Thesis--Universität Zürich. / Curriculum vitae.

Thesis (M.S.)--Kutztown University of Pennsylvania, 1996. / Source: Masters Abstracts International, Volume: 45-06, page: 3174. Typescript. Includes bibliographical references (leaves 84-85).

Proefschrift--Vrije Universiteit, Amsterdam. / "Stellingen": [4] p. inserted.

Thesis--Friedrich-Alexanders-Universität, 1918. / Lebenslauf.

Thesis (Ph.D.)--University of Wisconsin--Madison, 1993. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 199-207).

Following an examination of the properties of the conformal group in 4-space, a review is made of the procedure by which conformally covariant massless field equations are written in manifestly covariant form. By writing the Minkowski coordinates in terms of coordinates on the null hyperquadric of a 6-dimensional flat space with two timelike directions, the action of the group is linearized and field equations are written in rotationally covariant form in 6-dimensional space. It is then shown that extending the 6-coordinates off the null surface generalizes Minkowski space to a 5-dimensional space. Such a generalization necessitates employing a method of descent to 4-dimensional space from six dimensions which differs from the usual procedure, and allows one to encompass massive field theories in the manifest formalism. It is demonstrated that these massive fields can be understood as manifestations in Minkowski space of massless fields in 5-dimsnsional space. For the case of spinors, the field equation can accomodate precisely two species of particle having two different masses. An action principle is developed in the 6-space, and a method of field quantization is devised. As examples of the method, the special cases of spin-0, spin-1/2, and spin-1 fields are examined in detail, and minimal coupling of the spinor field equation is carried out. The formalism presented in this investigation provides a means by which one can apprehend a massive compensating field within the confines of a gauge invariant theory. The interactions which are obtained in Minkowski space include not only the usual couplings with massive vector or pseudovector fields, but as well the pseudoscalar coupling occurs automatically within this gauge invariant formulation. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Ph.D. (Mathematics) / Please refer to full text to view abstract

Cartan geometry provides a rich formalism from which to look at various geometrically motivated extensions to general relativity. In this manuscript, we start by motivating reasons to extend the theory of general relativity. We then introduce the reader to our technique, called the quotient manifold method, for extending the geometry of spacetime. We will specifically look at the class of theories formed from the various quotients of the conformal group. Starting with the conformal symmetries of Euclidean space, we construct a manifold where time manifests as a part of the geometry. Though there is no matter present in the geometry studied here, geometric terms analogous to dark energy and dark matter appear when we write down the Einstein tensor. Specifically, the quotient of the conformal group of Euclidean four-space by its Weyl subgroup results in a geometry possessing many of the properties of relativistic phase space, including both a natural symplectic form and nondegenerate Killing metric. We show the general solution possesses orthogonal Lagrangian submanifolds, with the induced metric and the spin connection on the submanifolds necessarily Lorentzian, despite the Euclidean starting point. By examining the structure equations of the biconformal space in an orthonormal frame adapted to its phase space properties, we also nd two new tensor fields exist in this geometry, not present in Riemannian geometry. The rst is a combination of the Weyl vector with the scale factor on the metric, and determines the time-like directions on the submanifolds. The second comes from the components of the spin connection, symmetric with respect to the new metric. Though this eld comes from the spin connection, it transforms homogeneously. Finally, we show in the absence of Cartan curvature or sources, the conguration space has geometric terms equivalent to a perfect fluid and a cosmological constant. We complete the analysis of this homogeneous space by transforming the known, general solution of the Maurer-Cartan equations into the orthogonal, Lagrangian basis. This results in a signature-changing metric, just as in the work of Spencer and Wheeler, however without any conditions on the curvature of the momentum sector. The Riemannian curvatures of the two submanifolds are directly related. We investigate the case where the curvature on the momentum submanifold vanishes, while the curvature of the configuration submanifold gives an effective energy-momentum tensor corresponding to a perfect fluid. In the second part of this manuscript, we look at the most general curved biconformal geometry dictated by the Wehner-Wheeler action. We use the assemblage of structure equations, Bianchi identities, and eld equations to show how the geometry of the manifolds self-organizes into trivial Weyl geometries, which can then be gauged to Riemannian geometries. The Bianchi identities reveal the strong relationships between the various curvatures, torsions, and cotorsions. The discussion of the curved case culminates in a number of simplifying restrictions that show general relativity as the base of the more general theory.

This thesis is a study of some properties of arcs which remain invariant under certain types of conformal representations. The study is carried on first in the conformal plane, then in conformal 3-space, and finally in conformal n-space. It is comprised of most of the research on this subject which has been carried on to date by Professors N.D. Lane and Peter Scherk. I have assisted Dr. Lane in the creation of some of the material which this thesis contains. / Thesis / Master of Arts (MA)

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