Up to now, continuous field theory has appeared as a tool in the study of critical phenomena. Conversely, techniques from statistical mechanics can be useful in field theory. In 1973, Wilson proposed a lattice analog of the Yang–Mills gauge model. Its major aim was to explain the confinement of quarks in quantum chromodynamics. The lattice implementation of a local symmetry yields a transparent geometric interpretation of the gauge potential degrees of freedom, the latter being replaced by group elements assigned to links. Strong coupling expansions predict a linearly rising potential energy between static sources. Complex phase diagrams emerge when gauge fields are coupled to matter fields, and new phenomena appear, such as the absence of local order parameters. The discretization of fermions leads also to interesting relations with topology. This chapter is devoted to the theoretical developments of these ideas.

Generalities

Presentation

Schematic as they are, statistical models have directly a physical background at any temperature. Lattices may represent the crystalline structure of solids. They play an important role at short distance, but become irrelevant in the critical region, except as a regulator for the field theory describing the approach to critical points. The opposite point of view can also be considered. A lattice is artificially introduced as a regulator for a continuous field theory. The lattice system has no physical meaning, but can be studied at any “temperature”, so that one can get information about its critical region, hopefully described by the initial continuous theory.