Monday, March 19, 2012

Modelling reality

Ever wondered why it is called the standard model of particle physics? And what a physicist has in mind, when she talks about models?

Models are the basic ingredient of what a theoretical physicist is doing. The problem is that we do not know the answer, we do not know the fundamental theory of everything. Thus, the best we can do is take what we know, and make a guess. The result of such a guess is a model. Such a model should describe what we see. Thus, the standard model of particle physics is the one model what we know about particle physics right now, as incomplete as it may be. It is called the standard one, because it is our best effort to describe nature so far, to model nature in terms of mathematics. There are also other standard models. We have one for how a sun functions, the standard model of the sun, or how the universe evolved, the standard model of cosmology.

Now, when I say, it is our best guess this implies that it is not necessarily right. Well, actually it is, in a sense. It was made the standard model, because it describes (or, if you read this in a couple of years, perhaps has described) our experiments as good as we can wish for. That means, we have found no substantial evidence against this model within the domain accessible in the experiment. This sentence has two important warning signs attached.

The one is about the domain. We do not know what is the final theory. But what we do know is the models. And any decent model will tell us, what it can describe, and what not. This also applies to the standard model. It tells us: 'Sorry guys, I cannot tell what is happening at very large energies, and on the matter of gravitation, well I stay away from this entirely.' This means that this standard model will only remain the standard model until we have figured out what is going on elsewhere. At higher energies, or what is up with gravitation. However, this does not mean that the standard model will be completely useless once we managed that. As with many standard models in the past, it likely will just become part of the large picture, and remain a well-trusted companion, at least in some area of physics. Happened to Newton's law, which was superseded by special relativity, and later by general relativity. Happened to Maxwell's theory of electromagnetism, which was superseded by Quantumelectrodynamics, and later by the standard model. Of course, there is once more no guarantee, and it may happen that we have to replace the standard model entirely, once we see the bigger picture. But this seems right now unlikely.

The other thing was about the experiment. Models are created to describe experiments (or observations, when we think about the universe). Their justification rests on describing experiments. We can have some experimental result, and cook up a model to explain it. Then we do a prediction, and make an experiment to test it. Either it works, and we go on. Or it does not, and then we discard the model. While people developed the standard model, this was a long, painful process during which many models have been developed, proposed, checked, and finally discarded. Only one winner remained, the model which we now call the standard model.

Ok, nice and cozy, and that how science works. But I was talking about methods the last couple of times, so what has this to do with it? Well, this should just prepare you for an entirely different type of models, to avoid confusion. Hopefully. Now the standard model is the model of particle physics. But, honestly, it is a monster. Just writing it down during a lecture requires something like fifteen minutes, two blackboards, and two months of preparation to explain all the symbols, abbreviations and notions involved to write it in such a brief version. I know, I have done it. If you want to solve it, things go often from bad to worse. That is where models come in once more.

Think of the following: You want to describe how electric current flows inside a block of, say, Aluminum. In principle, this is explained by the standard model. The nuclei of Aluminum come from the strong force, and the electrons from the electromagnetic one, and both are decorated with some weak interaction effects. If you really wanted to try describing this phenomena using the standard model, you would be very brave indeed. No physicist has yet tried to undertake such an endeavor. The reason is that the description using the standard model is very, very complicated, and actually most of it turns out to be completely irrelevant for the electric current in Aluminum. To manage complexity, therefore, physicists investigating aluminum do not use the standard model of particle physics in its full glory, but reduce it very, very much, and end up with a much simpler theory. This models Aluminum, but has forgotten essentially everything about particle physics. This is then a model of Aluminum. And it works nice and well for Aluminum. Applying it to, say, copper, will not work, as Aluminum nuclei have been put into it as elementary entities, to avoid the strong interactions. You would need a different model for copper then, or at least different parameters.

So, we threw away almost all of the power of the standard model. For what? Actually, for a price worth the loss: The final model of Aluminum is sufficiently simple to solve it. Most of our understanding of materials, technology, chemistry, biology (all described by the standard model of particle physics, in principle) rests on such simplified models. With only the standard model, we would not be able to accomplish anything useful for these topics, even knowing so much about particles. In fact, historically, the development was even the other way around. We started with simple models, describing few things, and generalized bit by bit.

Ok, you may say. You see the worth of simplified models for practical applications. But, you may ask, you surely do not simplify in particle physics? Well, unfortunately, we have to, yes. Even when only describing particles, the standard model is so complicated that we are not really able to solve it. So we very often make models only describing part of it. Most what we know about the strong interactions has been learned by throwing away most of the weak interactions, to have a simpler model. When talking about nuclear physics, we even reduce further. Also, when we talk about physics beyond the standard model, we often first create very simple-minded models, and in fact neglect the standard model part. Only, if we start to do experiments, we start to incorporate some parts of the standard model.

Again, we do this for the sake of manageability. Only by first solving simpler models, we understand how to deal with the big picture. In particle physics the careful selection of simplified models was what drove our insight since decades. And it will continue to do so. This strategy is called divide and conquer. It is a central concept in physics, but also in many other areas where you have to solve complicated problems.

Of course, there is always a risk. The risk is that we simplify the model too much. That we loose something important on the way. We try to avoid that, but it has happened, and will happen again. Therefore, one has to be careful with such simplifications, and double-check. Often, it turns out that a model makes very reliable predictions for some quantities, but fails utterly for others. Often, our intuition and experience tells us ahead what is a sensible question for a given model. But sometimes, we are wrong. Then experiment is one of the things which puts us back on track. Or that we are actually able to calculate something in the full standard model, and find a discrepancy compared to the simple model.

In the past, such simplified models were created by very general intuition, and including some of the symmetries of the original theory. Over time, we have also learned how to construct, more or less systematically, models. This systematic approach is referred to as effective field theory. This name comes about as it creates a (field) theory which is an effective (thus manageable) version of a more complicated field theory in a certain special case, e.g. low energies.

Thus, you see that models are in fact a versatile part of our tool kit. But they are only to some extent a method - we have still to specify how we perform calculations in them. And that will lead us then to the important concept of combining methods next time.