WHY WE CAN DO PHYSICS, part I

For instance, how is it possible that rolling of a tennis ball on a table can be easily described by Newton's laws? Isn't it the case that tennis balls and tables are objects composed of a wast number of molecules, each consisting of several atoms – bound states of negatively charged electrons and positive nucleus which, in turn consists of protons and neutrons bound together by an exchange of pions, all of them in fact being bound states of quarks and gluons, which may be nothing but different vibrational modes of a small string?

In other words, what's the miracle behind the fact that motion of a tennis ball can be modeled by a set of equations which were known already to Sir Isaac Newton in 17th century, while we don't have to pay any attention at all to some complicated (and, above all, completely unknown) equations of a fundamental theory of everything? An answer to these questions (which is simultaneously a reason why we can reasonably do any physics at all) is provided by the following two constructs: effective theory and renormalisation group. The goal of this story will be to facilitate these two concepts to  an interested reader assuming no knowledge of advanced maths. While in Part I we take a closer look at effective theories, in the second part we'll use a simple analogy of a bead on a wire to explain how renormalization helped physicists to solve the problem with infinities in quantum electrodynamics and what it has to say to what a quantum theory of gravity should (not) be like.

But first, let us go back to our tennis ball. We're going to probe its structure (account of which we gave in the introduction to some detail) by the favourite method of experimental particle physicists: we'll simply smash the ball against a wall (impenetrable and placed in vacuum) and see what we get back. After we reach a certain kinetic energy (value of which can be made precise by considering molecular structure of the ball's skin), the ball will no longer simply bounce off the wall and its skin will rupture. Should we keep ramping up the energy (until we reach an energy needed to severe molecular bonds) the ball would disintegrate into its constituent atoms. Increasing its initial energy further, we'd start to see individual nuclei and electrons coming out. And so on (there's a small issue with quarks and gluons as we'll see in the next episode).

We notice that what we have managed so far is to “translate” the structure of the tennis ball into an energy hierarchy, where at each step a new, increasingly more fundamental component appears as a result of our smashing experiment. At the same time, we see again that if we keep the initial speed of the ball low enough (for instance, if we just throw the ball by hand) it is completely acceptable to work within classical mechanics – there's no need to bring in more fundamental theories. Now it's also clear why this has to be the case: throwing the ball by hand, we simply cannot provide enough energy in order to tear the ball apart, allowing us to forget about its molecular structure. It is however good to point out that if, on the other hand, we somehow do succeed in supplying enough energy to the ball, classical mechanics will fail as the ball will rupture, which will require us to reconsider the situation and take a closer look at the microstructure of the ball's skin.

Now we can ask ourselves: what if we know what the fundamental theory looks like? Can we somehow reconstruct macroscopic Newton's laws from knowledge of the equations describing the motion and interactions of elementary particles? The answer to this question is, indeed, “theoretically yes”: there is a prescription, according to which we can, while investigating only low-energy phenomena (throwing a ball by hand), consistently ignore (or, if you want, average over) high-energy modes such as the degrees of freedom connected with individual molecules, protons or quarks constituting given macroscopic body. Classical mechanics then finally arises from this process as a so called effective theory. Similarly, if we look at the interaction between protons and neutrons in atomic nuclei, we don't need to concern ourselves with the full complexity of quantum chromodynamics (the theory describing quarks and gluons) as it is suffucient work with an effective theory explaining the interaction by an exchange of pions.

In the above paragraphs, we've found out why we don't have to go into the details of fundamental theories when trying to describe the everyday physics around us: we only need to consider effective theories which can be formulated empirically, as it was the case of Sir Newton and his laws of mechanics. At the same time, however, this doesn't look very promising from the viewpoint of finding fundamental theories: the process of construction of effective theories from the fundamental ones is manifestly irreversible, all details about high-energy modes being irreparably smeared out. If we want to study phenomena currently beyond reach of experiments, we have to resort to different strategies, example of which is the unification of theories. An extremely important lead in this quest is the so called renormalizability, about which (and other things) we'll have more to say next time.