Conserved quantities and renormalization group flows in two-dimensional field theory.

摘要

Several problems in two-dimensional field theory are investigated. The concepts of classical and quantum integrability in two space-time dimensions are presented in the Introduction and a number of algebraic structures associated with integrable systems are described. Some results of conformal field theory (CFT) and perturbed conformal field theory are reviewed.;In Chapter 2, the problem of interaction of two-level atoms in fibrillar geometry with electro-magnetic radiation is studied in perturbation theory. A new formalism is developed, representing the atomic spin operators with elementary fermions, and a resemblance between the structures of this model and quantum electrodynamics is established. Although the system studied is not itself integrable, it can be shown that the integrable quantum sine-Gordon model has some validity as an approximate theory.;The following two chapters study the properties of several multi-field generalizations of the sine-Gordon model. The Bukhvostov-Lipatov model is studied in Chapter 3. The classical integrability of the fermionic version of the model is established, both in the bulk and on the half line, by explicitly building a conserved charge of Lorentz spin 3. The quantum integrability of the more general double-cosine model is investigated using perturbed CFT. The analysis showed in particular that the conservation law is spoiled at the quantum level on the Bukhvostov-Lipatov submanifold of the parameter space. In Chapter 4 an N-field model is considered---its interaction term being a product of N cosines. For N ≥ 2 a conservation law of Lorentz spin 3 is found to first order in perturbed CFT on a manifold where the interaction becomes marginal. The integrability of the model on this manifold is further studied using renormalization techniques and for N = 2, 3, and 4, integrable points are found at which the model is equivalent to the bosonized Gross-Neveu model.;Finally, the renormalization properties of a class of integrable models---current-current perturbations to the Wess-Zumino-Witten (WZW) models---are studied in Chapter 5. A particular attention is given to superalgebra based models of this type.