Perturbation theory

Now, let us start with a look at the different methods in more detail. The first is the mainstay of theoretical physics, and the first thing everyone tries when encountering or designing a new theory: Perturbation theory, often abbreviated simply by PT.

The basic idea of perturbation theory is rather direct. When we have a theory, it is very often the case that we can solve a simpler version of it exactly. For example, if we have QED then we can solve exactly the case where the electromagnetic charge would be zero, because the particles then do not interact with each other. And free, non-interacting particles is something we can do very well. Of course, this is not what QED is really like. Otherwise, we could not see, as the electrons in our eyes would not react to the light made out of photons. To capture this, perturbation theory assumes that the interactions between electrons and photons is only a small alteration to the picture of free particle: A perturbation, and hence the name. Of course, finding a useful split depends on the theory in question, and many different types are actually in use.

Once such a setup is available, we have created powerful mathematical tools how to calculate then anything we want under this assumption. The most important principle is that we can reformulate what we mean by perturbation mathematically by stating that some quantity is small What this precisely means depends on the theory in question. In the above example of QED, it would be the electric charge.

We can then organize perturbation theory systematically by counting how often the small quantity appears in an expression. We then speak of the order of perturbation theory. If it appears the lowest possible number of times, which may be zero, we call this tree level. The reason for this name is that the mathematical expressions can be generated in the way a tree grows, i.e., in the form of starting somewhere and then moving on. In fact, in general this is often equivalent to a classical theory. This means that we treat the particles as quantum particles, but the interaction between them like a classical interaction, without additional quantum effects.

We can now increase the number of times the quantity appears. We also say that we calculate higher orders in the quantity, where order counts the number of times the quantity appears. If we calculate the contribution with the second-least number of times the quantity appears, we call this leading order correction. Since such a correction only appears in the quantum theory, we also call it a quantum correction. The contribution with third-least appearance is called next-to-leading order. If we further increase the order, we just add the corresponding number of times next-to in front, e.g. next-to-next-to-next-to-leading order. This seems to become quickly awkward, but no fear, no too high orders are often calculated.

The reason is that perturbation theory at higher order becomes rather complicated just from an organizational point of view. Quickly, perturbative expressions fill hundreds and thousands of pages with expressions, which have to be evaluated. The final end result will only fill a very few pages, if more than one at all. Over the time, we have developed very powerful methods to deal with this complexity. If you ever heard of Feynman diagrams, given that these have made their way even into some obscure corners of pop culture, then this is one of these tools. Its a very powerful graphical technique to organize perturbation theory in a very efficient way. And this is quite important. There are furthermore other ingenious methods to reduce the amount of calculations necessary. Nonetheless, in the end the expressions remain rather long, and it requires computers to evaluate them. This is in general straightforward to tell the computer what to do. But it is very challenging to do it in a way that the computer is not occupied for the next couple of years, but only a few days or less.

With these methods, we have went to order ten in QED, and for some quantities to order four in the standard model. This seems little, but because of the complexity required many people and decades of time. Still, some times experiments are so precise that the accuracy achieved by these calculations is not sufficient. But of course, there are also many cases where it is the other way around. In a way, it is a kind of arms race between theoreticians and experimentalists.

In the end, however, perturbation theory will not give you the full answer. You can mathematically prove that certain phenomena cannot be calculated using perturbation theory. You may be lucky, and using a different starting point, this can be circumvented for a certain quantity, but then other quantities will not be possible to access. Furthermore, we know that perturbation theory cannot be pursued to arbitrary order, but will collapse at a certain point, for mathematical reasons. Though also here progress has been made, we know that perturbation theory cannot provide the full answer to any question. Already as simple a quantity as the mass of the proton cannot be calculated in perturbation theory. Nonetheless, much of what we measure in experiments, say at LHC, can be very well and very accurately calculated with perturbation theory. Thus, perturbation theory remains to be one of the main tools in particle physics, and for very good reasons so.