This Week’s Finds – Lecture 10

 

This was the last of this year’s lectures on This Week’s Finds. You can see all ten lectures here. I will continue in September 2023.

This time I spoke about quaternions in physics and Dyson’s ‘three-fold way’: the way the real numbers, complex numbers and quaternions interact. For details, try my paper Division algebras and quantum theory.

One cute fact is how an electron is like a quaternion! More precisely: how quaternions show up in the spin-1/2 representation of SU(2) on ℂ².

Let me say a little about that here.

We can think of the group SU(2) as the group of unit quaternions: namely, 𝑞 with |𝑞| = 1. We can think of the space of spinors, ℂ², as the space of quaternions, ℍ. Then acting on a spinor by an element of SU(2) becomes multiplying a quaternion on the left by a unit quaternion!

But what does it mean to multiply a spinor by 𝑖 in this story? It’s multiplying a quaternion on the right by the quaternion 𝑖. Note: this commutes with left multiplications by all unit quaternions.

But there are some subtleties here. For example: multiplying a quaternion on the right by 𝑗 also commutes with left multiplication by unit quaternions. But 𝑗 anticommutes with 𝑖:

𝑖𝑗 = −𝑗𝑖

So there must be an ‘antilinear’ operator on spinors which commutes with the action of SU(2): that is, an operator that anticommutes with multiplication by 𝑖. Moreover this operator squares to -1.

In physics this operator is usually called ‘time reversal’. It reverses angular momentum.

You should have noticed something else, too. Our choice of right multiplication by 𝑖 to make the quaternions into a complex vector space was arbitrary: any unit imaginary quaternion would do! There was also arbitrariness in our choice of 𝑗 to be the time reversal operator.

So there’s a whole 2-sphere of different complex structures on the space of spinors, all preserved by the action of SU(2). And after we pick one, there’s a circle of different possible time reversal operators!

So far, all I’m saying is that quaternions help clarify some facts about the spin-1/2 particle that would otherwise seem a bit mysterious or weird.

For example, I was always struck by the arbitrariness of the choice of time reversal operator. Physicists usually just pick one! But now I know it corresponds to a choice of a second square root of -1 in the quaternions, one that anticommutes with our first choice: the one we call 𝑖.

At the very least, it’s entertaining. And it might even suggest some new things we could try: like ‘gauging’ time reversal symmetry (changing its definition in a way that depends on where we are), or even gauging the complex structure on spinors.