The n-Category Café

In 1932, Pascual Jordan tried to isolate some axioms that an ‘algebra of observables’ should satisfy. The unadorned phrase ‘algebra’ usually signals an associative algebra, but this not the kind of algebra Jordan was led to. In both classical and quantum mechanics, observables are closed under addition and multiplication by real scalars. In classical mechanics we can also multiply observables, but in quantum mechanics this becomes problematic. After all, given two bounded self-adjoint operators on a complex Hilbert space, their product is self-adjoint if and only if they commute!

However, in quantum mechanics one can still raise an observable to a power and obtain another observable. From squaring and taking real linear combinations, one can construct a commutative product: $$a \circ b = \frac{1}{2}((a+b)^2 - a^2 - b^2) = \frac{1}{2}(a b + b a) .$$ This product is not associative, but it is power-associative: any way of parenthesizing a product of copies of the same observable $a$ gives the same result. This led Jordan to define what is now called a formally real Jordan algebra: a real vector space with a bilinear, commutative and power-associative product 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 $A$ a partial ordering: if we write $a \le b$ when the element $b - a$ is a sum of squares, it says $$a \le b \; and \; b \le a \; \quad \implies \quad a = b .$$ So, in a formally real Jordan algebra it makes sense to speak of one observable being ‘greater’ than another.

In 1934, Jordan published a paper with von Neumann and Wigner classifying finite-dimensional formally real Jordan algebras. They began by proving that any such algebra is a direct sum of ‘simple’ ones. A formally real Jordan algebra is simple when its only ideals are $\{0\}$ and $A$ itself, where an ideal is a vector subspace $B \subseteq A$ such that $b \in B$ implies $a \circ b \in B$ for all $a \in A$.

And then, they proved:

Theorem: Every simple finite-dimensional formally real Jordan algebra is isomorphic to one on this list:

• The algebras $\mathrm{h}_n(\mathbb{R})$ of $n \times n$ self-adjoint real matrices with the product $a \circ b = \frac{1}{2}(a b + b a)$.
• The algebras $\mathrm{h}_n(\mathbb{C})$ of $n \times n$ self-adjoint complex matrices with the product $a \circ b = \frac{1}{2}( a b + b a)$.
• The algebras $\mathrm{h}_n(\mathbb{H})$ of $n \times n$ self-adjoint quaternionic matrices with the product $a \circ b = \frac{1}{2}(a b + b a)$.
• The algebras $\mathrm{h}_n(\mathbb{O})$ of $n \times n$ self-adjoint octonionic matrices with the product $a \circ b = \frac{1}{2}(a b + b a)$… but only for $n \le 3$!
• The spin factors, $\mathbb{R}^n \oplus \mathbb{R}$ with the product $$(x,t) \circ (x', t') = (t x' + t' x, x \cdot x' + t t').$$

Here we say a square matrix $T$ is self-adjoint if $T_{j i} = (T_{i j})^*$. Remember, last time we set up a theory of Hilbert spaces over $\mathbb{K} = \mathbb{R},\mathbb{C},$ or $\mathbb{H}$. In these cases, we can identify a self-adjoint $n \times n$ matrix with an operator $T : \mathbb{K}^n \to \mathbb{K}^n$ that is self-adjoint in the sense that $$\langle T v, w \rangle = \langle v, T w \rangle$$ for all $v,w \in \mathbb{K}^n$. In the octonionic case we do not know what Hilbert spaces and operators are, but we can still work with matrices. Curiously, in this case we cannot go beyond $3 \times 3$ self-adjoint matrices and still get a Jordan algebra. The $1 \times 1$ self-adjoint octonionic matrices are just the real numbers, and the $2 \times 2$ ones form a Jordan algebra that is isomorphic to a spin factor. The $3 \times 3$ self-adjoint octonionic matrices are the really interesting case: these form a 27-dimensional formally real Jordan algebra called the exceptional Jordan algebra.

What does all this mean for physics? The spin factors have an intriguing relation to special relativity, since $\mathbb{R}^n \oplus \mathbb{R}$ can be identified with $(n+1)$-dimensional Minkowski spacetime, and its cone of positive elements is then revealed to be none other than the future lightcone. Furthermore, we have some interesting coincidences:

• The Jordan algebra $\mathrm{h}_2(\mathbb{R})$ is isomorphic to the spin factor $\mathbb{R}^2 \oplus \mathbb{R}$.
• The Jordan algebra $\mathrm{h}_2(\mathbb{C})$ is isomorphic to the spin factor $\mathbb{R}^3 \oplus \mathbb{R}$.
• The Jordan algebra $\mathrm{h}_2(\mathbb{H})$ is isomorphic to the spin factor $\mathbb{R}^5 \oplus \mathbb{R}$.
• The Jordan algebra $\mathrm{h}_2(\mathbb{O})$ is isomorphic to the spin factor $\mathbb{R}^9 \oplus \mathbb{R}$.

This sets up a relation between the real numbers, complex numbers, quaternions and octonions and the Minkowski spacetimes of dimensions 3,4,6 and 10. These are precisely the dimensions where a classical superstring Lagrangian can be written down! Far from being a coincidence, this is the tip of a huge and still not fully fathomed iceberg, which John Huerta is digging into.

The exceptional Jordan algebra remains mysterious. Practically ever since it was discovered, physicists have looked for some application of this entity. For example, when it was first found that quarks come in three colors, Okubo and others hoped that $3 \times 3$ self-adjoint octonionic matrices might serve as observables for these exotic degrees of freedom. Alas, nothing much came of this. More recently, people have discovered some relationships between 10-dimensional string theory and the exceptional Jordan algebra, arising from the fact that the $2 \times 2$ self-adjoint octonionic matrices can be identified with 10-dimensional Minkowski spacetime. This lets us think of the exceptional Jordan algebra as built from scalars, spinors and vectors in 10d spacetime. But the full significance of this remains mysterious, at least to me.

In 1983, Zelmanov generalized the Jordan–von Neumann–Wigner classification to the infinite-dimensional case, working with Jordan algebras that need not be formally real. In any formally real Jordan algebra, the following peculiar law holds: $$(a^2 \circ b) \circ a = a^2 \circ (b \circ a).$$ Any vector space with a commutative bilinear product obeying this law is called a Jordan algebra. Zelmanov classified the simple Jordan algebras and proved they are all of three kinds: a kind generalizing the Jordan algebras of self-adjoint matrices, a kind generalizing the spin factors, and a kind generalizing the exceptional Jordan algebra.

This theorem is part of a massive development of Jordan algebra theory carried out by Zelmanov in the 1970’s and 1980’s. For a good introduction to this, try:

• Kevin McCrimmon, A Taste of Jordan Algebras, Springer, Berlin, 2004, 562 pages.

If you look at the size of this book, you may wonder why it’s called “a taste” — it looks like quite a large helping! But in fact it doesn’t contain a proof of Zelmanov’s classification theorem, which is apparently very deep; instead, after a very readable account of the basics, it goes through the proof of a preliminary result, in which Zelmanov showed that no infinite-dimensional simple Jordan algebras are exceptional: that is, they all live inside associative algebras, with a product of the form $a \circ b = \frac{1}{2}(a b + b a)$. There are, however, two 27-dimensional exceptional Jordan algebras. There’s the one described above, and its sister, which is defined the same way, but with the so-called split octonions taking the place of the octonions.

For a very enjoyable tour of Jordan algebras before Zelmanov did his thing, try:

• Kevin McCrimmon, Jordan algebras and their applications, Bull. Amer. Math. Soc. 84 (1978), 612–627. Freely available online from the AMS and Project Euclid.

As you’ll see, there are important connections between Jordan algebras and geometry that I haven’t mentioned here. This fun essay has been refashioned and expanded into a long introductory section in McCrimmon’s book.

Alas, Zelmanov’s classification theorem does not highlight the special role of the reals, complex numbers and quaternions. Indeed, when we drop the ‘formally real’ condition, a host of additional finite-dimensional simple Jordan algebras appear, beside those in Jordan, von Neumann and Wigner’s classification theorem. These seem to have little to do with the foundations of quantum theory, although the exceptional Jordan algebra built from split octonions does make an appearance in string theory.

So, do $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$ play some privileged role in the study of quantum systems with infinitely many degrees of freedom, like quantum fields — or not? The best result along these lines is the amazing theorem of Maria Pia Solèr:

• S. S. Holland Jr., Orthomodularity in infinite dimensions: a theorem of M. Solèr, Bull. Amer. Math. Soc. 32 (1995), 205–234. Also available as arXiv:math/9504224.

Starting with simple hypotheses that don’t even mention topology or the real numbers, she is led to three choices: infinite-dimensional Hilbert spaces over $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Zounds!

Since David Corfield has expressed interest in this theorem, I may try to describe it in a future post. But my goal for next time is to tell you how the result of Jordan, von Neumann and Wigner reappears as a classification of certain cones: cones that can be used to describe nonnegative observables, but also ‘mixed states’. And here is where we’ll meet state-observable duality.