An application of the theory of moments to Euclidean relativistic quantum mechanical scattering
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, $\mathcal{H}$ mode...
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ndltd-uiowa.edu-oai-ir.uiowa.edu-etd-73832019-10-13T04:53:54Z An application of the theory of moments to Euclidean relativistic quantum mechanical scattering Aiello, Gordon J. 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*} 2017-12-15T08:00:00Z dissertation application/pdf https://ir.uiowa.edu/etd/5902 https://ir.uiowa.edu/cgi/viewcontent.cgi?article=7383&context=etd Copyright © 2017 Gordon J. Aiello Theses and Dissertations eng University of IowaPolyzou, Wayne Nicholas, 1952- Jørgensen, Palle E. T., 1947- Euclidean Quantum Scattering Mathematical Physics Moment Problem Quantum Theory Scattering Theory Theoretical Physics Applied Mathematics |
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English |
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Others
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Euclidean Quantum Scattering Mathematical Physics Moment Problem Quantum Theory Scattering Theory Theoretical Physics Applied Mathematics |
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Euclidean Quantum Scattering Mathematical Physics Moment Problem Quantum Theory Scattering Theory Theoretical Physics Applied Mathematics Aiello, Gordon J. An application of the theory of moments to Euclidean relativistic quantum mechanical scattering |
description |
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*} |
author2 |
Polyzou, Wayne Nicholas, 1952- |
author_facet |
Polyzou, Wayne Nicholas, 1952- Aiello, Gordon J. |
author |
Aiello, Gordon J. |
author_sort |
Aiello, Gordon J. |
title |
An application of the theory of moments to Euclidean relativistic quantum mechanical scattering |
title_short |
An application of the theory of moments to Euclidean relativistic quantum mechanical scattering |
title_full |
An application of the theory of moments to Euclidean relativistic quantum mechanical scattering |
title_fullStr |
An application of the theory of moments to Euclidean relativistic quantum mechanical scattering |
title_full_unstemmed |
An application of the theory of moments to Euclidean relativistic quantum mechanical scattering |
title_sort |
application of the theory of moments to euclidean relativistic quantum mechanical scattering |
publisher |
University of Iowa |
publishDate |
2017 |
url |
https://ir.uiowa.edu/etd/5902 https://ir.uiowa.edu/cgi/viewcontent.cgi?article=7383&context=etd |
work_keys_str_mv |
AT aiellogordonj anapplicationofthetheoryofmomentstoeuclideanrelativisticquantummechanicalscattering AT aiellogordonj applicationofthetheoryofmomentstoeuclideanrelativisticquantummechanicalscattering |
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1719265354638688256 |