## Theses and Dissertations

Dissertation

Fall 2017

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Applied Mathematical and Computational Sciences

Wayne N. Polyzou

Palle E. T. Jorgensen

Wayne N Polyzou

#### Second Committee Member

Palle E Jorgensen

Paul S Muhly

#### Fourth Committee Member

Vincent G Rodgers

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

viii, 115 pages

#### Bibliography

Includes bibliographical references (pages 112-115).