In this thesis we shall be concerned with the study of models which arise as a consequence of adopting discrete regularisations for various Euclidean space quantum field theories. Specifically, we employ a random triangulation of the continuum space, and define the fields only over nodes or links of the mesh. Lattice field theories, together with the Renormalisation Group, are introduced in the first chapter. Continuum physics is shown to depend on the positions and stabilities of zeroes of the β-function, which in turn requires a knowledge of the critical behaviour of the associated statistical model. In Chapter 2. we examine a theory of Dirac fermions in 2 + 1 dimensions on a random lattice. We investigate the behaviour of the 2-pt function and fermion condensate in the absence of any background gauge field. The results indicate certain doubling problems, generic to regular lattice formulations of fermion field theories, are evaded, at least at tree graph level. We then go on to examine the fermion vacuum currents in the presence of background fields with non-zero winding number. We are able to demonstrate the existence of a Chern-Simon's topological term in the gauge field effective action which yields parity violating vacuum currents. The magnitude of these are in agreement with certain continuum calculations. The final chapter concerns the properties of random surfaces. The particular class of models chosen originate as discretisations of Polyakov's string. The partition function is approximated by a sum over all possible random triangulations and an integral over vertex positions. The sum over random lattices is intended to mimick the functional integral over intrinsic metrics encountered in the continuum, and the model may also be pictured as 2D quantum gravity coupled to a scalar field. We consider the phase structure of the models when two forms of extrinsic curvature are added to the standard action. Monte-Carlo simulation indicates that with one type of curvature term a strong 2^nd order phase transition exists at finite coupling, leading to a new continuum limit for the model possessing long-range correlation properties. With the other type a much weaker higher order transition is observed. In this case the surface will be crumpled at long distance. We discuss the implications of these results for continuum surfaces.

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 5-matrices, which have the property that they are factorized into two-particle 5-matrices. 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 Yang-Baxter 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 5-matrices with Yang-Baxter structure, and comments on their features, both intrinsic and in relation to purely elastic 5-matrices. 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 5-matrices, but based on quantum-deformed algebras rather than classical algebras. In chapter five we describe the structure of the 5-matrices 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 Yang-Baxter dependent 5-matrices, 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 R-matrices computed in the text, and a summary of known solutions of the Yang-Baxter equation.

Thesis (Ph.D.)--McMaster University (Canada), 1993. / Source: Dissertation Abstracts International, Volume: 54-12, Section: B, page: 6250. Adviser: R. K. Bhaduri.

Thesis (Ph.D.) in Physics--University of Maine, 2007. / Includes vita. Includes bibliographical references (leaves 109-112).

Thesis (Ph.D.) - University of St Andrews, October 2007.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Dr. rer. nat.)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2001. / Includes bibliographical references (p. 131-133).

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Daniel S. Freed. Vita. Includes bibliographical references. Also available from UMI.

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: David Lowe. Includes bibliographical references (leaves 111-122). Also available online.