Title:

On the algebraic structure of factorized Smatrices

This thesis investigates the algebraic structure of certain quantum field theories in one space and one time dimension. These theories are integrable  essentially, highly constrained and therefore soluble. Thus, instead of having to use perturbative techniques, it is possible to conjecture their exact 5matrices, which have the property that they are factorized into twoparticle 5matrices. In particular, there are two types of such theory: in one, scattering is purely elastic, whilst in the other, there is additional structure dictated by the YangBaxter equation. This thesis explores the algebraic structure of the latter and its links with the former. We begin, in chapter one, with an informal summary of the development of the subject, followed by a more mathematical exposition in chapter two. Chapter three constructs explicitly some exact factorized 5matrices with YangBaxter structure, and comments on their features, both intrinsic and in relation to purely elastic 5matrices. In particular, there is an unexplained close correspondence between the mass spectra and particle fusings in the two types of theory. The next three chapters attempt to shed some light on these features. Chapter four constructs similar 5matrices, but based on quantumdeformed algebras rather than classical algebras. In chapter five we describe the structure of the 5matrices when the particles they describe transform in irreducible representations of classical algebras. This leads us to consider the Yangian algebra, the representation theory of which underlies YangBaxter dependent 5matrices, and which we therefore review briefly. We begin chapter six by reviewing the work which shows that the Yangian is also the charge algebra of the integrable quantum field theory, and subsequently show that the Yangian is also to a great extent present in the corresponding classical theory. We conclude with a brief seventh chapter describing the outlook for further research, followed by appendices containing respectively details of the Lagrangians of some integrable quantum field theories, a continuum formulation of the quantum inverse problem, explicit expressions for some of the Rmatrices computed in the text, and a summary of known solutions of the YangBaxter equation.
