Being formal

One of the topics I am working on is about basic properties of so-called gauge symmetries. I just have published a new paper on it, and here I want to describe what it is about.

A gauge symmetry is, very roughly stated, a useful tool for which we pay the price of a very redundant description. Pictorially speaking, we can say the same thing with very many different words. This may sound awful. However, in practice, it seems to work just like a charm. So what is there still to investigate?

Well, knowing how to use something in one way, and understanding it fully are two very different things. And actually, we are not, on a very strict and formal level, absolutely sure that we know how to use gauge symmetry. Though this is likely the case. But the situation is nonetheless for two reasons not really satisfactory.

The first is more a question of approach. When we use something, we would really like to know what we are actually doing. The second is that if we would understand it better, there may very well be ways to use it much better than we currently do. So there are reasons for understanding it better.

But what is it what I actually want to do?

It all starts when going back to the meaning of symmetries. Symmetries introduce redundant directions, meaning that when you have a symmetry you have more directions to point then there are actually. That is in general very helpful on a technical level.

But here enters the problem. If we have directions, we should be able to say 'go in this direction'. To do so, we introduce coordinate systems. Now comes the catch: For a local symmetry this is easier said than done, especially when it comes to the gauge symmetries of the standard model.

The problem is somewhat abstract. When you think about directions, then usually you think about left, right, up or down, and so on. This is true if you think about our usual space around us. But not everything has the same geometry. Especially, symmetries can also have a direction of bending. This is still not a major issue. But, there are some symmetries where the bending of directions becomes so strong, that some directions bend back on themselves or meet others, when going too far. And this is a problem. If they bend back, or even worse, bend on a different direction, what is direction anyway? I can start walking in one direction, and then I am in another direction. Sounds like a catastrophe, right?

Well, the reason it sounds like that is that we insisted to define directions once and for all. This is what we are used to. A direction is a direction is a direction. Unfortunately, not everything is so straight and something, especially gauge symmetries, have additional directions which are warped, and can intersect each other. The problem then arises how to orient oneself, if directions change. The answer to this is that it is necessary to give up directions which are always the same. Rather, you need to define directions only in some area around you, and when you move, you may need to change them.

The aforementioned paper now investigates this bending of directions. In a sense, it tries to map how far it is possible to go in a fixed direction, before this direction changes. Finally, it attempts to draw a map of where these directional changes occur. That sounds now pretty graphical, but the reality is once more mathematically involved. But in the end, this map hopefully will help to setup useful collections of coordinate systems, and a dictionary telling you where to use which coordinate system to get your directions.

The details are pretty involved. But the rough outline of what I did was to put myself at many points, create there coordinate systems, follow the directions they give and check when they started to make no more sense - when they hit other directions or themselves. Then I got a list of collisions, and where they occurred. And from this I could get a map of collisions. What I did not yet do is to make something useful out of the map. That comes next.