Date of Award

Fall 10-1-2021

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics

First Advisor

Fleming , George

Abstract

We have extended two recently developed theoretical methods, the Quantum Finite Elements (QFE) and the Euclidean-signature semi-classical method (ESSCM). The QFE is a technique for constructing lattice field theories (LFTs) on curved Riemannian manifolds. We extended the applicability of the QFE to formulating LFTs on certain three and four dimensional Riemannian manifolds such as $\mathbb{S}^{3}$ and $\mathbb{R} \cross \mathbb{S}^{3}$. This was done by first constructing a novel simplicial approximation to $\mathbb{S}^{3}$. Then, after correctly computing the weights of the links and vertices that make up this simplicial approximation, we defined a Laplacian on it, whose low lying spectrum was observed to approach the known continuum limit as we further refined our simplicial complex. To facilitate a comparison between the QFE and the bootstrap, we calculated an estimate of the fourth-order Binder cumulant using CFT data extracted from the conformal bootstrap. The ESSCM is a methodology for facilitating the use of already known mathematical theorems/results to approach Lorentzian signature problems in bosonic field theory and quantum gravity in terms of their Euclidean-signature analogs. We further developed this method by applying it in a novel fashion to quantum cosmological models with matter sources. In particular, for the Taub models, we proved for the first time the existence of a countably infinite number of well behaved ‘excited’ state solutions when $\Lambda$ is present. Both methods are promising and have applications for field theory, beyond standard model physics, and quantum gravity.

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