The n-Category Café

I wrote a long paper about this:

• Getting to the bottom of Noether’s theorem.

My starting-point was that self-adjoint complex matrices form not only a Jordan algebra with product

$$A \circ B = \frac{1}{2} (A B + B A)$$

but also a Lie algebra with bracket

$$-i [A, B] = -i(A B - B A)$$

The commutator of two self-adjoint matrices is skew-adjoint, but we can multiply it by $i$ to get something self-adjoint. That’s what is going on in Heisenberg’s equation. But this trick doesn’t work for other Jordan algebras, at least not automatically, so there is a lot to say.

I just bumped into a nice paper on this issue that I hadn’t seen before:

• Paul Townsend, The Jordan formulation of quantum mechanics: a review.

The idea here is to replace the commutator in Heisenberg’s equation by an associator:

$$(A, B, C) = (A \circ B) \circ C - A \circ (B \circ C)$$

This is well-defined whenever our observables are elements in a Jordan algebra. Jordan algebras are always commutative, but rarely associative!

Here’s the trick. Let $\mathfrak{h}_n(\mathbb{C})$ be the Jordan algebra of self-adjoint $n \times n$ complex matrices, and let’s start with Heisenberg’s equation

$$\frac{d}{d t} O(t) = -i [H, O(t)]$$

where $H \in \mathfrak{h}_n(\mathbb{C})$. Suppose we can write

$$H = -4i [X, Y]$$

for some $X, Y \in \mathfrak{h}_n(\mathbb{C})$. In this case we can use a really cool identity to express the commutator in Heisenberg’s equation in terms of an associator:

$$[[X, Y], A] = -\frac{1}{4}(X, A, Y)$$

This holds in any associative algebra if you define $[X,Y] = X Y - YX$, $X \circ Y = \tfrac{1}{2} (X Y + Y X)$ and $(X, A, Y) = (X \circ A) \circ Y - X \circ (A \circ Y)$. It’s easy to check: just expand out both sides and compare them!

Using this identity, we get

$$\frac{d}{d t} O(t) = (X, O(t), Y)$$

Now we’re describing dynamics using only operations that are available in any Jordan algebra!

This raises the question of when a self-adjoint complex matrix $H$ can be written as $-4i [X, Y]$ for self-adjoint matrices $X, Y$. This is true whenever $H$ is traceless, since $\mathfrak{su}(n)$ is a compact simple real Lie algebra, and every element of such a Lie algebra is a commutator (as shown by Akhieser).

But any self-adjoint complex matrix $H$ is of the form $H' + \lambda I$ where $H'$ is traceless, so writing $H' = -4i[X,Y]$ we have

$$[H, O(t)] = [H' + \lambda I, O(t)] = [H, O(t)] = -4i [[X,Y], O(t)] = i (X, O(t), Y)$$

so we can rewrite Heisenberg’s equation as

$$\frac{d}{d t} O(t) = (X, O(t), Y)$$

Moreover, in any Jordan algebra, any pair of elements $X, Y$ determines a derivation $(X, \cdot, Y)$: see Section I.7 of Jacobson’s Structure and Representations of Jordan Algebras. In the finite-dimensional case there is no difficulty with exponentiating any derivation to obtain a one-parameter group of automorphisms. Thus, for any elements $X, Y$ of a finite-dimensional Jordan algebra, the solution of the above equation always determines a one-parameter group of Jordan algebra automorphisms! And this is just what we’d want for describing how observables change with time.

The are two obvious next questions: one mathematical and one more philosophical.

First, how many one-parameter groups of Jordan algebra automorphisms do we actually get out of solutions to

$$\frac{d}{d t} O(t) = (X, O(t), Y)$$

In the case of $\mathfrak{h}_n(\mathbb{C})$, we get them all, since it’s already known that we get them all from Heisenberg’s equation

$$\frac{d}{d t} O(t) = - i [H , O(t)]$$

What about $\mathfrak{h}_n(\mathbb{R})$ and $\mathfrak{h}_n(\mathbb{H})$? I’m actually more interested in the exceptional Jordan algebra $\mathfrak{h}_3(\mathbb{O})$, and here it seems we get them all! This was shown in a paper that’s fairly hard to find even though it’s available for free online:

• Piero Truini and Lawrence Christian Biedenharn, A comment on the dynamics of $M_3^8$, Hadronic Journal 4 (no. 3) (1980/81), 995–1017.

It starts on page 214 of the PDF file.

The editor of this journal is somewhat disreputable, which puts off some people I’m talking to. But you can’t judge a paper by the journal it appeared in. Truini and Biedenharn are good — Biedenharn is famous for helping discover an identity, the Biedenharn–Elliott identity, that amounts to the pentagon identity for the category of representations of $\text{SU}(2)$! And the paper looks fine, as far as I can tell.

Second, the more philosophical question: what does it mean to describe dynamics using not one observable, the Hamiltonian, but two? Perhaps the best way to tackle this is to try doing it, and seeing how it works. Note that this method is not just good for dynamics, but for any Lie group of symmetries.