ABSTRACT
For studying the properties of QCD at large distances it is necessary to regularize the theory in such a way that this regularization is not connected with the usual expansion into a series in terms of the Feynman diagrams, which is admissible only in the case of a small coupling constant. For this purpose Wilson [122] evolved the gauge theory on a lattice, in which the discrete space-time was introduced instead of space-time continuum. Notice that the idea of replacement of the field variable (determined in each point of space-time) with a lattice variable (assigned only in some points of space-time) is probably older than the idea of the field variable itself. In some physical problems, such as the description of electrons in crystals, a lattice spacing a is a physical quantity. The introduction of lattice is a mathematical trick. It is leading to the natural scheme of cut-off because the lengths of waves that are less than twice the spacing do not have sense, that is, the region of changing the momenta is limited by the value pi/a. Consequently, the introduction of the lattice provides a cut-off removing the ultraviolet divergences, which are the scourge of the quantum field theory (QFT). Like any regularization, the lattice cut-off must be eliminated after the renormalization. Physical results can be obtained only in a continuous limit, when the lattice spacing is taken to zero. Most often the cubic lattice is used, the points of which (called sites) are situated at the vertices of cubes filling the space. The shortest interval between two neighboring cites is called a link, the length of a link-a lattice spacing. The simplest example of QFT on a lattice is the scalar field theory, for which only the
values on lattice sites are considered, and derivatives appearing in the motion equations are approximated by finite differences. The values of fields in lattice sites are the dynamic variables of a task. As in all practical applications the sites of finite size are considered, the QFT on a lattice changes into the theory with a finite value of degrees of freedom that are determined by the number of sites. For a satisfactory description of continuous field configuration it is necessary for the lattice spacing to be much less than the typical scale of changing the fields (in the case of smooth configurations this can be achieved by a decrease of the lattice spacing). In lattice formulating of a spinor field its values are also ascribed to the lattice sites while the values of a vector field are attributed to the links. On a lattice the QFT becomes well-defined mathematically and can be studied by dif-
ferent methods. The lattice perturbation theory reproduces all the usual results of other regularization schemes although it meets some technical difficulties. In QCD the discrete space-time is particularly well suited for a strong coupling expansion. It is remarkable that
Particle and
in this limit the confinement (or color confinement) is automatic and the theory reduces to one of quarks on the ends of strings with a finite energy per unit length.
