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We define a Yang-Mills model in 1+1 dimensions coupled to a real scalar field and we study the Yang-Mills flow equations for this simple model. Yang-Mills flows have not been thoroughly studied, especially in a physical context, but may be able to provide valuable insight into both particle physics as well as gravity. We study our model using both the Hamiltonian equations and Euler-Lagrange equations, and we calculate the flow numerically using a simple finite difference method for the case of an Abelian Lie group and static fields. We are able to find several analytic solutions to the equations of motion and the numerical calculation of the flow suggests most non-constant solutions are unstable. We also find that the flow depends upon the relative values of the coupling constant and the mass of the scalar field. The results found with this simple model provide a starting point for the study of Yang-Mills flow in the context of more complicated (but more physical) models such as the Abelian Higgs.

Thesis (Ph. D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1992. / Includes bibliographical references (leaves 117-120).

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2008. / Includes bibliographical references (p. 123-131). Also available in electronic version.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1982. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 113-119).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1983. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 80-82).

Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisors: Vadim Kaplunovsky and Willy Fischler. Vita. Includes bibliographical references.

Gauge theories which have a phase transition could be useful in the study of quark confinement. One of the simplest theories containing a phase transition is the Coleman-Weinberg model of massless scalar electrodynamics. The calculation of the renormalized effective potential for the Coleman-Weinberg model is reviewed in detail using the path integral formalism. The effective potential is evaluated at the one-loop level to show that the model exhibits dynamical symmetry breaking at zero temperature. The divergent parts are shown to be renormalizable to two-loop order. The temperature dependence of the effective potential is then calculated to one-loop in order to demonstrate that the symmetry of the model is restored at high temperature, indicating a phase transition. Finally, for models which exhibit this type of behaviour, applications to SU(n) theories of quarks are discussed. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

This dissertation is concerned with variational problems whose field variables are functions on a product manifold M x G of two manifolds M and G. These field variables transform as type (0,1) tensor fields on M and are denoted by ψ(h)ᵅ (h = 1, ..., n = dim M, α = 1, ..., r = dim G). The dependence of ψ(h)ᵅ on the coordinates of G is given by a generalized gauge transformation that depends on a local map h:M → G. The requirement that a Lagrangian that is defined in terms of these field variables be independent of the coordinates of G and the choice of the map h endows G with a local Lie group structure. The class of Lagrangians that exhibits this type of invariance may be characterized by three invariance identities. These identities, together with an arbitrary solution of a system of partial differential equations, may be used to define field strengths associated with the ψ(h)ᵅ as well as connection and curvature forms on M. The former may be used to express the Euler-Lagrange equations in a particularly simple form. An energy-momentum tensor may also be defined in the usual manner; however additional conditions must be imposed in order to guarantee the existance of conservation laws resulting from this tensor. The above analysis may be repeated for the case that the field variables behave as type (0,2) tensor fields under coordinate transformations on M. For these field variables, the Euler-Lagrange expressions may be expressed as a product of a covariant divergence with the components λʰ of a type (1,0) vector field on M. An unexpected consequence of this construction is the fact that the Euler-Lagrange equations that result for the vector field λʰ are satisfied whenever the Euler-Lagrange equations associated with the field variables are satisfied.

by Nga-Wai Liu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Bibliography: leaves 161-166. / Chapter Chapter 0 --- Introduction / Chapter 0.1 --- Historical background I ´ؤ The Atiyah-Singer index theorem --- p.1 / Chapter 0.2 --- Historical background II ´ؤGauge theory --- p.3 / Chapter 0.3 --- Arrangement of the thesis --- p.5 / Chapter Chapter 1 --- Fredholm operators / Chapter 1.1 --- Basic propetries --- p.7 / Chapter 1.2 --- Compact operators --- p.8 / Chapter 1.3 --- Homotopy- invariance of the index --- p.9 / Chapter 1.4 --- Family of Fredholm operators ´ؤ Index bundle --- p.13 / Chapter 1.5 --- Wiener-Hopf operators --- p.19 / Chapter Chapter 2 --- K-theory / Chapter 2.1 --- K-theory of compact spaces --- p.24 / Chapter 2.2 --- K-theory with compact support --- p.28 / Chapter 2.3 --- Bott periodicity theorem --- p.32 / Chapter 2.4 --- Difference construction --- p.44 / Chapter 2.5 --- Thom isomorphism theorem on K-theory --- p.51 / Chapter Chapter 3 --- Operators on manifolds / Chapter 3.1 --- Differential operators on Euclidean spaces --- p.54 / Chapter 3.2 --- Differential operators on manifolds --- p.55 / Chapter 3.3 --- Pseudodifferential operators on Euclidean spaces --- p.58 / Chapter 3.4 --- Pseudodifferential operators on manifolds --- p.62 / Chapter 3.5 --- Elliptic operators --- p.70 / Chapter 3.6 --- Tensor products --- p.76 / Chapter Chapter 4 --- Atiyah-Singer index theorem / Chapter 4.1 --- The topological index --- p.84 / Chapter 4.2 --- The analytical index --- p.87 / Chapter 4.3 --- The Atiyah-Singer index theorem --- p.89 / Chapter 4.4 --- Characteristic classes --- p.95 / Chapter 4.5 --- Thorn isomorphisms --- p.98 / Chapter 4.6 --- Cohomological formulation of the topological index --- p.101 / Chapter Chapter 5 --- Geometric preliminaries / Chapter 5.1 --- "Connections on principal bundles, and associated bundles" --- p.104 / Chapter 5.2 --- Gauge transformations --- p.109 / Chapter 5.3 --- Riemannian geometry --- p.112 / Chapter 5.4 --- Bochner-Weitzenboch formula --- p.116 / Chapter 5.5 --- Characteristic classes via curvature forms --- p.121 / Chapter 5.6 --- Holonomy --- p.126 / Chapter Chapter 6 --- Gauge theory / Chapter 6.1 --- The Yang-Mills functionals --- p.128 / Chapter 6.2 --- Instantons on S4 --- p.131 / Chapter 6.3 --- Moduli of self-dual connections --- p.142 / Chapter 6.4 --- Manifold structure for Moduli of self-dual connections --- p.153 / References --- p.161