Theoretical physics is the most math-heavy of disciplines. We don’t use all that math because we like to be intimidating, but because it’s the most useful and accurate description of nature we know.
I am often asked to please explain this or that mathematical description in layman terms – and I try to do my best. But truth is, it’s not possible. The mathematical description is the explanation. The best I can do is to summarize the conclusions we have drawn from all that math. And this is pretty much how popular science accounts of theoretical physics work: By summarizing the consequences of lots of math.
This, however, makes science communication in theoretical physics a victim of its own success. If readers get away thinking they can follow a verbal argument, they’re left to wonder why physicists use all that math to begin with. Sometimes I therefore wish articles reporting on recent progress in theoretical physics would on occasion have an asterisk that notes “It takes several years of lectures to understand how B follows from A.”
One of the best examples for the power of math in theoretical physics – if not the best example to illustrate this – are spin 1/2 particles. They are usually introduced as particles that have to be rotated twice to return to the same initial state. I don’t know if anybody who didn’t know the math already has ever been able to make sense of this explanation – certainly not me when I was a teenager.
But this isn’t the only thing you’ll stumble across if you don’t know the math. Your first question may be: Why have spin 1/2 to begin with?
Well, one answer to this is that we need spin 1/2 particles to describe observations. Such particles are fermionic and therefore won’t occupy the same quantum state. (It takes several years of lectures to understand how B follows from A.) This is why for example electrons – which have spin 1/2 – sit in shells around the atomic nucleus rather than clumping together.
But a better answer is “Why not?” (Why not?, it turns out, is also a good answer to most why-questions that Kindergartners come up with.)
Mathematics allows you to classify everything a quantum state can do under rotations. If you do that you not only find particles that return to their initial state after 1, 1/2, 1/3 and so on of a rotation – corresponding to spin 1, 2, 3... etc – you also find particles that return to their initial state after 2, 2/3, 2/5 and so on of a rotation – corresponding to spin 1/2, 3/2, 5/2 etc. The spin, generally, is the inverse of the fraction of rotations necessary to return the particle to itself. The one exception is spin 0 which doesn’t change at all.
So the math tells you that spin 1/2 is a thing, and it’s there in our theories already. It would be stranger if it nature didn’t make use of it.
But how come that the math gives rise to such strange and non-intuitive particle behaviors? It comes from the way that rotations (or symmetry transformations more generally) act on quantum states, which is different from how they act on non-quantum states. A symmetry transformation acting on a quantum state must be described by a unitary transformation – this is a transformation which, most importantly, ensures that probabilities always add up to one. And the full set of all symmetry transformations must be described by a “unitary representation” of the group.
Symmetry groups, however, can be difficult to handle, and so physicists prefer to instead work with the algebra associated to the group. The algebra can be used to build up the group, much like you can build up a grid from right-left steps and forwards-backwards steps, repeated sufficiently often. But here’s where things get interesting: If you use the algebra of the rotation group to describe how particles transform, you don’t get back merely the rotation group. Instead you get what’s called a “double cover” of the rotation group. It means – guess! – you have to turn the state around twice to get back to the initial state.
I’ve been racking my brain trying to find a good metaphor for “double-cover” to use in the-damned-book I’m writing. Last year, I came across the perfect illustration in real life when we took the kids to a Christmas market. Here it is:
I made a sketch of this for my book:
The little trolley has to make two full rotations to get back to the starting point. And that’s pretty much how the double-cover of the rotation group gives rise to particles with spin 1/2. Though you might have to wrap your head around it twice to understand how it works.
I later decided not to use this illustration in favor of one easier to generalize to higher spin. But you’ll have to buy the-damned-book to see how this works :p