101. Confinement. Extremely naive introduction

So, as I have promised last time, I am starting to collect my thoughts on the problem of quark confinement here. Actually, if things will go well with writing, I shall want to make a little review paper out of it.

So, why should you, fellow reader, even be interested in the problem of quark confinement or may be even eager to solve it? Let us first postulate that you are a young physicist or possibly even ambitious one   (Well, do we have any other kind of young physicists?) There is than a very good chance that you are aware of Clay Institute Millenium Prize Problems. Among them is one called “Yang Mills Theory“. If you solve it, you will be 1 million US dollars richer,* since the sponsors of the Clay Mathematics Institute are willing to pay it to a person smart enough to solve the problem.

The problem is not explanation of the quark confinement but the closely related one of the mass gap in the Yang-Mills theory. In essence, it is unclear why elementary excitations of the Yang-Mills field which are massless at the classical level acquire (small) mass at the quantum level.

(By the way, in this series of posts I am going to also discuss the origin of the mass gap in several 2-dimensional quantum field theories like non-linear sigma model and the principal chiral field. The latter one is especially interesting and somewhat resembles the behaviour of the Yang-Mills QFT in 4 dimensions.)

What if you are not a physicist but, say, a biologist or you work in finances? Why should you be interesting to hear about confinement? I would say, one good reason is the following: the problem of quark confinement is the problem two generations of theoretical physicists (very good ones, too) have broken their teeth dealing with. Although there are lots of ideas, as you will see, the ultimate understanding of quark confinement seems to be as far as, say, theoretical understanding of developed turbulence (note that the list of seven Millenium Prize problems also contains the problem of existence and smoothness of general solutions for the Navier-Stokes equation )

It actually seems that the confinement problem is currently in the heart of theoretical and mathematical physics. For example, in order to solve it you will probably have to understand string theory sufficiently well since YM does behave as a string theory at strong couplings. Unfortunately, we don’t even have an idea what this string theory is.

But enough advertisement… I am sure you got it. So, what is the problem of quark confinement? It can be formulated as follows. It is well known from the early 1970s that strong interactions are described by quantum chromodynamics (or QCD) – Yang-Mills quantum field theory of self-interacting gauge bosons and fermions coupled to them, the theory of color interaction (here comes the title from, chromo is color in Greek). The fermions (called quarks) are rather exotic, in particular, they carry fractional electric charge.

QCD is well tested at small distances where it is weakly coupled, so we are very much sure that it is correct theory of strong interactions. Although it seems that the correct degrees of freedom at small distances are quarks and gauge bosons carrying interaction between quarks, there was no single quark observed in Nature to this day. While, as it seems, quarks are nice degrees of freedom at energies much larger than so called (of the order few hundred MeV), at smaller energies all physical states are colorless and no particles with fractional electric charge are observed in Nature. Somehow quarks are getting bounded with each other, but a particular mechanism responsible for this binding remains unclear. What we know for sure is that quantum chromodynamics (QCD) is sufficient to explain the quark confinement: in particular, the latter is observed in lattice simulations of QCD.

That’s all for the introduction. Next time I am going to start discussing criteria of confinement and unfortunately shall have to be much more technical. I don’t suppose that you know QCD or non-perturbative methods in QFT (instantons, monopoles, Montonen-Olive conjecture, strong coupling expansion, large N expansion, Migdal-Makeenko) and string theory, but shall – that you’ve learned some kind of QFT 101, i.e., you are aware of such things as renormalization group, functional integrals, etc.

Finally, here are some books that may be of help for you:

1. Greiner, Schramm, Stein, “Quantum chromodynamics”. I have an impression that this is currently the best QCD textbook.
2. Polyakov, “Gauge fields and strings”. Polyakov is one of the creators of string theory and well known expert in confinement. In particular, it was he who solved confinement problem in abelian compact YM and have introduced ‘t Hooft-Polyakov monopoles and instanton gas.
3. Smilga, “Lectures on quantum chromodynamics”. Smilga is a very well known expert in QCD in general and QCD sum rules in particular. His book is actually very easy to read (not the first reading though – Greiner, Schramm, Stein is much better for it – but perfect as a second textbook).