Thesis (Ph. D.)--University of Oregon, 1999. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 238-241). Also available for download via the World Wide Web; free to University of Oregon users. Address:http://wwwlib.umi.com/cr/uoregon/fullcit?p9940438.

The contribution of this thesis is a novel framework for rewriting in higher categories. Its theoretical foundation is the theory of quasistrict higher categories and the practical realisation is a proof assistant Globular. The framework introduces the notions of diagrams and signatures as new mutually-recursive structures that give the algebraic basis for the approach. These structures are related the notion of an n-polygraph, but allow reasoning about quasistrict higher categorical structures in a way amenable to computer implementation. Building on this, we propose a new definition of a quasistrict 4-category, and prove a result that in a quasistrict 4-category, an adjunction of 1-morphisms gives rise to a coherent adjunction satisfying the butterfly equations.

It is not an exaggeration to say that the knowledge of the interatomic potential energy is the stepping stone to the physical properties of matter. In the case of gases the most important commutation on the path of development is between the interatomic potential energy on the one hand and the equation of state and the transport properties – viz. conductivity, viscosity and diffusion – on the other hand. However, for the solid state, the role of interatomic forces in crystal structures should not be overlooked. It is clear that the development is reversible in that the potential energy could be determined from one or more of the physical properties mentioned above and vice versa. A great many of the existing interatomic potential energies in the literature are of this phenomenological kind obtained by fitting the experimental results to an analytic function. Theoretically it should be possible to determine this energy from first principles and thereby arrive at values for the physical properties for comparison with the experimental results. The latter is the objective of this investigation.

Bessel beams are optical fields which falls into the category of non-diffracting beams. Vector Bessel beams are vector beams possessing cylindrical symmetry. Cylindrically symmetric beams tend to have a tight focal point during propagation. The tight focal beam nature of vector Bessel beams makes them a good potential in various facets of science such as biological optical trapping, wireless communications, remote sensing, microscopy etc. In this research work, vector Bessel beams were generated using the phase of an Axicon that was encoded into a spatial light modulator. Firstly, scalar Bessel beams which possess linear polarization were generated and converted to circularly polarized vector beams by the use of a q-plate. The orbital angular momentum (OAM) modes that are embedded in the vortex beams were detected using modal decomposition technique. This was implemented for both the scalar and vector case using a quarter wave plate. The measure of the degree of non-separability of the vector Bessel beams using tomographic quantum tools was also implemented where the density matrix was reconstructed. The concurrence and fidelity which explore the measure of vectorness of both scalar and vector Bessel beams were calculated from the density matrix. The obtained results show that the spatial modes and polarization are coupled in the vector case as expected.

In this work we have undertaken the study of some aspects of higher spin relativistic wave equations of the first order. These aspects are their definiteness of charge, their second quantization and their causal propagation. The work is divided into three parts. In part one we have studied the Bhabha equations based on the sixteen and twenty dimensional representations and showed that the charge associated with them is indefinite. In part two we have studied in detail the Gel'fand Yaglom equations and showed that certain equations can have definite charge. Finally in part three we have studied the propagation in the external electromagnetic field of certain higher spin relativistic wave equations.

We give a brief review of geometric quantisation up to and including the Blattner-Kostant-Sternberg kernal. In general this leads to symmetric operators that are not essentially self-adjoint so motivating a study of Hermitian operators as observables in a generalised quantum mechanics. We show that a generalised squaring axiom can reproduce the results of Blattner-Kostant-Sternberg quantisation. We also show that quantisation with respect to polarisations with compact leaves gives results that conflict with the nonlocal nature of quantum mechanics. We develop a front form quantum mechanics of a free scalar particle using geometric quantisation. The front and instant forms are related via unitary maps derived from the pairing which intertwines quantisations with respect to the forms. The front form position operator has a maximally symmetric component so we are compelled to work within the framework of a generalised quantum mechanics; the result in there being no Hegerfeldt type instantaneous spreading of initially localised wavefunctions in the front form. Finally we show that this model can be lifted to a many particle free field theory.

Clustered quantum materials provide a new platform for the experimental study of many- body entanglement. Here we address a simple model featuring N interacting spins in a transverse field. The field can induce an entanglement transition (ET). We calculate the magnetisation, low-energy gap and neutron-scattering cross-section and find that the ET has distinct signatures, detectable at temperatures as high as 10% of the interaction strength. Unlike a quantum critical point, the signatures of the ET are stronger for smaller clusters.

Please read the abstract in the section 00front of this document / Thesis (PhD (Physics))--University of Pretoria, 2007. / Physics / unrestricted

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