Wednesday, April 11, 2012

Groundwork

The first topic of my research is both very fundamental and very abstract. It has something to do with coordinate systems coordinate systems, but not the ones to describe where event takes place in space and time. A while ago, I have discussed local symmetries. Saying something has a local symmetry means that I can change things at different places in different ways. Back then, I used a grid of billiard balls. I could rotate each of the balls differently, but because the balls are perfect spheres (we over-painted the numbers), this did not change the way the grid looked.

Now, lets assume that for some obscure purpose you would like to keep track of what is going on with the balls. That you want to know the position on each ball in some way. You can do this by introducing coordinates on every ball. Think of painting a point on every ball (actually, you will need two, which are not lying on opposite points on the balls). These points destroy the local symmetry. You can now keep track of all rotations by looking at the points. In exchange for loosing the symmetry you now always know when the balls are rotated. Especially, you can now always talk about a ball pointing in some direction, because you can use the points on the ball to identify a direction.

If you now go to a particle physics theory, you also have such local symmetries. A consequence of these local symmetries was that the amount of numbers you needed to describe a photon was very small. But is was very inconvenient to use this minimal amount. At least that was what I said. If you go to something more complicated than the photons, like the gluons, it becomes even more inconvenient to use the smallest possible number.

Ok, wait a minute. What does these two things have to do with coordinate systems, you my say. And the answer is, actually, quite a lot. The procedure of using more numbers than necessary requires that you use a coordinate system to measure these additional numbers. That is something which we call introducing a gauge or fixing a gauge in particle physics. We have to know the size and the direction of the additional elements we bring into the discussion. That sounds awfully technical, and indeed it is. But at the same time it is very often very convenient. I fear, here I have to ask you to take this statement on faith. Even the simplest case where one sees how powerful this is fills a page with formulas. In fact, most of the page would be filled with the calculations which try to not use the additional coordinates. And only a few lines at the bottom would use the additional coordinates. In calculations with the full standard model, the reduction becomes incredibly large. Thus, we do it often. Only for very few calculations, in particular those done by a computer in a simulation, we can afford to do without.

This was so far only the prelude to my actual research topic. As I have warned, it is very abstract. The topic has now to do with how such coordinate systems can be chosen. So far, it seemed to be quite straightforward to do so. For the balls, we just make points on them, one ball at a time. That is, what we call a local coordinate system. This means, the choice of the coordinates can be done independently at every place. In this case, by working on each ball separately.

But the theories used for particle physics are strange. Imagine a hideous, malevolent demon. Whenever you make a point on one ball, he sneaks behind your back and changes the points you have already made, depending on where you make right now a point. How would you fight such a creature? You would take many, many pencils, and then construct a sophisticated device such that you can make the points on every ball at the same time. With this you trick the demon. What you have done was choosing your coordinate systems all at once everywhere. Since you have constructed the machine, you have no longer the possibility to make an individual choice on each ball just when you pick it up. The choice is fixed for all balls by the machine. That is what we call a global choice of coordinate systems.

Though we do not have demons running around the particles, at least, as far as we know, we have a very similar problem in particle physics. The mathematical structure of the theories in the standard model requires us to make global choices when introducing our additional coordinates. Seems to be not so complicated at first sight, but it is. Constructing appropriate machines is very complicated. Especially it turns out to be very complicated to construct a machine which works wells with every method. But when you want to combine methods, you must use the same coordinates. And thus build a machine which works with more than one method. And this is baffling complicated. In fact, so complicated that we have been struggling with it ever since the problems has been recognized. And that was in the late 1970s.

Why does this not stop us completely in our tracks? The reason is that in perturbation theory the global choice becomes again local. That means, whenever you can do perturbation theory, you can make a much simpler local choice. That is something we can do perfectly. And since perturbation theory is so helpful in many cases, this problem is not a show stopper. The reason is that perturbation theory only admits small changes. And small changes means that we never go far away from a single point we made on the ball. Thus, the demon can never sneak up on us.

Furthermore, when we do simulations, we know how to evade the problem. However, the price we pay is that things become obscure. Everything is just a black box, and in the end numbers come out. But for many problems, this is quite satisfactory, so this is also often fine.

But then there remain some problems where we just cannot evade the demon. We have to fight it. and that is where I enter the scene. One of my research topics is to understand how to make the same global choice with different methods. That is something I have been working on since 2008, and it proved and proves to be a formidable challenge. So far, the current state is that I have made some proposals for machines working with more than one method. Now, I have to understand whether these proposals make sense, and if yes, if they are simple enough to be used. It is and remains a persistent question, and one which will accompany me probably for the rest of my scientific life. But sometimes you just have to bite it, and do things like that. Its pure technical, almost mathematical. There is no physics in it - coordinate systems are choices of humans and not of nature. It is the type of ground work you have to do occasionally in science. It is part of building the tools, which you use then later for doing exciting physics, learning how nature works.