Document Type

Dissertation

Date of Degree

Fall 2017

Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

Wayne N. Polyzou

Second Advisor

Palle E. T. Jorgensen

First Committee Member

Wayne N Polyzou

Second Committee Member

Palle E Jorgensen

Third Committee Member

Paul S Muhly

Fourth Committee Member

Vincent G Rodgers

Fifth Committee Member

Richard L Baker

Abstract

One recipe for mathematically formulating a relativistic quantum mechanical scattering theory utilizes a two-Hilbert space approach, denoted by $\mathcal{H}$ and $\mathcal{H}_{0}$, upon each of which a unitary representation of the Poincaré Lie group is given. Physically speaking, $\mathcal{H}$ models a complicated interacting system of particles one wishes to understand, and $\mathcal{H}_{0}$ an associated simpler (i.e., free/noninteracting) structure one uses to construct 'asymptotic boundary conditions" on so-called scattering states in $\mathcal{H}$. Simply put, $\mathcal{H}_{0}$ is an attempted idealization of $\mathcal{H}$ one hopes to realize in the large time limits $t\rightarrow\pm\infty$.

The above considerations lead to the study of the existence of strong limits of operators of the form $e^{iHt}Je^{-iH_{0}t}$, where $H$ and $H_{0}$ are self-adjoint generators of the time translation subgroup of the unitary representations of the Poincaré group on $\mathcal{H}$ and $\mathcal{H}_{0}$, and $J$ is a contrived mapping from $\mathcal{H}_{0}$ into $\mathcal{H}$ that provides the internal structure of the scattering asymptotes.

The existence of said limits in the context of Euclidean quantum theories (satisfying precepts known as the Osterwalder-Schrader axioms) depends on the choice of $J$ and leads to a marvelous connection between this formalism and a beautiful area of classical mathematical analysis known as the Stieltjes moment problem, which concerns the relationship between numerical sequences $\{\mu_{n}\}_{n=0}^{\infty}$ and the existence/uniqueness of measures $\alpha(x)$ on the half-line satisfying

\begin{equation*}

\mu_{n}=\int_{0}^{\infty}x^{n}d\alpha(x).

\end{equation*}

Keywords

Euclidean Quantum Scattering, Mathematical Physics, Moment Problem, Quantum Theory, Scattering Theory, Theoretical Physics

Pages

viii, 115 pages

Bibliography

Includes bibliographical references (pages 112-115).

Copyright

Copyright © 2017 Gordon J. Aiello

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