Exact S-matrices for quantum affine Toda solitons and their bound states

• Main Author: Gandenberger, G. M.
• Published: University of Cambridge 1996
• Subjects: 518
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.599294

The primary aim of this dissertation is to construct and study exact <I>S</I>-matrices for (1+1)-dimensional integrable field theories with quantum affine symmetry. We introduce a general scheme of how to use trigonometric <I>R</I>-matrices for the construction of exact <I>S</I>-matrices. This scheme is then applied to a specific class of relativistic field theories defined in (1+1)-dimensional Minkowski space, the so-called affine Toda field theories with imaginary coupling constant. The most important feature of these theories is the fact that their classical equations of motion admit soliton solutions. As a step towards the consistent quantisation of these theories we attempt to construct <I>S-</I>matrices for the quantum scattering of affine Toda solitons. Apart from the solitons there are also bound states of solitons in affine Toda field theories. By using the bootstrap principle we derive the <I>S</I>-matrices for the scattering of bound states. We focus in particular on the scalar bound states which are the analogues of the breathers in Sine-Gordon theory, and show that the <I>S</I>-matrices for the lowest breathers in the theory are identical to the <I>S-</I>matrices for the fundamental quantum particles. We also provide evidence for the consistency of the conjectured <I>S</I>-matrices through a detailed examination of their pole structure. We find that a large number of poles can be explained in terms of higher order diagrams, many of which involve a generalised Coleman-Thun mechanism. The layout of this thesis is as follows. In the introduction the axioms of analytic <I>S-</I>matrix theory are reviewed and some of the main features of affine Toda field theories are introduced. In chapter 2 we provide an introduction to the theory of quantised universal enveloping algebras and their <I>R-</I>matrices, where there are trigonometric solutions of the Yang-Baxter equation. We also describe the main features of quantum affine symmetries in two-dimensional field theories. Chapters 3, 4 and 5 deal with the detailed discussion of soliton <I>S-</I>matrices and bound states in <I>a<SUB>n</SUB></I><SUP>(1)</SUP>, <I>d<SUB>n</SUB></I><SUP>(2)</SUP><I><SUB>+</SUB></I><SUB>1, </SUB><I>b<SUB>n</SUB></I><SUP>(1) </SUP>and <I>a</I><SUB>2n</SUB><SUP>(2) </SUP>affine Toda field theories. Special attention is given to the two cases of <I>a</I><SUB>2</SUB><SUP>(1) </SUP>and <I>d</I><SUB>3</SUB><SUP>(2)</SUP>, and the pole structures of their proposed <I>S</I>-matrices are examined in great detail. We also provide a conjecture of the complete quantum spectrum of these theories. The dissertation concludes with a summary of results and some remarks about open questions and unsolved problems. In the appendices we provide detailed proofs and calculations omitted from the main part of the thesis. We also attempt to construct integral representations of <I>S-</I>matrix scalar factors and give tables containing relevant data for affine Lie algebras.