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The aim of this thesis is to present and discuss some mathematical and physical problems in the theory of weak, electromagnetic and strong interactions. Our main concern is a parallel development of mathematical and physical concepts and when it is possible, an attempt to bridge the abstract mathematical formulations with physical ideas. A central role in this thesis is played by a general construction scheme, which enables us to calculate explicitly all the mathematical quantities like matrix elements, Clebsch-Gordan series, Clebsch-Gordan coefficients which are necessary for a Grand Unification model construction. In this content, we have followed two basic principles: simplicity and applicability. To meet the first principle, all the construction methods developed are based on first principles and basic concepts of the Lie algebras and its representation theory, like roots and weights. Moreover, the requirement of applicability is met with the implementation of all the algorithms into computer programs. In the physical area, we have concentrated on the problem of mass. The lepton mass spectrum us studied in a theory of weak and electromagnetic interactions, while the mass problem of the SO(10) Grand Unified theory is analysed as a direct application of our Lie group construction scheme.

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In this project we introduce the general idea of geometric quantization and demonstratehow to apply the process on a few examples. We discuss how to construct a line bundleover the symplectic manifold with Dirac’s quantization conditions and how to determine if we are able to quantize a system with the help of Weil’s integrability condition. To reducethe prequantum line bundle we employ real polarization such that the system does notbreak Heisenberg’s uncertainty principle anymore. From the prequantum bundle and thepolarization we construct the sought after Hilbert space.

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Thesis (M.A.)--University of North Carolina at Wilmington, 2006. / Includes bibliographical references (leaf: [41])

Thesis (M.S.)--Brigham Young University. Dept. of Electrical and Computer Engineering, 2004. / Includes bibliographical references (p. 103-104).

The development of electronics has been growing at a fast rate in recent years. More and more ideas have been searched and are increasing at a faster rate. However, there is more detail work in the nanolevel or nanostructure yet to be understood. Thus, more and more semiconductor physicists have move to the new field of study in nanostructures. Nanostructures are the future of electronic devices. By understanding nanostructure electronic devices, electronics is the key for the progress of any modern equipment and advancement. This comes about when electronic transport of a nanostructure is thoroughly understood. Thus, future electronic devices can utilize the development of conductance through components having dimensions on the nanometer scale.The objective of the proposed research project is to study electronic transport in a ring with an infinite potential barrier at the center and a modulated external potential in one of the arms. The relative phase between the two paths in this structure can be controlled by applying electrostatic potential in one of the arms. One can compare these types of systems with optical interferometers, where the phase difference between the two arms is controlled by changing the refractive index of one arm through the electro-optic effect. By modulating the potential in one arm of the ring, we will study the interference effect on conductance. The method of finding the conductance of a nanostructure will be using the recursive Green's function method. This includes finding transverse eigenvalues, eigenfunctions, and hopping integrals to determine Green's propagators. A FORTRAN 77 computer program is used for numerical calculations.These remarkable ultra-small and ultra-clean quantum systems are currently achieved due to significant technological advancement in fabrication. For ultra-small quantum devices, the theoretical understanding of device performance must be based on quantum carrier transport of confined electrons and holes in the channel. This theoretical research will lead to the understanding of the effects of geometry and impurities on transport of the carriers in the nanochannels. / Department of Physics and Astronomy

I investigate some algebras and calculi naturally associated with the symplectic and metric Clifford algebras. In particular, I reformulate the well known Lepage decomposition for the symplectic exterior algebra in geometrical form and present some new results relating to the simple subspaces of the decomposition. I then present an analogous decomposition for the symmetric exterior algebra with a metric. Finally, I extend this symmetric exterior algebra into a new calculus for the symmetric differential forms on a pseudo-Riemannian manifold. The importance of this calculus lies in its potential for the description of bosonic systems in Quantum Theory.