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*}
| Identifer | oai:union.ndltd.org:uiowa.edu/oai:ir.uiowa.edu:etd-7383 |
| Date | 15 December 2017 |
| Creators | Aiello, Gordon J. |
| Contributors | Polyzou, Wayne Nicholas, 1952-, Jørgensen, Palle E. T., 1947- |
| Publisher | University of Iowa |
| Source Sets | University of Iowa |
| Language | English |
| Detected Language | English |
| Type | dissertation |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | Copyright © 2017 Gordon J. Aiello |
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