) / \{\pm 1\} ,$$

we get isomorphisms

$$% latex2html id marker 1619 \begin{array}{ccl} \PSL (2,\... ...\SO _0(6,1) \\ \PSL (2,\O)& \iso &\SO _0(9,1) . \end{array}$$

Putting this together with our earlier observations, it follows that $\PSL (2,\K)$ acts as conformal transformations of $\KP^1$.

We conclude with some words about how all this relates to spinors. The machinery of Clifford algebras and spinors extends effortlessly from the case of inner product spaces to vector spaces equipped with an indefinite metric. In particular, the Lorentz group $\SO _0(n+1,1)$ has a double cover called $\Spin (n+1,1)$, and this group has certain representations called spinor representations. When $n = 1,2,4$ or $8$, we actually have

$$\Spin (n+1,1) \iso \SL (2,\K)$$

where $\K$ is the normed division algebra of dimension $n$. The fundamental representation of $\SL (2,\K)$ on $\K^2$ is the left-handed spinor representation of $\Spin (n+1,1)$. Its dual is the right-handed spinor representation. Moreover, the interaction between vectors and spinors that serves as the basis of supersymmetric theories of physics in spacetimes of dimension 3, 4, 6 and 10 is just the action of $\h _2(\K)$ on $\K^2$ by matrix multiplication. In a Feynman diagram, this is represented as follows:

In the case $\K = \C$, Penrose [72] has described a nice trick for getting points on the heavenly sphere from spinors. In fact, it also works for other normed division algebras: if $(x,y) \in \K^2$ is nonzero, the hermitian matrix

$$% latex2html id marker 1621 \left( \begin{array}{c} x \\ y... ...^\ast & x y^\ast \\ y x^\ast & y y^\ast \end{array} \right)$$

is nonzero but has determinant zero, so it defines a point on the heavenly sphere. If we restrict to spinors of norm one, this trick reduces to the Hopf map. This clarifies the curious double role of $\KP^1$ as both the heavenly sphere in special relativity and a space of propositions in the quantum logic associated to the Jordan algebra $\h _2(\K)$: any point on the heavenly sphere corresponds to a proposition specifying the state of a spinor!

home