Sigma model

Abstract: We examine some problems associated with the low-momentum behavior of gauge theories and other renormalizable field theories. Our main interest is in the infrared structure of unbroken non-Abelian gauge theories and how this is affected by the presence of other heavy fields coupled to the massless gauge fields. It is shown in the context of a simple model of gauge mesons coupled to massive fermions that the heavy fields decouple at low momenta except for their contribution to renormalization effects. This result is used to discuss the mass-shell structure of the fermion propagator. The decoupling theorem is then stated for a general renormalizable theory and applied to some interesting examples. One is a more general gauge theory which makes use of the Higgs mechanism and attempts to unify the elementary particle forces. Another is the connection of the linear and nonlinear $\ensuremath{\sigma}$ models in the limit ${m}_{\ensuremath{\sigma}}\ensuremath{\rightarrow}\ensuremath{\infty}$.

Abstract: The interacting disordered electron problem is reviewed with emphasis on the quantum phase transitions that occur in a model system and on the field-theoretic methods used to describe them. An elementary discussion of conservation laws and diffusive dynamics is followed by a detailed derivation of the extended nonlinear sigma model, which serves as an effective field theory for the problem. A general scaling theory of metal-insulator and related transitions is developed, and explicit renormalization-group calculations for the various universality classes are reviewed and compared with experimental results. A discussion of pertinent physical ideas and phenomenological approaches to the metal-insulator transition not contained in the sigma-model approach is given, and phase-transition aspects of related problems, like disordered superconductors and the quantum Hall effect, are discussed. The review concludes with a list of open problems.