3.3 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, . 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

It is easy to see that the group of such transformations is isomorphic to : complex matrices with determinant 1, modulo scalar multiples of the identity. Less obviously, it is also isomorphic to the Lorentz group : the identity component of the group of linear transformations of that preserve the Minkowski metric

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 where has . 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 . This allows us to identify the heavenly sphere with , and it implies that the Lorentz group acts as conformal transformations of . 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

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 and 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 -dimensional real inner product space , let the **spin
factor** be the Jordan algebra freely generated by modulo
relations

Polarizing and applying the commutative law, we obtain

so is isomorphic to with the product

Though Jordan algebras were invented to study quantum mechanics, the
spin factors are also deeply related to special relativity. We can
think of
as **Minkowksi spacetime**, with
as space and as time. The reason is that is naturally
equipped with a symmetric bilinear form of signature , the **Minkowski metric**:

The group of linear transformations preserving the Minkowski metric is called , and the identity component of this is called the

When is at least 2-dimensional, we can build a projective space from the Jordan algebra . The result is none other than the heavenly sphere! To see this, note that aside from the elements 0 and 1, all projections in are of the form where 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 . 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 be a normed division algebra of dimension . Then
the Jordan algebra is secretly a spin factor! There is an
isomorphism

given by

Furthermore, the determinant of matrices in is well-defined even when is noncommutative or nonassociative:

and clearly we have

for all .

These facts have a number of nice consequences. First of all, since the
Jordan algebras
and are isomorphic, so are
their associated projective spaces. We have seen that the former space
is the heavenly sphere
, and that the latter is
. It follows that

This gives another proof of something we already saw in Section 3.1: is an -sphere. But it shows more. The Lorentz group 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 in a natural way.

Secondly, it follows that the determinant-preserving linear
transformations of form a group isomorphic to .
How can we find some transformations of this sort? If , this
is easy: when
and
, we again have
, and

This gives a homomorphism from to . This homomorphism is two-to-one, since both and act trivially, and it maps onto the identity component of . It follows that is a double cover of . The exact same construction works for , so is a double cover of .

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

The Lie algebra comes born with a representation:
its **fundamental representation** as real-linear operators on ,
given by

whenever actually corresponds to a traceless matrix with entries in . Tensoring the fundamental representation with its dual, we get a representation of on the space of matrices , given by

whenever is a traceless matrix with entries in . Since is hermitian whenever is, this representation restricts to a representation of on . This in turn can be exponentiated to obtain a representation of the group on .

Now let us return to the case . One can prove that the
representation of on is determinant-preserving
simply by checking that

when lies in and 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

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

we get isomorphisms

Putting this together with our earlier observations, it follows that acts as conformal transformations of .

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 has
a double cover called , and this group has certain
representations called spinor representations. When or ,
we actually have

where is the normed division algebra of dimension . The fundamental representation of on is the left-handed spinor representation of . 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 on by matrix multiplication. In a Feynman diagram, this is represented as follows:

In the case , 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 is nonzero, the hermitian matrix

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 as both the heavenly sphere in special relativity and a space of propositions in the quantum logic associated to the Jordan algebra : any point on the heavenly sphere corresponds to a proposition specifying the state of a spinor!

© 2001 John Baez