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Existence of point processes through families of commuting Hermitian operators. / Lin Mei

Swansea University Author: Lin Mei

Abstract

This dissertation is devoted to problems of existence and physical interpretation of some point processes. In the first part of the dissertation, we introduce the notion of the correlation measure of a family of commuting Hermitian operators. Let X be a locally compact, second countable Hausdorff to...

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Published: 2007
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
URI: https://cronfa.swan.ac.uk/Record/cronfa42505
first_indexed 2018-08-02T18:54:52Z
last_indexed 2018-08-03T10:10:20Z
id cronfa42505
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spelling 2018-08-02T16:24:29.4937951 v2 42505 2018-08-02 Existence of point processes through families of commuting Hermitian operators. 3757cdf19dadebce8cef7787887bffb5 NULL Lin Mei Lin Mei true true 2018-08-02 This dissertation is devoted to problems of existence and physical interpretation of some point processes. In the first part of the dissertation, we introduce the notion of the correlation measure of a family of commuting Hermitian operators. Let X be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators a(Delta) indexed by all measurable, relatively compact sets Delta m X. For such a family, we introduce the notion of a correlation measure and prove that, if this correlation measure exists and satisfies some condition of growth, then there exists a point process over X having the same correlation measure (in the sense of the classical theory of point processes). Furthermore, the operators a(Delta) can be realised as multiplication operators in the L2-space with respect to this point process. In particular, our result extends the criterion of existence of a point process from [6, 15], to the case of the topological space X, which is a standard underlying space in the theory of point processes. In the second part of the dissertation, we consider some important applications of our general results. We discuss particle densities of the quasi-free representation of the CAR and CCR, which lead to fermion (determinantal), and boson (permanental) point processes. We also discuss convolutions of these particle densities, which lead to point processes whose correlation functions are given through the Vere-Jones a-determinants. E-Thesis Mathematics. 31 12 2007 2007-12-31 COLLEGE NANME Mathematics COLLEGE CODE Swansea University Doctoral Ph.D 2018-08-02T16:24:29.4937951 2018-08-02T16:24:29.4937951 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Lin Mei NULL 1 0042505-02082018162459.pdf 10801735.pdf 2018-08-02T16:24:59.7900000 Output 2540087 application/pdf E-Thesis true 2018-08-02T16:24:59.7900000 false
title Existence of point processes through families of commuting Hermitian operators.
spellingShingle Existence of point processes through families of commuting Hermitian operators.
Lin Mei
title_short Existence of point processes through families of commuting Hermitian operators.
title_full Existence of point processes through families of commuting Hermitian operators.
title_fullStr Existence of point processes through families of commuting Hermitian operators.
title_full_unstemmed Existence of point processes through families of commuting Hermitian operators.
title_sort Existence of point processes through families of commuting Hermitian operators.
author_id_str_mv 3757cdf19dadebce8cef7787887bffb5
author_id_fullname_str_mv 3757cdf19dadebce8cef7787887bffb5_***_Lin Mei
author Lin Mei
author2 Lin Mei
format E-Thesis
publishDate 2007
institution Swansea University
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description This dissertation is devoted to problems of existence and physical interpretation of some point processes. In the first part of the dissertation, we introduce the notion of the correlation measure of a family of commuting Hermitian operators. Let X be a locally compact, second countable Hausdorff topological space. We consider a family of commuting Hermitian operators a(Delta) indexed by all measurable, relatively compact sets Delta m X. For such a family, we introduce the notion of a correlation measure and prove that, if this correlation measure exists and satisfies some condition of growth, then there exists a point process over X having the same correlation measure (in the sense of the classical theory of point processes). Furthermore, the operators a(Delta) can be realised as multiplication operators in the L2-space with respect to this point process. In particular, our result extends the criterion of existence of a point process from [6, 15], to the case of the topological space X, which is a standard underlying space in the theory of point processes. In the second part of the dissertation, we consider some important applications of our general results. We discuss particle densities of the quasi-free representation of the CAR and CCR, which lead to fermion (determinantal), and boson (permanental) point processes. We also discuss convolutions of these particle densities, which lead to point processes whose correlation functions are given through the Vere-Jones a-determinants.
published_date 2007-12-31T19:40:26Z
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score 11.048302

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