3. Octonionic Projective Geometry

Projective geometry is a venerable subject that has its origins in the study of perspective by Renaissance painters. As seen by the eye, parallel lines — e.g., train tracks — appear to meet at a 'point at infinity'. When one changes ones viewpoint, distances and angles appear to change, but points remain points and lines remain lines. These facts suggest a modification of Euclidean plane geometry, based on a set of points, a set of lines, and relation whereby a point 'lies on' a line, satisfying the following axioms:

• For any two distinct points, there is a unique line on which they both lie.
• For any two distinct lines, there is a unique point which lies on both of them.
• There exist four points, no three of which lie on the same line.
• There exist four lines, no three of which have the same point lying on them.

A structure satisfying these axioms is called a projective plane. Part of the charm of this definition is that it is 'self-dual': if we switch the words 'point' and 'line' and switch who lies on whom, it stays the same.

We have already met one example of a projective plane in Section 2.1: the smallest one of all, the Fano plane. The example relevant to perspective is the real projective plane, $\RP^2$. Here the points are lines through the origin in $\R^3$, the lines are planes through the origin in $\R^3$, and the relation of 'lying on' is taken to be inclusion. Each point $(x,y) \in \R^2$ determines a point in $\RP^2$, namely the line in $\R^3$ containing the origin and the point $(x,y,-1)$:

There are also other points in $\RP^2$, the 'points at infinity', corresponding to lines through the origin in $\R^3$ that do not intersect the plane $\{z = -1\}$. For example, any point on the horizon in the above picture determines a point at infinity.

Projective geometry is also interesting in higher dimensions. One can define a projective space by the following axioms:

• For any two distinct points $p,q$, there is a unique line $pq$ on which they both lie.
• For any line, there are at least three points lying on this line.
• If $a,b,c,d$ are distinct points and there is a point lying on both $ab$ and $cd$, then there is a point lying on both $ac$ and $bd$.

Given a projective space and a set $S$ of points in this space, we define the span of $S$ to be the smallest set $T$ of points containing $S$ such that if $a$ and $b$ lie in $T$, so do all points on the line $ab$. The dimension of a projective space is defined to be one less than the minimal cardinality of a set that spans the whole space. The reader may enjoy showing that a 2-dimensional projective space is the same thing as a projective plane [40].

If $\K$ is any field, there is an $n$-dimensional projective space called $\KP^n$ where the points are lines through the origin in $\K^{n+1}$, the lines are planes through the origin in $\K^{n+1}$, and the relation of 'lying on' is inclusion. In fact, this construction works even when $\K$ is a mere skew field: a ring such that every nonzero element has a left and right multiplicative inverse. We just need to be a bit careful about defining lines and planes through the origin in $\K^{n+1}$. To do this, we use the fact that $\K^{n+1}$ is a $\K$-bimodule in an obvious way. We take a line through the origin to be any set

$$% latex2html id marker 1575 L = \{ \alpha x \; \colon\; \alpha \in \K \}$$

where $x \in \K^{n+1}$ is nonzero, and take a plane through the origin to be any set

$$% latex2html id marker 1576 P = \{ \alpha x + \beta y \; \colon \; \alpha,\beta \in \K \}$$

where $x,y \in \K^{n+1}$ are elements such that $\alpha x + \beta y = 0$ implies $\alpha,\beta = 0$.

Given this example, the question naturally arises whether every projective $n$-space is of the form $\KP^n$ for some skew field $\K$. The answer is quite surprising: yes, but only if $n > 2$. Projective planes are more subtle [84]. A projective plane comes from a skew field if and only if it satisfies an extra axiom, the 'axiom of Desargues', which goes as follows. Define a triangle to be a triple of points that don't all lie on the same line. Now, suppose we have two triangles $xyz$ and $x'y'z'$. The sides of each triangle determine three lines, say $LMN$ and $L'M'N'$. Sometimes the line through $x$ and $x'$, the line through $y$ and $y'$, and the line through $z$ and $z'$ will all intersect at the same point:

The axiom of Desargues says that whenever this happens, something else happens: the intersection of $L$ and $L'$, the intersection of $M$ and $M'$, and the intersection of $N$ and $N'$ all lie on the same line:

This axiom holds automatically for projective spaces of dimension 3 or more, but not for projective planes. A projective plane satisfying this axiom is called Desarguesian.

The axiom of Desargues is pretty, but what is its connection to skew fields? Suppose we start with a projective plane $P$ and try to reconstruct a skew field from it. We can choose any line $L$, choose three distinct points on this line, call them $0, 1$, and $\infty$, and set $\K = L - \{\infty\}$. Copying geometric constructions that work when $P = \RP^2$, we can define addition and multiplication of points in $\K$. In general the resulting structure $(\K,+,0,\cdot,1)$ will not be a skew field. Even worse, it will depend in a nontrivial way on the choices made. However, if we assume the axiom of Desargues, these problems go away. We thus obtain a one-to-one correspondence between isomorphism classes of skew fields and isomorphism classes of Desarguesian projective planes.

Projective geometry was very fashionable in the 1800s, with such worthies as Poncelet, Brianchon, Steiner and von Staudt making important contributions. Later it was overshadowed by other forms of geometry. However, work on the subject continued, and in 1933 Ruth Moufang constructed a remarkable example of a non-Desarguesian projective plane using the octonions [69]. As we shall see, this projective plane deserves the name $\OP^2$.

The 1930s also saw the rise of another reason for interest in projective geometry: quantum mechanics! Quantum theory is distressingly different from the classical Newtonian physics we have learnt to love. In classical mechanics, observables are described by real-valued functions. In quantum mechanics, they are often described by hermitian $n\times n$ complex matrices. In both cases, observables are closed under addition and multiplication by real scalars. However, in quantum mechanics, observables do not form an associative algebra. Still, one can raise an observable to a power, and from squaring one can construct a commutative but nonassociative product:

$$a \circ b = {1\over 2}((a+b)^2 - a^2 - b^2) = {1\over 2}(ab + ba) .$$

In 1932, Pascual Jordan attempted to understand this situation better by isolating the bare minimum axioms that an 'algebra of observables' should satisfy [53]. He invented the definition of what is now called a formally real Jordan algebra: a commutative and power-associative algebra satisfying

$$a_1^2 + \cdots + a_n^2 = 0 \quad \implies \quad a_1 = \cdots = a_n = 0$$

for all $n$. The last condition gives the algebra a partial ordering: if we write $a \le b$ when the element $b - a$ is a sum of squares, it says that $a \le b$ and $b \le a$ imply $a = b$. Though it is not obvious, any formally real Jordan algebra satisfies the identity

$$a \circ (b \circ a^2) = (a \circ b) \circ a^2$$

for all elements $a$ and $b$. Any commutative algebra satisfying this identity is called a Jordan algebra. Jordan algebras are automatically power-associative.

In 1934, Jordan published a paper with von Neumann and Wigner classifying all formally real Jordan algebras [55]. The classification is nice and succinct. An ideal in the Jordan algebra $A$ is a subspace $B \subseteq A$ such that $b \in B$ implies $a \circ b \in B$ for all $a \in A$. A Jordan algebra $A$ is simple if its only ideals are $\{0\}$ and $A$ itself. Every formally real Jordan algebra is a direct sum of simple ones. The simple formally real Jordan algebras consist of 4 infinite families and one exception.

1. The algebra $\h _n(\R)$ with the product $a \circ b = {1\over 2}(ab + ba)$.
2. The algebra $\h _n(\C)$ with the product $a \circ b = {1\over 2}(ab + ba)$.
3. The algebra $\h _n(\H)$ with the product $a \circ b = {1\over 2}(ab + ba)$.
4. The algebra $\R^n \oplus \R$ with the product
   
   $$(v,\alpha) \circ (w, \beta) = (\alpha w + \beta v, \langle v,w\rangle + \alpha \beta).$$
   
   

5. The algebra $\h _3(\O)$ with the product $a \circ b = {1\over 2}(ab + ba)$.

Here we say a square matrix with entries in the $\ast$-algebra $A$ is hermitian if it equals its conjugate transpose, and we let $\h _n(A)$ stand for the hermitian $n\times n$ matrices with entries in $A$. Jordan algebras in the fourth family are called spin factors, while $\h _3(\O)$ is called the exceptional Jordan algebra. This classification raises some obvious questions. Why does nature prefer the Jordan algebras $\h _n(\C)$ over all the rest? Or does it? Could the other Jordan algebras — even the exceptional one — have some role to play in quantum physics? Despite much research, these questions remain unanswered to this day.

The paper by Jordan, von Neumann and Wigner appears to have been uninfluenced by Moufang's discovery of $\OP^2$, but in fact they are related. A projection in a formally real Jordan algebra is defined to be an element $p$ with $p^2 = p$. In the familiar case of $\h _n(\C)$, these correspond to hermitian matrices with eigenvalues $0$ and $1$, so they are used to describe observables that assume only two values — e.g., 'true' and 'false'. This suggests treating projections in a formally real Jordan algebra as propositions in a kind of 'quantum logic'. The partial order helps us do this: given projections $p$ and $q$, we say that $p$ 'implies' $q$ if $p \le q$.

The relation between Jordan algebras and quantum logic is already interesting [30], but the real fun starts when we note that projections in $\h _n(\C)$ correspond to subspaces of $\C^n$. This sets up a relationship to projective geometry [91], since the projections onto 1-dimensional subspaces correspond to points in $\CP^n$, while the projections onto 2-dimensional subspaces correspond to lines. Even better, we can work out the dimension of a subspace $V \subseteq \C^n$ from the corresponding projection $p \maps \C^n \to V$ using only the partial order on projections: $V$ has dimension $d$ iff the longest chain of distinct projections

$$0 = p_0 < \cdots < p_i = p$$

has length $i = d$. In fact, we can use this to define the rank of a projection in any formally real Jordan algebra. We can then try to construct a projective space whose points are the rank-1 projections and whose lines are the rank-2 projections, with the relation of 'lying on' given by the partial order $\le$.

If we try this starting with $\h _n(\R)$, $\h _n(\C)$ or $\h _n(\H)$, we succeed when $n \ge 2$, and we obtain the projective spaces $\RP^n$, $\CP^n$ and $\HP^n$, respectively. If we try this starting with the spin factor $\R^n \oplus \R$ we succeed when $n \ge 2$, and obtain a series of 1-dimensional projective spaces related to Lorentzian geometry. Finally, in 1949 Jordan [54] discovered that if we try this construction starting with the exceptional Jordan algebra, we get the projective plane discovered by Moufang: $\OP^2$.

In what follows we describe the octonionic projective plane and exceptional Jordan algebra in more detail. But first let us consider the octonionic projective line, and the Jordan algebra $\h _2(\O)$.

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