Since the above characterization appeared [1] in 1965 we have witnessed great progress in quantum field theory, our description of fundamental particles and their interactions. This book displays her in informal dress, robust and ready to give results, rigorous, while at a pedestrian mathematical level. By approximating space–time by a collection of points on a lattice we get a number of benefits:

• it serves as a precise but simple definition of quantum fields, which has its own beauty;

• it brings to the fore and clarifies essential aspects such as renormalization, scaling, universality, and the role of topology;

• it makes a fruitful connection to statistical physics;

• it allows numerical simulations on a computer, giving truly nonperturbative results as well as new physical intuition into the behavior of the system.

This book is based on notes of a lecture course given to advanced undergraduate students during the period 1984–1995. An effort was made to accomodate those without prior knowledge of field theory.

We introduce here quarks and gluons. The analogy with electrodynamics at short distances disappears at larger distances with the emergence of the string tension, the force that confines the quarks and gluons permanently into bound states called hadrons.

Subsequently we introduce the simplest relativistic field theory, the classical scalar field.

QED, QCD, and confinement

Quantum electrodynamics (QED) is the quantum theory of photons (γ) and charged particles such as electrons (e±), muons (μ±), protons (p), pions (π±), etc. Typical phenomena that can be described by perturbation theory are Compton scattering (γ + e– → γ + e–), and pair annihilation/production such as e+ + e– → μ+ + μ–. Examples of non-perturbative phenomena are the formation of atoms and molecules. The expansion parameter of perturbation theory is the fine-structure constant α = e2/4π.

Quantum chromodynamics (QCD) is the quantum theory of quarks (q) and gluons (g). The quarks u, d, c, s, t and b (‘up’, ‘down’, ‘charm’, ‘strange’, ‘top’ and ‘bottom’) are analogous to the charged leptons νe, e, νμ, μ, ν, and. In addition to electric charge they also carry ‘color charges’, which are the sources of the gluon fields. The gluons are analogous to photons, except that they are self-interacting because they also carry color charges. The strength of these interactions is measured by αs = g2/4π (alpha strong), with g analogous to the electromagnetic charge e.

In many physical problems, especially when fluctuations of scales of different orders of magnitude are essential, there is no small parameter which could simplify a study. A typical example is QCD where the effective coupling, describing strong interaction at a given distance, becomes large at large distances so that the interaction really becomes strong.

't Hooft [Hoo74a] proposed in 1974 to use the dimensionality of the gauge group SU(N) as such a parameter, considering the number of colors, N, as a large number and performing an expansion in 1/N. The motivation was an expansion in the inverse number of field components N in statistical mechanics where it is known as the 1/N-expansion, and is a standard method for nonperturbative investigations.

The expansion of QCD in the inverse number of colors rearranges diagrams of perturbation theory in a way which is consistent with a string picture of strong interaction, the phenomenological consequences of which agree with experiment. The accuracy of the leading-order term, which is often called multicolor QCD or large-N QCD, is expected to be of the order of the ratios of meson widths to their masses, i.e. about 10–15%.

While QCD is simplified in the large-N limit, it is still not yet solved. Generically, it is a problem of infinite matrices, rather than of infinite vectors as in the theory of second-order phase transitions in statistical mechanics.

Matrix models first appeared in statistical mechanics and nuclear physics [Wig51, Dys62] and turned out to be very useful in the analysis of various physical systems where the energy levels of a complicated Hamiltonian can be approximated by the distribution of eigenvalues of a random matrix. The statistical averaging is then replaced by averaging over an appropriate ensemble of random matrices. This idea has been applied, in particular, in studying the low-energy chiral properties of QCD [SV93, VZ93].

Matrix models possess some features of multicolor QCD described in Chapter 11 but are simpler and can often be solved as N → ∞ (i.e. in the planar limit) using the methods proposed for multicolor QCD. For the simplest case of the Hermitian one-matrix model, the genus expansion in 1/N can be constructed.

The Hermitian one-matrix model is related to the problem of enumeration of graphs. Its explicit solution at large N was first obtained by Brézin, Itzykson, Parisi and Zuber [BIP78] and inspired a lot of activity in this subject. Further results in this direction are linked to the method of orthogonal polynomials [Bes79, IZ80, BIZ80].

A very interesting application of the matrix models along this line is for the problem of discretization of random surfaces and two-dimensional quantum gravity [Kaz85, Dav85, ADF85, KKM85]. The continuum limits of these matrix models are associated with lower-dimensional conformal field theories and exhibit properties of integrable systems.