Geometric Phase in Eigenspace Evolution of Invariant and Adiabatic Action Operators

The theory of geometric phase is generalized to a cyclic evolution of the eigenspace of an invariant operator with $N$-fold degeneracy. The corresponding geometric phase is interpreted as a holonomy inherited from the universal connection of a Stiefel U(N)-bundle over a Grassmann manifold. Most significantly, for an arbitrary initial state, this geometric phase captures the inherent geometric feature of the state evolution. Moreover, the geometric phase in the evolution of the eigenspace of an adiabatic action operator is also addressed, which is elaborated by a pullback U(N)-bundle. Several intriguing physical examples are illustrated.Comment: Added Refs. and corrected typos; 4 page

Variational principles for involutive systems of vector fields

In many relevant cases -- e.g., in hamiltonian dynamics -- a given vector field can be characterized by means of a variational principle based on a one-form. We discuss how a vector field on a manifold can also be characterized in a similar way by means of an higher order variational principle, and how this extends to involutive systems of vector fields.Comment: 31 pages. To appear in International Journal of Geometric Methods in Modern Physics (IJGMMP

Municipal Water Resources Analysis for Area Potentially Impacted by MX Missile Complex in Utah

Scope of Report: This report analyzes the impact of the proposed MX Missile complex upon existing municipal water supply and waste treatment systems serving selected communitites either near the perimeter or within the Utah portion of the proposed MX complex boundary. As can be seen from the location map in Figure 1, possible sites for elements within the total MX missile complex have been identified in 14 Utah desert valleys in the five counties, from north to south, of Tooele, Juab, Millard, Beaver, and Iron. The 60,000 people, who live in these counties according to the 1975 census, are largely located in their eastern ends of the base of a series of mountain ranges with numerous peaks over 10,000 feet. Sites closer to these mountains have a more dependable and higher quality water supply from the snowpack runoff. Surface runoff evaporates or infiltrates underground and waters generally become more saline as one moves further west into the desert. The desert ranges, separting the 14 valleys, are lowever, generate much less runoff, and streams flow only for short periods, during spring snowmelt or summer thunderstorm, to recharge aquifers along the basin margins. Interstate 15, the main highway from Salt Lake to Las Vegas, passes through the towns of Nephi, Fillmore, Beaver, Parowan, and Cedar City and the best farming country in the region along the base of the mountain ranges at the eastern edge of these counties. About 20 miles further west, the Union Pacific Railroad corridor passes through the towns of Delta and Milford and several small villages of population less than 50 as it roughly demarcates the farming country to the east from the desert valleys being considered as MX missile sites further west. The 100-mile wide strip between the Union Pacific Corridor and the Nevada border is extremely sparsely inhabited with the largest single community begin the 60 people who live at Garrison. Generally, nature provides more water on the basin margins along the eastern sides of these five counties. However, because the water is more readily available and easier to develop there, almost all available supplies are fully appropriated and new users can only obtain water by purchasing prior rights. Further west, surface water (and therefore early development) has been very limited, and significant amounts of groundwater remain upappropriated. Much would have to be pumped from deeper aquifers. The specific communities assigned for analysis of their water supply and wastewater treatment systems in this study are Delta, Milford and Cedar City plus an overview of impact upon the water supply situation in the smaller communities of Hinckley, Deseret, Oasis (all a few miles southwest of Delta) and Garrison, near the Utah-Nevada border. The locations of these cities and villages in relation to the potential MX storage sites are shown in Figure 1. The report begins by presenting the pertinent hydrologic information, particularly groundwater hydrology, for areas immediately adjacent to the communities of interest. The hydrology of the other valleys where the MX sites are contemplated is not within the scope of this report. The second major section of the report is a description of the existing municipal water systems for these seven communities, their current water requirements, their capacity without any expansion, and, finally, and assessment of the expansion in water rights and various components of each system which would be required to serve an assumed MX related growth scenario in each region. The final section is a similar analysis of existing wasteawter collection and treatment facilities and of how they would be affected by the growth scenarios. In addition to possible MX related growth, the Delta area is also facing probable construction of a very large coal-fired power generating complex known as the Intermountain Power Project (IPP). The water and wastewater demand projections are based upon the assumed normal growth without MX (including the proposed Intermountain Power Project (IPP) impact in the Delta area) plus MX related growth. The MX-related population gorwth projected for Utah amounts to a population increase of 30,000 (employees, dependents and indirect) by 1987 at the peak of MX consturction. The population increase would assume to be distributed by community as follows

Chern-Simons Reduction and non-Abelian Fluid Mechanics

We propose a non-Abelian generalization of the Clebsch parameterization for a vector in three dimensions. The construction is based on a group-theoretical reduction of the Chern-Simons form on a symmetric space. The formalism is then used to give a canonical (symplectic) discussion of non-Abelian fluid mechanics, analogous to the way the Abelian Clebsch parameterization allows a canonical description of conventional fluid mechanics.Comment: 12 pages, REVTeX; revised for publication in Phys Rev D; email to [email protected]

Phases in Strongly Coupled Electronic Bilayer Liquids

The strongly correlated liquid state of a bilayer of charged particles has been studied via the HNC calculation of the two-body functions. We report the first time emergence of a series of structural phases, identified through the behavior of the two-body functions.Comment: 5 pages, RevTEX 3.0, 4 ps figures; Submitted to Phys. Rev. Let

Self-organization of (001) cubic crystal surfaces

Self-organization on crystal surface is studied as a two dimensional spinodal decomposition in presence of a surface stress. The elastic Green function is calculated for a $(001)$ cubic crystal surface taking into account the crystal anisotropy. Numerical calculations show that the phase separation is driven by the interplay between domain boundary energy and long range elastic interactions. At late stage of the phase separation process, a steady state appears with different nanometric patterns according to the surface coverage and the crystal elastic constants

Chern-Simons States at Genus One

We present a rigorous analysis of the Schr\"{o}dinger picture quantization for the $SU(2)$ Chern-Simons theory on 3-manifold torus$\times$line, with insertions of Wilson lines. The quantum states, defined as gauge covariant holomorphic functionals of smooth $su(2)$-connections on the torus, are expressed by degree $2k$ theta-functions satisfying additional conditions. The conditions are obtained by splitting the space of semistable $su(2)$-connections into nine submanifolds and by analyzing the behavior of states at four codimension $1$ strata. We construct the Knizhnik-Zamolodchikov-Bernard connection allowing to compare the states for different complex structures of the torus and different positions of the Wilson lines. By letting two Wilson lines come together, we prove a recursion relation for the dimensions of the spaces of states which, together with the (unproven) absence of states for spins\s>{_1\over^2}level implies the Verlinde dimension formula.Comment: 33 pages, IHES/P

Hamiltonian reduction of SU(2) Dirac-Yang-Mills mechanics

The SU(2) gauge invariant Dirac-Yang-Mills mechanics of spatially homogeneous isospinor and gauge fields is considered in the framework of the generalized Hamiltonian approach. The unconstrained Hamiltonian system equivalent to the model is obtained using the gaugeless method of Hamiltonian reduction. The latter includes the Abelianization of the first class constraints, putting the second class constraints into the canonical form and performing a canonical transformation to a set of adapted coordinates such that a subset of the new canonical pairs coincides with the second class constraints and part of the new momenta is equal to the Abelian constraints. In the adapted basis the pure gauge degrees of freedom automatically drop out from the consideration after projection of the model onto the constraint shell. Apart from the elimination of these ignorable degrees of freedom a further Hamiltonian reduction is achieved due to the three dimensional group of rigid symmetry possessed by the system.Comment: 25 pages Revtex, no figure

Cohomology of bundles on homological Hopf manifold

We discuss the properties of complex manifolds having rational homology of $S^1 \times S^{2n-1}$ including those constructed by Hopf, Kodaira and Brieskorn-van de Ven. We extend certain previously known vanishing properties of cohomology of bundles on such manifolds.As an application we consider degeneration of Hodge-deRham spectral sequence in this non Kahler setting.Comment: To appear in Proceedings of 2007 conference on Several complex variables and Complex Geometry. Xiamen. Chin

K\"{a}hler-Einstein metrics on strictly pseudoconvex domains

The metrics of S. Y. Cheng and S.-T. Yau are considered on a strictly pseudoconvex domains in a complex manifold. Such a manifold carries a complete K\"{a}hler-Einstein metric if and only if its canonical bundle is positive. We consider the restricted case in which the CR structure on $\partial M$ is normal. In this case M must be a domain in a resolution of the Sasaki cone over $\partial M$. We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a K\"{a}hler-Einstein manifold. We are able to mostly determine those normal CR 3-manifolds which can be CR infinities. Many examples are given of K\"{a}hler-Einstein strictly pseudoconvex manifolds on bundles and resolutions.Comment: 30 pages, 1 figure, couple corrections, improved a couple example