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Quasi-free states on algebras of multicomponent commutation relations / Nedal Othman

Swansea University Author: Nedal Othman

DOI (Published version): 10.23889/SUthesis.62993

Abstract

Let X = R² and let V be a finite-dimensional complex inner product space. Let C : X² → L (V ⊗²) be a continuous function such that, for each (x, y) ∈ X², C(x, y) is a unitary operator in V ⊗², C∗(x, y) = C(y, x), and the functional Yang-Baxter equa-tion is satisfied. The dissertation deals with the...

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Published: Swansea University 2023
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Lytvynov, Eugene, Finkelshtein, Dmitri.
URI: https://cronfa.swan.ac.uk/Record/cronfa62993
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fullrecord <?xml version="1.0"?><rfc1807><datestamp>2023-03-20T15:22:25.5868464</datestamp><bib-version>v2</bib-version><id>62993</id><entry>2023-03-20</entry><title>Quasi-free states on algebras of multicomponent commutation relations</title><swanseaauthors><author><sid>a5da4c291c531185ea524821b1afd370</sid><firstname>Nedal</firstname><surname>Othman</surname><name>Nedal Othman</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2023-03-20</date><deptcode>SMA</deptcode><abstract>Let X = R&#xB2; and let V be a finite-dimensional complex inner product space. Let C : X&#xB2; &#x2192; L (V &#x2297;&#xB2;) be a continuous function such that, for each (x, y) &#x2208; X&#xB2;, C(x, y) is a unitary operator in V &#x2297;&#xB2;, C&#x2217;(x, y) = C(y, x), and the functional Yang-Baxter equa-tion is satisfied. The dissertation deals with the multicomponent commutation relations governed by the function C, see [A. Liguori, M. Mintchev, Comm. Math. Phys. 169 (1995) 635&#x2013;652]. We introduce the &#x2217;-algebra of the C-multicomponent commutation re-lations (C-MCR algebra). We propose definitions of a gauge-invariant quasi-free state andof a strongly quasi-free state on the C-MCR algebra, A. Under restrictive assumptions on the function C, we construct a class of gauge-invariant quasi-free states on A, which, for some functions C, are also strongly quasi-free. We show that, when dim V = 1 (i.e., when we deal with the anyon commutation relations), among all gauge-invariant quasi- free states on A, only the Fock state is strongly quasi-free. In the case dim V = 2 (i.e., when we deal with two-component systems), we present a non-trivial class of examples of function C to which our theory is applicable, and hence, we can construct gauge-invariant quasi-free states, or even strongly quasi-free states on A.</abstract><type>E-Thesis</type><journal/><volume/><journalNumber/><paginationStart/><paginationEnd/><publisher/><placeOfPublication>Swansea University</placeOfPublication><isbnPrint/><isbnElectronic/><issnPrint/><issnElectronic/><keywords>Fock space, deformed commutation relations, multicomponent quantum system, anyon, plekton</keywords><publishedDay>20</publishedDay><publishedMonth>3</publishedMonth><publishedYear>2023</publishedYear><publishedDate>2023-03-20</publishedDate><doi>10.23889/SUthesis.62993</doi><url/><notes>Copyright: The author, Nedal Othman, 2023</notes><college>COLLEGE NANME</college><department>Mathematics</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>SMA</DepartmentCode><institution>Swansea University</institution><supervisor>Lytvynov, Eugene, Finkelshtein, Dmitri.</supervisor><degreelevel>Doctoral</degreelevel><degreename>Ph.D</degreename><apcterm>Not Required</apcterm><funders/><projectreference/><lastEdited>2023-03-20T15:22:25.5868464</lastEdited><Created>2023-03-20T13:04:57.5576848</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>Nedal</firstname><surname>Othman</surname><order>1</order></author></authors><documents><document><filename>62993__26903__41512bc0edc64e0c8ab31a2531f89a4c.pdf</filename><originalFilename>Final version.Nedal Othman.PhD.2023.pdf</originalFilename><uploaded>2023-03-20T13:15:29.3971675</uploaded><type>Output</type><contentLength>890376</contentLength><contentType>application/pdf</contentType><version>E-Thesis &#x2013; open access</version><cronfaStatus>true</cronfaStatus><documentNotes>Copyright: The author, Nedal Othman, 2023.</documentNotes><copyrightCorrect>true</copyrightCorrect><language>eng</language></document></documents><OutputDurs/></rfc1807>
spelling 2023-03-20T15:22:25.5868464 v2 62993 2023-03-20 Quasi-free states on algebras of multicomponent commutation relations a5da4c291c531185ea524821b1afd370 Nedal Othman Nedal Othman true false 2023-03-20 SMA Let X = R² and let V be a finite-dimensional complex inner product space. Let C : X² → L (V ⊗²) be a continuous function such that, for each (x, y) ∈ X², C(x, y) is a unitary operator in V ⊗², C∗(x, y) = C(y, x), and the functional Yang-Baxter equa-tion is satisfied. The dissertation deals with the multicomponent commutation relations governed by the function C, see [A. Liguori, M. Mintchev, Comm. Math. Phys. 169 (1995) 635–652]. We introduce the ∗-algebra of the C-multicomponent commutation re-lations (C-MCR algebra). We propose definitions of a gauge-invariant quasi-free state andof a strongly quasi-free state on the C-MCR algebra, A. Under restrictive assumptions on the function C, we construct a class of gauge-invariant quasi-free states on A, which, for some functions C, are also strongly quasi-free. We show that, when dim V = 1 (i.e., when we deal with the anyon commutation relations), among all gauge-invariant quasi- free states on A, only the Fock state is strongly quasi-free. In the case dim V = 2 (i.e., when we deal with two-component systems), we present a non-trivial class of examples of function C to which our theory is applicable, and hence, we can construct gauge-invariant quasi-free states, or even strongly quasi-free states on A. E-Thesis Swansea University Fock space, deformed commutation relations, multicomponent quantum system, anyon, plekton 20 3 2023 2023-03-20 10.23889/SUthesis.62993 Copyright: The author, Nedal Othman, 2023 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University Lytvynov, Eugene, Finkelshtein, Dmitri. Doctoral Ph.D Not Required 2023-03-20T15:22:25.5868464 2023-03-20T13:04:57.5576848 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Nedal Othman 1 62993__26903__41512bc0edc64e0c8ab31a2531f89a4c.pdf Final version.Nedal Othman.PhD.2023.pdf 2023-03-20T13:15:29.3971675 Output 890376 application/pdf E-Thesis – open access true Copyright: The author, Nedal Othman, 2023. true eng
title Quasi-free states on algebras of multicomponent commutation relations
spellingShingle Quasi-free states on algebras of multicomponent commutation relations
Nedal Othman
title_short Quasi-free states on algebras of multicomponent commutation relations
title_full Quasi-free states on algebras of multicomponent commutation relations
title_fullStr Quasi-free states on algebras of multicomponent commutation relations
title_full_unstemmed Quasi-free states on algebras of multicomponent commutation relations
title_sort Quasi-free states on algebras of multicomponent commutation relations
author_id_str_mv a5da4c291c531185ea524821b1afd370
author_id_fullname_str_mv a5da4c291c531185ea524821b1afd370_***_Nedal Othman
author Nedal Othman
author2 Nedal Othman
format E-Thesis
publishDate 2023
institution Swansea University
doi_str_mv 10.23889/SUthesis.62993
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 Let X = R² and let V be a finite-dimensional complex inner product space. Let C : X² → L (V ⊗²) be a continuous function such that, for each (x, y) ∈ X², C(x, y) is a unitary operator in V ⊗², C∗(x, y) = C(y, x), and the functional Yang-Baxter equa-tion is satisfied. The dissertation deals with the multicomponent commutation relations governed by the function C, see [A. Liguori, M. Mintchev, Comm. Math. Phys. 169 (1995) 635–652]. We introduce the ∗-algebra of the C-multicomponent commutation re-lations (C-MCR algebra). We propose definitions of a gauge-invariant quasi-free state andof a strongly quasi-free state on the C-MCR algebra, A. Under restrictive assumptions on the function C, we construct a class of gauge-invariant quasi-free states on A, which, for some functions C, are also strongly quasi-free. We show that, when dim V = 1 (i.e., when we deal with the anyon commutation relations), among all gauge-invariant quasi- free states on A, only the Fock state is strongly quasi-free. In the case dim V = 2 (i.e., when we deal with two-component systems), we present a non-trivial class of examples of function C to which our theory is applicable, and hence, we can construct gauge-invariant quasi-free states, or even strongly quasi-free states on A.
published_date 2023-03-20T04:23:27Z
_version_ 1763663945299656704
score 11.036706

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