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Being Discrete about Yang and Mills: Basic Techniques of Euclidean Lattice Gauge Theory
Author(s) -
Edgar Robert C.
Publication year - 1985
Publication title -
fortschritte der physik/progress of physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.2190330202
Subject(s) - lattice field theory , lattice gauge theory , gauge theory , physics , yang–mills theory , euclidean geometry , lattice (music) , crossover , abelian group , mathematical physics , theoretical physics , duality (order theory) , quantum field theory , mathematics , pure mathematics , geometry , acoustics , artificial intelligence , computer science
We discuss the formulation of gauge‐invariant quantum field theories (without dynamical matter fields) as statistical mechanics systems on four‐dimensional Euclidean lattices. Approximation methods including strong‐ and weak‐coupling expansions, mean‐field theory and Monte Carlo simulations are reviewed in detail, and Abelian duality transformations are derived. New models are discussed. An action is defined on 2 × 1 rectangular loops of links and its properties are investigated. It is found to result in phase transitions in 2, 3 and 4 dimensions with Z (2) and SU (2) gauge groups. A large class of models with Z ( N ) symmetry realised on plaquettes is investigated, and several phase diagrams are presented. A mixed model with interactions through both plaquettes and rectangles is found to have a line of phase transitions and a critical point associated with the crossover region in the Wilson SU (2) model.