The Monte Carlo method in lattice gauge theories

Applications of the Monte Carlo method in lattice gauge theories, including applications in quantum chromodynamics, are reviewed. The lattice formulation of gauge theories, the corresponding concepts, and the corresponding methods are introduced. The Monte Carlo method as it is applied to lattice gauge theories is described. Some specific calculations by the Monte Carlo method and their results are examined. The phase structure of lattice gauge theories with Abelian groups Z$_N$ and U(1) (a lattice formulation of a compact electrodynamics) is discussed. The non-Abelian groups SU(2), SU(3) (a lattice formulation of quantum chromodynamics), and others are also discussed. The procedure for calculating quantities referring to the continuum limit by the Monte Carlo method is discussed for quantum chromodynamics. A detailed analysis is made of results calculated for the continuum theory: string tensions and interaction potentials, which show that quarks are confined; glueball mass spectra; and the temperature of the transition from the phase of hadronic matter to the phase of a quark-gluon plasma. Masses calculated for hadrons consisting of quarks are briefly discussed.