An Idiot's Guide to My Work (Part One)

Idiot's Guide as in a Guide written by an Idiot, you understand. I don't expect it to be helpful to other Idiots.

So I'm busy avoiding writing a thesis entitled Rational R-matrices and the Exceptional Lie Algebras. (Actually I'm not being too successful at the avoidance: I might even be nearly finished. In a month or two.) As "algebra" is a word that seems to make most people's eyes glaze over, I think I'll leave all the maths for a bit and talk instead about particle physics. But not the hard bits, so don't worry.

Prevailing scientific wisdom states that everything is made up of small particle-like things (only particle-like because of quantum, y'see). I'm not going to delve into it, because it's largely irrelevant and I don't really understand it all myself. In fact, all I really need is for you to believe that people might be interested in what particles do and how they interact, like in places like CERN where they smash particles into one another to see what they get.



Now, as theorists, we try and say something about what happens when particles collide like this. In other words, we try and describe the "Something happens" bit. (We use Maths! for this of course. In particular, we model the particles as existing in some vector space and look at algebras acting on that space. Which algebra we pick dictates how the particles can interact.) There are lots of different possible ways particles could interact, but we're interested not only in sensible ways, but in easy-to-describe ways and want to end up with maths we can actually work with. (This is the big secret of science, I suppose. We pick a theory not only because it works [sort of] but because it's understandable and easy to make predictions with. Shh, don't tell anyone.) A particularly nice set of rules for particle interaction are what we call integrable theories - amongst other things these say we need only consider two particles at a time.

So we look at a situation where two particles go in, and two come out. The thing that describes what happens in the middle is what physicists called a scattering matrix - or S-matrix for short.



This has to satisfy various conditions for the theory to be integrable, but the only one I'm really interested in is the consistency condition for three particle interactions. Because we are only looking at two particles at a time, if three particles collide we split that up into three sets of two-particle interactions. The consistency condition is that it doesn't matter what order we do it in:



This is called the Yang-Baxter Equation and my thesis pretty much consists of constructing and checking solutions to it. Now the eagle-eyes among you will have noticed my thesis title mentions R-matrices, not S-matrices - that's because I'm a mathematician, and not really interested in scattering matrices at all. Fooled you! But you can chalk it up to historical reasons and a difference in terminology if you like, and still think about particles interacting whenever I bleat on about R-matrices.

Ok, that's Part One done. Not sure when (if ever) I'll get round to doing Part Two - I have a thesis to avoid finish writing after all.