Monday, March 26, 2012

Methods. United, they are strong.

In the last few postings, I have collected a number of methods: perturbation theory, simulations, and the abstract equations of motion. I have furthermore gave you a bit of a taste of one of our most important strategies: divide and conquer. Or, more bluntly, if the original problem is too complicated, first try a simpler one, which resembles it. This lead us to a stack of models, which bit by bit always included more details of the world.

This list is by no means complete. Over the years, decades, and centuries, physicists have developed many methods. I could probably fill a blog all by its own just by giving a brief introduction to each of them. I will not do this here. Since the man purpose of this blog is to write about my own research, I will just contend myself with this list of methods. These are, right now, those which I use myself.

You may now ask, why do I use more than one method? What is the advantage in this? To answer this, lets have a look at my work-flow. Well, actually this is similar to what many people in theoretical particle physics do, but with some variations on the choice of methods and topics.

The ultimate goal of my work is to understand the physics encoded in the standard model of particle physics, and to get a glimpse of what else may be out there. Not an easy task at all. One, which many people work on, many hundreds, probably even thousands nowadays. And not something to be done in an afternoon, not at all. We know the standard model, more or less, since about forty years at the time of this writing. We think essentially as long as it exists about what else there might be in particle physics.

Thus, the first thing I do is to make the things more manageable. I do this, by making a simpler model of particles. I will give some examples of these simpler models in the next few entries. For now, lets say, I just keep a few of the particles, and one or two of their interactions, not more. This looks much more like something I can deal with. Ok, so now I have to treat this chunk of particles happily playing around with each other.

To get a first idea of what I am facing, I usually start off with perturbation theory, if no one else did this before me. This gives me an idea of what is going on, when the interactions are weak. This hides much of the interesting stuff, but it gives me a starting point. Also, very many insights of perturbation theory can be gained with a sheet of paper an a pencil (and many erasers), and probably a good table of mathematical formulas. Thus, I can be reasonably sure that what I do is right. Thus, whatever I will do next, it has to reduce to what I just did now when the interactions become weak.

Now I turn to the things, which really interest me. What happens, when the interactions are not weak? When they are strong? To get an idea of this, the next step is to perform some simulations of the theory. This will give me a rough idea, of what is going on. How the theory behaves. What kind of interesting phenomena will occur. Armed with this knowledge, I have already gained quite a lot of understanding of the model. I usually know then what are the typical way the particles arrange themselves. How their interaction changes, when looking at it from different directions. What the fate of the symmetries is. And lot more of details.

With this, I stand at a crossroad. I can either go on, and deepen my understanding by improving my simulations. Or, I can make use of the equations of motion to understand the internal workings a bit better. What usually decides for the latter is then that many questions about how a theory works can be best answered when going to extremes. Going to very long or very short distances when poking the particles. Looking a very light or very heavy particles. Simulations cannot do this with an affordable amount of computing time. So I formulate my equations. Then I have to make approximations, as they are usually too complicated. For this, I use the knowledge gained from the simulations. And then I solve the equations, thereby learning more about how the model works.

When I am done to my satisfaction, then I can either enlarge the model somewhat, by adding some more particles or interactions, or go a different model. Hopefully, at the end I arrive at the standard model.

What sounds so very nice and straightforward up to here is not. The process I describe is an ideal. Even if it should work out like this, I am talking about the several years of work. But usually it does not. I run across all kind of difficulties. It could turn out that my approximations for the equations of motion have been too bold, and I can get no sensible solution. Then I have to do more simulations, to improve the approximations. Or the calculations with the equations of motion tell me that I was looking at the wrong thing in my simulations. That the thing I was looking at was deceiving me, and gave me a wrong idea about what is going on. Or it can turn out that the model cannot be simulated efficiently enough, and I would have to wait a couple of decades to get a result. Then, I have to learn more about my model. Possibly, I even have to change it, and start from a different model. This often requires quite a detour to get back to the original model. This may even take many years of work. And then, it may happen that the different method give different results, and I have to figure out, what is going on, and what to improve.

You see, working on a problem means for me to go over the problem many times, comparing the different results. Eventually, it is the fact that the different methods have to agree in the end what guides my progress. Thus, a combination of different methods, each with their specific strengths and weaknesses, is what permits me to make progress. In the end, reliability is what counts. And with this nothing cuts it like a set of methods all pointing to the same answer.