Date of Award

Spring 5-2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Computational and Data Sciences

First Advisor

Jeff Tollaksen

Second Advisor

Yakir Aharonov

Third Advisor

Daniel Alpay

Abstract

This is a dissertation in two parts. In the first one, the Aharonov-Bohm effect is investigated. It is shown that solenoids (or flux lines) can be seen as barriers for quantum charges. In particular, a charge can be trapped in a sector of a long cavity by two flux lines. Also, grids of flux lines can approximate the force associated with continuous two-dimensional distributions of magnetic fields. More, if it is assumed that the lines can be as close to each other as desirable, it is explained how the classical magnetic force can emerge from the Aharonov-Bohm effect. Continuing, the quantization of the source of the magnetic field, and not just of the degrees of freedom of the particle interacting with it, is considered. Special attention is given to the cases where the source has a relatively small spreading and is post-selected. As it will be discussed, in those cases, the weak value plays a role in the determination of the effective vector potential "experienced" by the particle. In the second part of this work, notions from functional analysis are extended to Banach algebras and completions of Grassmann algebras. A notion of analyticity is given to the functions of a single Banach algebra variable. This notion allows the introduction of holomorphic polynomials, power series, and rational functions. With that, the analogous of Hilbert spaces of power series are also considered. Finally, closures of Grassmann algebras with respect to the 1 and the 2-norms are explored. The analogous of the complex analysis in the open disk or a half-plane (usually referred to as Schur analysis) is presented in the 1-norm closure. Also, a Wiener-like algebra, interpolation problems, and a process known as the Schur algorithm are studied in this setting. Now, an inner product between two elements can be introduced in the 2-norm closure, revealing similarities between this space and the non-commutative Fock-Bargamann-Segal space. It is, then, defined a class of stochastic processes. To conclude, the derivatives of these processes are analyzed in an analogous of the space of stochastic distributions.

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