Phase transitions in lattice gauge theories

The phase structure of gauge theories on a lattice is discussed. The concept of order and disorder which stems from the theory of phase transitions in statistical mechanics is emphasized. In particular the analogies between four-dimensional gauge theories and two-dimensional magnetic systems are analyzed. Order and disorder parameters, as well as correlation functions, are compared. In particular a remarkable analogy is exhibited between the two-dimensional Planar ferromagnet and the four-dimensional Abelian gauge theory. The behavior of gauge theories with dynamical matter (Higgs) fields is also considered. It is shown that, if the matter field carries the fundamental unit of charge and if there are enough of them so that the gauge symmetry is completely broken, then the confinement regime of the pure gauge theory is continuously connected to the spontaneously broken (or Higgs) phase. In addition, it is found that a phase with free charges and no Higgs mechanism may also exist in certain conditions. Finally, the role of local (gauge) symmetries in random magnetic systems is discussed. This has special relevance for the spin glass problem. Attention is focused on the concept of gauge-invariant disorder or frustration. The energetics of these defects and the behavior of correlation functions and order parameters in simple distribution of frustrations are studied