Lately, I’ve been doing a bit of reading about the use of group theory in particle physics. I need to do this because I’m meant to be teaching it in the next few weeks. Now, the education research says I can still teach something without being an expert in it – I just need to be able to inspire my students to learn it themselves. That’s re-assuring, because I am certainly no expert in particle physics. Of all the area of physics, it’s the one I’ve always had most trouble grasping. Perhaps it’s because you can’t neatly summarize it with one equation – I’m not sure.

Anyway, one thing I am sure about is that finding a textbook on group theory in physics (and particle physics in particular) that is accessible to mortals like me is a bit of a mission. The books I’ve looked at in our library, and there are lots of them, tend to give the impression that the whole thing is insanely complicated. After a fair bit of reading, I’m beginning to get the hang of some of it, but it is really nasty stuff.

As I’ve already said, group theory is about looking at symmetries. In particle physics, we can see lots of symmetries of various forms, so groups sit naturally here. An example (probably the simplest one) is the symmetry between the neutron and the proton. These two particles make up the nucleus of an atom. At school we’re told that the proton has a positive charge, a neutron has no charge, and that the two have near-identical masses. In fact, it’s not just (nearly) the same mass that the proton and neutron share – they are pretty-well identical, except for their charge.

This leads to the question of whether, in some fundamental way, the proton and neutron are actually different manifestations of the same thing. Heisenberg developed this idea with his ‘isospin’ theory. It turns out that there’s a clever mathematical way of describing isospin, using what’s known as the SU(2) group. This group contains the underlying physics – when we look at its symmetries the neutron and proton states naturally ‘drop-out’, in the same way that (some of) the gaits of a quadruped ‘drop-out’ of the analysis of the symmetries of a rectangle.

But it gets better than that. Not only does this group describe neutrons and protons it describes other, not so well known particles – the pions, as a neat mathematical combination of two nucleons. (By nucleon we mean a neutron or proton). The whole physics of the way that neutrons and protons interact through exchange of pions is encapsulated in the SU(2) group. Really neat!

Unfortunately, the realm of particle physics is rather bigger than neutrons, protons and pions, and so our SU(2) group doesn’t get us terribly far. But it makes a start, and certainly helps us (by which I mean me) to see why particle physicists like to bleat on about symmetry groups.

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