LATTICE GAUGE THEORY The general goal of this research is to further the understanding of non-Abelian gauge fields in nonperturbative regions. I am particularly interested in the interplay of gauge symmetry, vacuum structure, and hadron structure. Non-Abelian gauge theories are the foundation of the Standard Model. In particular, it is generally accepted that the SU(3) gauge theory of quarks and gluons describes the strong interactions. For those physical processes that are controlled by short-distance interactions and that are relatively insensitive to large-distance effects, the results of perturbative calculations and experiments can be compared quantitatively. For most reactions, this is not the case because large-distance, nonperturbative effects are important. Confinement is only the most dramatic example; structure functions, hadronization, and strong interaction corrections to weak decays are important, practical issues. The perturbative calculations that are valid at short distance where the effective coupling is weak require a gauge fixing that obscures to some considerable extent the fundamental role of gauge symmetry. Lattice gauge theory is an approximation that is complementary to continuum perturbation theory in that it maintains manifest gauge invariance while sacrificing Poincare symmetry. The space-time symmetry is restored as the correlation length for lattice fields diverges (as measured in lattice units) and the lattice spacing approaches zero (as measured in physical units). This occurs at a second order phase transition. The long-distance, nonperturbative effects are more accessible in the lattice approximation. Within the framework of lattice gauge theory, one can employ either approximate, analytical methods or a direct attack by numerical simulation. My work has exploited both approaches. Recently it has been mostly analytical although much of the motivation is from my previous numerical results. The zero-temperature, pure Yang-Mills gauge theory has no adjustable parameters. This is part of the reason that it is so difficult to approximate. A study of the theory at finite temperature introduces a parameter T that can be varied in a controlled way so as to probe different aspects of the theory. In particular, confinement, which is apparently present at low temperature, is absent at sufficiently high temperature. A study of the theory as the temperature is changed can yield useful information about the features that distinguish these two phases and thus contribute to an understanding of the vacuum state. One recent project is concerned with SU(2) gauge fields near the finite-temperature phase transition. It is an analytical study of the critical properties of a flux model that can be used above the critical temperature. Another completed project offers an improved physical understanding of the order parameter for the phase transition. A new project deals with the internal structure of the flux tube as the temperature increases. The second new project relates to a different subject: localization. This is a phenomenon in condensed matter physics. A sufficiently random external field eliminates conduction states for electrons. Is there an analogue in relativistic quantum field theory and in particular in QCD? ``Critical Exponent for the Density of Percolating Flux'', Phys. Rev. D 49, 2597 (1994). The goal is to find a model that can describe the important physics of the gauge field just above the transition temperature. This region is so far from the regions of validity of strong and weak coupling expansions that neither can be expected to provide an adequate description. Flux tube models have been used successfully in the past to describe the confined phase of finite-temperature gauge fields. Fluctuating tubes of flux connect fundamental sources. There are also thermal fluctuations that are small lumps of flux --- i.e. closed loops or other more complicated closed patterns. In the high-temperature, deconfined phase, the situation is rather different. There is a single percolating network of flux along with a finite density of flux lumps. As T approaches T_c from above, the density of the infinite network approaches zero. It is interesting to consider the relative contributions of the finite and the infinite networks to the critical, singular parts of physical quantities such as the internal energy. I devised a simplified model, which allows an analytical approach that separates the contributions from the percolating flux network and the finite lumps. Flux is moving from the infinite cluster to the finite clusters much more rapidly than the total density is decreasing. ``Phase of the Wilson Line'', UCD-94-21. This paper discusses some aspects of the global Z(N) symmetry of finite-temperature, SU(N), lattice gauge theory without matter fields. It contributes to the recent discussion of the physics of the phase of the Wilson line expectation value. There is a low-temperature phase, which is presumed to confine fundamental sources, and a high-temperature phase, which does not. The order parameter is the finite-temperature expectation value of the Wilson line. The line carries a nontrivial representation of the Z(N) symmetry. In the confining phase, the Wilson line is zero, and the ensemble is Z(N) symmetric. In the high T phase, the Wilson line takes one of N distinct values proportional to the Nth roots of unity in Z(N). The Z(N) symmetry is broken. In a Hamiltonian approach, the line is a projection operator that forces the gauge field to be in a fundamental rather than a singlet state at the spatial position of the line. Thus, it is usually interpreted as the exponential of the excess free energy due to the source. But this is necessarily greater than or equal to 0. So, there is a need for a refined physical interpretation that is consistent with negative or complex values for the QWilson line in the nonconfining, high-temperature phase. My approach to the Z(N) symmetry and to the infinite-volume limit resolves the problem. In terms of the physical variables of quantum, finite-temperature, Hamiltonian gauge fields, there is no Z(N) symmetry to be broken. The physical system is the same in each of the N pure phases that can be chosen when the line is not zero. The confining and nonconfining phases are distinguished by the presence of percolating flux in the nonconfining phase. ``Fluctuating vs. Spreading Flux at the SU(2), Finite-Temperature, Deconfining Phase Transition''. At low temperature, quarks are confined, and at high temperature, they are not. The phase transition that separates these regions is second order for pure Yang-Mills, SU(2), gauge theory. In the usual picture of the confining phase, there is a tube of color-electric flux that connects sources and leads to a potential that is linear in the separation. As the temperature increases, the flux tube fluctuates and grows longer. At the critical temperature, these effects have become so large that the position of one end of the tube looses knowledge of the position of the other end. The potential becomes a constant at large separation. The flux tube stays intact while it fluctuates. This picture correctly accounts for some of the properties of the phase transition. The flux tube is thought to be created by coherent, supercurrents of color-magnetic monopoles that encircle the tube. This suggests another possibility for the phase transition: with increasing temperature, the coherent currents dissipate, and the color-electric flux spreads out becoming like a dipole, Coulomb field rather than a tube. Evidently, these two physical pictures are quite different. I will explore the similarities and differences between these two pictures. One possibility is a numerical attack that measures correlations that distinguish the two physical pictures. Analytical approximations and models may also shed some light on the problem. In particular, the Nielsen-Olesen magnetic vortices might be an interesting dual model to consider. Also, the behavior of the real physical system of vortices in a type II superconductor could be instructive. ``Quark Localization in QCD''. In the context of condensed matter physics, there is the phenomenon of localization: Electrons interacting with a sufficiently random potential have no extended states. Roughly speaking, there are bound states but no conduction bands. The response of the system to applied fields is drastically altered. In the context of quantum field theory, fermions, such as quarks, interact with other fluctuating fields, such as the color gauge field. Thus, I wonder if these fluctuations could lead to a version of localization for the fermion states. This would completely alter the nature of the excitations in the theory. In weakly coupled QED, there are ordinary electrons, so there does not seem to be localization in that case. On the other hand, QCD is particularly interesting because the size of the gauge field fluctuations grows with increasing length scale. Also, of course, there are no low-lying levels of individual quarks to immediately put the lie to the possibility of localization. For this project, it is also true that both analytic and numerical techniques can contribute. There are already well-developed numerical procedures for calculating the behavior of quarks in background gauge fields that could be applied here. It is also worth considering the effects of instantons, which are known to lead to bound states and other important changes in the quark spectrum.