Last Seminar on This Week’s Finds
Posted by John Baez
On Thursday December 2nd I gave the last of this year’s seminars on This Week’s Finds. You can see videos of all ten here. I will continue in September 2023.
In my last one, I spoke about Dyson’s ‘three-fold way’: the way the real numbers, complex numbers and quaternions interact in representation theory and quantum mechanics. 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:
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.
Re: Last Seminar on This Week’s Finds
A video of my last talk is now available here.