Searching for structure

This time I want to report on a new bachelor thesis, which I supervise. In this project we try to understand a little better the foundations of so-called gauge symmetries. In particular we address some of the ground work we have to lay for understanding our theories.

Let me briefly outline the problem: Most of the theories in particle physics include some kind of redundancy I.e., there are more things in it then we actually see in experiments. The surplus stuff is actually not real. It is just a kind of mathematical device to make calculations simpler. It is like a ladder, which we bring to climb a wall. We come, use the ladder, and are on top. The ladder we take again with us, and the wall remains as it was. The ladder made live simpler. Of course, we could have climbed the wall without it. But it would have been more painful.

Unfortunately, theories are more complicated than wall climbing.

One of the problems is that we usually cannot solve problems exactly. And as noted before, this can mess up the removal of the surplus stuff.

The project the bachelor student and I am working on has the following basic idea: If we can account for all of the surplus stuff, we should be able to know whether our approximations did something wrong. It is like preparing an engine. If something is left afterwards it is usually not a good sign. Unfortunately, things are again more complicated. For the engine, we just have to look through our workspace to see whether anything is left. But how to do so for our theories? And this is precisely the project.

So, the project is essentially about listing stuff. We start out with something we know is real and important. For this, we take the most simplest thing imaginable: Nothing. Nothing means in this case just an empty universe, no particles, no reactions, no nothing. That is certainly a real thing, and one we want to include in our calculations.

Of this nothing, there are also versions where some of the surplus stuff appears. Like some ghost image of particles. We actually know how to add small amounts of ghost stuff. Like a single particle in a whole universe. But these situations are not so very interesting, as we know how to deal with them. No, the really interesting stuff happens if well fill the whole universe with ghost images. With surplus stuff which we add just to make life simpler. At least originally. And the question is now: How can we add this stuff systematically? As the ghost stuff is not real, we know it must fulfill special mathematical equations.

Now we do something, which is very often done in theoretical physics: We use an analogy. The equations in question are not unique to the problem at hand, but appear also in quite different circumstances, although with a completely different meaning. In fact, the same equations describe how in quantum physics one particle is bound to each other. In quantum physics, depending on the system at hand, there may be one or more different ways how this binding occurs. You can count the number, and there is a set which one can label by whole numbers. Incidentally, this feature is where the name quantum originates from.

Returning to our original problem, we do the following analogy: Enumerating the ghost stuff can be cast into the same form as enumerating the possibilities of binding two particles together in quantum mechanics. The actual problem is only to find the correct quantum system which is the precise analogous one to our original problem. Finding this is still a complicated mathematical problem. Finding only one solution for one example is the aim of this bachelor thesis. But already finding one would be a huge step forward, as so far we do not have one at all. Having it will probably be like having a first stepping stone for crossing a river. From understanding it, we should be able to understand how to generate more. Hopefully, we will eventually understand how to create arbitrary such examples. And thus solve our enumeration problem. But this is still in the future. For the moment, we do the first step.