Partial Ordering of Gauge Orbit Types for SU(n)-Gauge Theories

The natural partial ordering of the orbit types of the action of the group of local gauge transformations on the space of connections in space-time dimension d<=4 is investigated. For that purpose, a description of orbit types in terms of cohomology elements of space-time, derived earlier, is used. It is shown that, on the level of these cohomology elements, the partial ordering relation is characterized by a system of algebraic equations. Moreover, operations to generate direct successors and direct predecessors are formulated. The latter allow to successively reconstruct the set of orbit types, starting from the principal type.Comment: 35 pages, LaTe

Semiclassical Time Evolution and Trace Formula for Relativistic Spin-1/2 Particles

We investigate the Dirac equation in the semiclassical limit \hbar --> 0. A semiclassical propagator and a trace formula are derived and are shown to be determined by the classical orbits of a relativistic point particle. In addition, two phase factors enter, one of which can be calculated from the Thomas precession of a classical spin transported along the particle orbits. For the second factor we provide an interpretation in terms of dynamical and geometric phases.Comment: 8 pages, no figure

High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.Comment: 26 pages, latex2

Nonlinear stochastic evolution equations of second order with damping

Convergence of a full discretization of a second order stochastic evolution equation with nonlinear damping is shown and thus existence of a solution is established. The discretization scheme combines an implicit time stepping scheme with an internal approximation. Uniqueness is proved as well.Comment: This is the version of the article accepted for publication. The final publication is available at http://link.springer.co

Semiclassical approximations for Hamiltonians with operator-valued symbols

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter $\varepsilon\ll 1$ controls the separation of time scales and the limit $\varepsilon\to 0$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time $\varepsilon\to 0$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the $\varepsilon$-dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn, coming from an \epsi-dependent classical Hamilton function and an $\varepsilon$-dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order $\varepsilon^2$. In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics. Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation.Comment: Final version to appear in Commun. Math. Phys. Results have been strengthened with only minor changes to the proofs. A section on the Hofstadter model as an application of the general theory was added and the previous section on other applications was remove

Gauge Orbit Types for Theories with Classical Compact Gauge Group

We determine the orbit types of the action of the group of local gauge transformations on the space of connections in a principal bundle with structure group O(n), SO(n) or $Sp(n)$ over a closed, simply connected manifold of dimension 4. Complemented with earlier results on U(n) and SU(n) this completes the classification of the orbit types for all classical compact gauge groups over such space-time manifolds. On the way we derive the classification of principal bundles with structure group SO(n) over these manifolds and the Howe subgroups of SO(n).Comment: 57 page

Moyal star product approach to the Bohr-Sommerfeld approximation

The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order $\hbar^2$ (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in $\hbar$, starting with $\hbar^0$, with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.Comment: 27 pages, no figure

Stratification of the orbit space in gauge theories. The role of nongeneric strata

Gauge theory is a theory with constraints and, for that reason, the space of physical states is not a manifold but a stratified space (orbifold) with singularities. The classification of strata for smooth (and generalized) connections is reviewed as well as the formulation of the physical space as the zero set of a momentum map. Several important features of nongeneric strata are discussed and new results are presented suggesting an important role for these strata as concentrators of the measure in ground state functionals and as a source of multiple structures in low-lying excitations.Comment: 22 pages Latex, 1 figur

Zitterbewegung and semiclassical observables for the Dirac equation

In a semiclassical context we investigate the Zitterbewegung of relativistic particles with spin 1/2 moving in external fields. It is shown that the analogue of Zitterbewegung for general observables can be removed to arbitrary order in \hbar by projecting to dynamically almost invariant subspaces of the quantum mechanical Hilbert space which are associated with particles and anti-particles. This not only allows to identify observables with a semiclassical meaning, but also to recover combined classical dynamics for the translational and spin degrees of freedom. Finally, we discuss properties of eigenspinors of a Dirac-Hamiltonian when these are projected to the almost invariant subspaces, including the phenomenon of quantum ergodicity

An acid-gas removal system for upgrading subquality natural gas

The objective of this project is to develop systems to reduce the cost of treating subquality natural gas. Based on over 1,000 laboratory experiments on vapor-liquid equilibria and mass transfer and simulation studies, the use of N-Formyl Morpholine as a solvent together with structured packings has the following advantages: high capacity for H{sub 2}S and CO{sub 2} removal; little or no refrigeration required; less loss of hydrocarbons (CH{sub 4}, C{sub 2}-C{sub 6}); and dehydration potential. To verify these findings and to obtain additional data base for scale-up, a field test unit capable of processing 1MMSCF/d of natural gas has been installed at the Shell Western E and P Inc. (SWEPI) Fandango processing plant site. The results of the testing at the Fandango site will be presented when available