Blessing and bane: Redundancy

We have just recently published a new paper. It is part of my research on the foundations of theoretical particle physics. To fully appreciate its topic, it is necessary to say a few words on an important technical tool: Redundancy.

Most people have heard the term already when it comes to technology. If you have a redundant system, you have two or more times the same system. If the first one fails, the second takes over, and you have time to do repairs. Redundancy in theoretical physics is a little bit different. But it serves the same ends: To make life easier.

When one thinks about a theory in particle physics, one thinks about the particles it describes. But if we would write down a theory only using the particles which we can observe in experiment, these theories would become very quickly very complicated. Too complicated, in fact, in most cases. Thus people have very early on found a trick. If you add artificially something more to the theory, it becomes simpler. Of course, we cannot just simply add it really, because otherwise we would have a different theory. What we really do is, we start with the original theory. Then we add something additional. We make our calculations. And from the final result we remove then what we added. In this sense, we added a redundancy to our theory. It is a mathematical trick, nothing more. We imagine a theory with more particles, and by removing in the end everything too much, we end up with the result for our original problem.

Modern particle physics would not be imaginable without such tricks. It is one of the first things we learn when we start particle physics, the power of redundancy. A particular powerful case is to add additional particles. Another one is to add something external to the system. Like opening a door. It is the latter kind with which we had to deal.

Now, what has this to do with our work? Well, redundancies are a powerful tool. But one has to be careful with them nonetheless. As I have written, we remove at the end everything we added too much. The question is, can this be done? Or becomes everything so entwined that this is no longer possible? We have looked at especially was such a question.

To do this, we regarded a theory of only gluons, the carrier of the strong force. There has been a rather long debate in the scientific community how such gluons move from one place to another. A consensus has only recently started to emerge. One of the puzzling things were that you could prove mathematical certain properties of their movement. Surprisingly, numerical simulations did not agree with this proof. So what was wrong?

It was an example of reading the fine-print carefully enough. The proof made some assumptions. Making assumptions is not bad. It is often the only way of making progress: make an assumption, and see whether everything fits together. Here it did not. When studying the assumptions, it turned out that one had to do with such redundancies.

What was done, was essentially adding an artificial sea of such gluons to the theory. At the end, this sea was made to vanish, to get the original result. The assumption was that the sea could be removed without affecting how the gluons move. What we found in our research was that this is not correct. When removing the sea, the gluons cling to it in a way that for any sea, no matter how small, they still moved differently. Thus, removing the sea little by little is not the same as starting without the sea in the first place. Thus, the introduction of the sea was not permissible, and hence we found the discrepancy. There have been a number of further results along the way, where we learned a lot more about the theory, and about gluons, but this was the essential result.

This may seem a bit strange. Why should an extremely tiny sea have such a strong influence? I am talking here about a difference of principle, not just a number.

The reason for this can be found in a very strange property of the strong force, which is called confinement: A gluon cannot be observed individually. When the sea is introduced, it offers the gluons the possibility to escape into the sea, a loophole of confinement. It is then a question of principle: Any sea, no matter how small, provides such a loophole. Thus, there is always an escape for the gluons, and they can therefore move differently. At the same time, if there is no sea to begin with, the gluons remain confined. Unfortunately, this loophole was buried deep into the mathematical formalism, and we had to first find it.

This taught us an important lesson that, while redundancies are a great tool, one has to be careful with them. If you do not introduce your redundancies carefully enough, you may alter the system in a way too substantial to be undone. We now know what to avoid, and can go on, making further progress.