The exact S-matrices of affine Toda field theories
This thesis is concerned with exact solutions to various massive field theories in 1+1 dimensions. Two approaches are described. The first, abstract and non-Lagrangian, relies on the considerable understanding that there now is of massless two-dimensional field theories. A perturbative scheme can be developed within which various exact statements may be made. Chapter 1 contains a review of this technique, together with some work applying it in various simple situations. The particular structures studied turn out to have a deep connection with certain Lie algebras, a fact which is discussed in the concluding three sections of the chapter. A complementary approach is to study specific, classically integrable, lagrangians in the hope that their quantum versions will also permit an exact treatment. Motivated to some extent by the findings of chapter 1, the remainder of the thesis is devoted to a particular class of models known as affine Toda field theories. Mixtures of perturbative and non-perturbative ideas are employed. The non-perturbative elements are to be found in analytic S-matrix theory, reviewed in chapter 2, while various features of the classical theory necessary for a perturbative quantum treatment are derived in chapter 3. Making use of this information, chapter 4 proposes exact expressions for the S-matrices for a large subset of the Toda theories, which are then checked in perturbation theory. Finally, the relevance or otherwise of the Toda S-matrices to the perturbations of massless theories studied in chapter 1 is discussed, and some possible directions for future work are mentioned.