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In order to have a new perspective on the long-standing problem of the mass gap in Yang-Mills theory, we study the quantum Yang-Mills theory in the presence of topologically nontrivial backgrounds in this paper. The topologically stable gauge fields are constrained by the form invariance condition and the topological properties. Obeying these constraints, the known classical solutions to the Yang-Mills equation in the 3- and 4-dimensional Euclidean spaces are recovered, and the other allowed configurations form the nontrivial topological fluctuations at quantum level. Together, they constitute the background configurations, upon which the quantum Yang-Mills theory can be constructed. We demonstrate that the theory mimics the Higgs mechanism in a certain limit and develops a mass gap at semi-classical level on a flat space with finite size or on a sphere., Comment: 52 + 41 pages, 11 figures; v2: published version with minor changes

We solve the equations of motion of a complex $\phi^4$ theory coupled to some given gauge field background. The solutions are given in both cylindrical and spherical coordinates and have finite energy., Comment: 10 pages, 3 figure, Discussion section enlarged, two more graphs

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.\udKeywords: liquid flow; hydrodynamic Euler equations; diffeomorphism group; Lie-Poisson structure; isentropic hydrodynamic invariants; vortex invariants; charged liquid fluid dynamics; symmetry reduction

We consider a method for deriving relativistic two-body wave equations for fermions in the coordinate representation. The Lagrangian of the theory is reformulated by eliminating the mediating fields by means of covariant Green's functions. Then, the nonlocal interaction terms in the Lagrangian are reduced to local expressions which take into account retardation effects approximately. We construct the Hamiltonian and two-fermion states of the quantized theory, employing an unconventional “empty” vacuum state, and derive relativistic two-fermion wave equations. These equations are a generalization of the Breit equation for systems with scalar, pseudoscalar, vector, pseudovector and tensor coupling.

A string action which is essentially a Willmore functional is presented and studied. This action determines the physics of a surface in Euclidean three space which can be used to model classical string configurations. By varying this action an equation of motion for the mean curvature of the surface is obtained which is shown to govern certain classical string configurations. Several classes of classical solutions for this equation are discussed from the symmetry group point of view and an application is presented.

The topology of the universe is discussed in relation to the singularity problem. We explore the possibility that the initial state of the universe might have had a structure with 3-Klein bottle topology, which would lead to a model of a nonsingular oscillating (cyclic) universe with a well-defined boundary condition. The same topology is assumed to be intrinsic to the nature of the hypothetical primitive constituents of matter (usually called preons) giving rise to the observed variety of elementary particles. Some phenomenological implications of this approach are also discussed., Comment: 21 pages, 9 figures; v.4: final version

Die vorliegende Arbeit befasst sich mit der Behandlung der Spin-Bahn Kopplung im Rahmen der Periodische-Bahnen-Theorie. Letztere stellt eine Beziehung zwischen fluktuierendem Anteil des Quantenspektrums und Eigenschaften periodischer Bahnen des zugehörigen klassischen Systems dar. Sie ist gewissermaßen die Verallgemeinerung der Bohr-Sommerfeld-Quantisierung für klassisch nicht integrable Systeme. Der von Gutzwiller ursprüngliche gefundene Zusammenhang -die sogenannte Spurformel- beschränkt sich auf skalare Wellengleichungen und lässt sich nicht unmittelbar auf mehrkomponentige Systeme anwenden, wie sie etwa bei Vorhandensein einer Spin-Bahn Kopplung auftreten. Diese Arbeit stützt sich auf verallgemeinerte Theorien, bei denen die Kopplung auf zwei unterschiedliche Arten (schwache und starke Kopplung) integriert werden kann. Erstmals wird gezeigt, wie diese an konkreten Problemstellungen angewandt werden können. Deren Qualität kann durch Vergleich mit quantenmechanischen Resultaten beurteilt werden. Bei den untersuchten Systemen handelt es sich vorwiegend um 2 bzw. 3-dim. harmonische Oszillatoren, die unterschiedlichen Situationen wie Deformation oder Magnetfeld ausgesetzt wurden. Die verwendete Spin-Bahn Kopplung entspricht der Form wie sie typischerweise in der Kernphysik oder in 2-dim. Halbleitersystemen (Rashba-Term) auftritt. Die ausgewählten Modelle erlauben in vielen Fällen einen analytischen Zugang. Besondere Aufmerksamkeit wird dem Problem der Mode-Conversion gewidmet. Es handelt sich hier um ein bislang ungelöstes Problem, das in Verbindung mit der Methode der starken Kopplung auftreten kann. An speziell konstruierten Systemen kann dieses Phänomen beobachtet werden und mit dem Auftreten eines Spinflips im klassischen System in Verbindung gebracht werden., The dissertation is concerned with spin-orbit coupling in the context of periodic-orbit theory (POT). POT relates the oscillating part of the level density to properties of periodic orbit in the corresponding classical system. In a way this is a generalization of the Bohr-Sommerfeld quantization for systems that are classically non integrable. The relationship is called trace formula and was introduced by Gutzwiller. However in its original form it is restricted to scalar wave equations and cannot be applied in a direct way to a multicomponent system like systems with spin-orbit coupling. This work is based on generalized theories, where the coupling can be integrated in two limits (weak and strong coupling). For the first time these methods are used to calculate the full trace formula for a given problem. The validity of the approximation is verified by comparing to the exact quantum mechanical results. The systems that are investigated are mostly 2 resp. 3-dim harmonic oscillators which are in addition modified by deformation or magnetic field. The used spin-orbit coupling typically appears in nuclear physics or in 2-dim. semiconductor systems (Rashba term). In many cases an anlytical approach is possible. Special attention is attended to the problem of mode conversion. Up to now this is an unsolved problem and can occour within the method of strong coupling. Special prepared systems allow an observation of mode conversion and can be understood as a spin flip in the classical system.

We show that longwavelength excitations of the quark-gluon plasma are described by simple kinetic equations which represent the exact equations of motion at leading order in $g$. Properties of the so-called ``hard thermal loops'', i.e. the dominant contributions to amplitudes with soft external lines, find in this approach a natural explanation. In particular, their generating functional appears here as the effective action describing long wavelength excitations of the plasma., January 8, 1993; 8 pages; SPhT/93-1