The tools of the trade

By now, I have collected and presented you quite a number of the basic ingredients of the standard model (and beyond). You should be now well equipped to get a good understanding of what I am doing. Therefore, I can come back to the original idea of this blog, and can discuss some aspects of my own research. At times, and when need be, I will add further more general entries.

Before I can enter the subjects of my research, I have to present another important part of the work of a theoretical physicist: The methods she or he is using. Each methods has its distinct advantages and drawbacks. As a result, a given problem can often be addressed by multiple methods. If this is the case, it is also possible to combine the different methods.

The latter is of particular importance because of an insight of singular importance in physics: Any problem of fundamental interest in particle physics so far is so complicated that we were not (yet) able to find an exact solution. At first, this appears like a very depressing insight. It is usually a cultural shock for students when they enter research, as up to then one is usually only exposed to simple problems which an be solved exactly, for reasons of a pedagogical and manageable presentation. At times, one acquires the insight that this horrible complexity of real problems is just a natural consequence of the richness of physics, even of the very elementary particles which lie at the heart of our current understanding of the universe. Nonetheless, physicists strive for getting better and better and ultimately exact solutions, and perhaps this holy grail of a theoretician can be reached someday. For now, however, this is not the case, and we have to live with the fact that despite our methods working often exceptionally well, they can never give you the full answer. But for some questions they can provide answers, which are ten or more digits precise. And this is quite encouraging.

For the topics I am interested in such enormously good results have not been achieved. The reason for this is that problems become simpler the weaker the interactions are. The method perfectly suited for this is perturbation theory, the first method I will be introducing shortly.

However, if the interaction is weak not so much interesting is happening. Particles ignore each other most of the time, and if they meet, they, well interact weakly, and just scatter a bit off each other. If the interactions become stronger, interesting things start to happen. Bound states form, particles condense, and much more. That is where my interest lies.

The downside of this is that if the interactions between particles become strong, it becomes very hard to find a mathematical handle to treat them. That is the challenge, and the reason why rather few exact results are available. One solution is then to use brute force and just simulate the physics using a sufficiently large computer. That has provided us with very deep insights, and has become an invaluable tool in modern theoretical physics. For the type of problems I am most interested in such simulation methods are called lattice gauge theory, for reason I will explain later.

There are two major alternatives to such brute force simulations. One is the use of models and the other are so-called functional methods. In both cases the idea is to simplify the problem while capturing everything of interest.

Models, a term which I use here in a very broad sense, underlies the idea to find a simplified version of the theory at hand, sufficiently simplified to be easier to handle. Such theories than have often a very narrow range of applicability (for very similar reasons as the standard model itself ). However, if they are constructed very carefully such models very often help to understand not only broad features but often even quantitatively what is going on.

Functional methods are a different approach. The basic feature of theses methods are a set of equations which are in principle exact. Unfortunately, this set is often infinite, and in general approximations are needed to find solutions to them. If the approximations are good, it is possible to describe very much successfully with these equations and at the same time get deeper insight. Also, the approximations can be improved step-by-step, and thus permit eventually a full solution to the theory. I.e., at least in principle.

There are, of course, many other methods available, but these are the most important ones for my own research, and, except for models, I use them essentially on a day-by-day basis. The important methodological aspect in this is the combination of all the methods, and this results in something which is much more than just the sum of its parts.