{"id":{"repo_id":"uwo","oai_identifier":"oai:uwo.scholaris.ca:20.500.14721/29183"},"canonical_url":"https://search.dev.ndltd.org/etd/uwo/oai:uwo.scholaris.ca:20.500.14721/29183","repository":{"repo_id":"uwo","name":"Western University","base_url":"https://uwo.scholaris.ca/server/oai/request"},"display":{"title":"Ricci Curvature of Noncommutative Three Tori, Entropy, and Second Quantization","abstract":"In noncommutative geometry, the metric information of a noncommutative space is encoded in the data of a spectral triple $(mathcal{A}, mathcal{H},D)$, where $D$ plays the role of the Dirac operator acting on the Hilbert space of spinors. Ideas of spectral geometry can then be used to define suitable notions such as volume, scalar curvature, and Ricci curvature. In particular, one can construct the Ricci curvature from the asymptotic expansion of the heat trace $textrm{Tr}(e^{-tD^2})$. In Chapter 2, we will compute the Ricci curvature of a curved noncommutative three torus. The computation is done for both conformal and a non-conformal perturbation of the flat metric. By applying Connes' pseudodifferential calculus for the noncommutative tori, we explicitly compute the second density of the heat trace expansion for the perturbed Laplacians on both functions and $1-$forms. On the other hand, in noncommutative geometry one also wants to get a good notion of an action functional which depends only on the spectrum of $D$, called spectral action functional. It is known that such a functional can be expressed as $textrm{Tr(f(D))}$ for some function $f$. In chapter 3, we show that the von Neumann entropy, average energy, and negative free energy of the Gibbs state of the second quantized Dirac operator $dGamma D$ has a spectral action functional interpretation of the original Dirac operator $D$. To be able to carry on the computations, we have to incorporate the chemical potential $mu$. All those spectral action coefficients can be given in terms of the modified Bessel functions.","abstract_html":"In noncommutative geometry, the metric information of a noncommutative space is encoded in the data of a spectral triple $(mathcal{A}, mathcal{H},D)$, where $D$ plays the role of the Dirac operator acting on the Hilbert space of spinors. Ideas of spectral geometry can then be used to define suitable notions such as volume, scalar curvature, and Ricci curvature. In particular, one can construct the Ricci curvature from the asymptotic expansion of the heat trace <span class=\"etd-inline-math\">textrm{Tr}(e<sup>-tD<sup>2</sup></sup>)</span>. In Chapter 2, we will compute the Ricci curvature of a curved noncommutative three torus. The computation is done for both conformal and a non-conformal perturbation of the flat metric. By applying Connes&#x27; pseudodifferential calculus for the noncommutative tori, we explicitly compute the second density of the heat trace expansion for the perturbed Laplacians on both functions and $1-$forms. On the other hand, in noncommutative geometry one also wants to get a good notion of an action functional which depends only on the spectrum of $D$, called spectral action functional. It is known that such a functional can be expressed as $textrm{Tr(f(D))}$ for some function $f$. In chapter 3, we show that the von Neumann entropy, average energy, and negative free energy of the Gibbs state of the second quantized Dirac operator $dGamma D$ has a spectral action functional interpretation of the original Dirac operator $D$. To be able to carry on the computations, we have to incorporate the chemical potential $mu$. All those spectral action coefficients can be given in terms of the modified Bessel functions.","abstract_has_math":true,"creators":["Dong, Rui"],"institution":"The University of Western Ontario","degree_name":"Ph D","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Khalkhali, Masoud"],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-08-06","date_published":"2019-08-06","updated_at":"2026-07-27T21:56:18Z","subjects":["Noncommutative Geometry","Spectral Triples","Second Quantization","Spectral Geometry","Differential Geometry","Modified Bessel Functions","Chemical Potential","Entropy","Ricci Curvature","Scalar Curvature"],"languages":["en_ca"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14721/29183","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Khalkhali, Masoud"]},{"key":"dc:creator","label":"Author","values":["Dong, Rui"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-07-10T16:18:04Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-07-10T16:18:04Z"]},{"key":"dc:date.issued","label":"Date","values":["2019-08-06"]},{"key":"dc:publisher","label":"Institution","values":["The University of Western Ontario"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph D"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Noncommutative Geometry","Spectral Triples","Second Quantization","Spectral Geometry","Differential Geometry","Modified Bessel Functions","Chemical Potential","Entropy","Ricci Curvature","Scalar Curvature"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_ca"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14721/29183"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."]},{"key":"dc:description.abstract","label":"Abstract","values":["In noncommutative geometry, the metric information of a noncommutative space is encoded in the data of a spectral triple $(mathcal{A}, mathcal{H},D)$, where $D$ plays the role of the Dirac operator acting on the Hilbert space of spinors. Ideas of spectral geometry can then be used to define suitable notions such as volume, scalar curvature, and Ricci curvature. In particular, one can construct the Ricci curvature from the asymptotic expansion of the heat trace $textrm{Tr}(e^{-tD^2})$. In Chapter 2, we will compute the Ricci curvature of a curved noncommutative three torus. The computation is done for both conformal and a non-conformal perturbation of the flat metric. By applying Connes' pseudodifferential calculus for the noncommutative tori, we explicitly compute the second density of the heat trace expansion for the perturbed Laplacians on both functions and $1-$forms. On the other hand, in noncommutative geometry one also wants to get a good notion of an action functional which depends only on the spectrum of $D$, called spectral action functional. It is known that such a functional can be expressed as $textrm{Tr(f(D))}$ for some function $f$. In chapter 3, we show that the von Neumann entropy, average energy, and negative free energy of the Gibbs state of the second quantized Dirac operator $dGamma D$ has a spectral action functional interpretation of the original Dirac operator $D$. To be able to carry on the computations, we have to incorporate the chemical potential $mu$. All those spectral action coefficients can be given in terms of the modified Bessel functions."]},{"key":"dc:title","label":"Title","values":["Ricci Curvature of Noncommutative Three Tori, Entropy, and Second Quantization"]}]}],"canonical_facts":{"dc:contributor.advisor":["Khalkhali, Masoud"],"dc:creator":["Dong, Rui"],"dc:date.accessioned":["2025-07-10T16:18:04Z"],"dc:date.available":["2025-07-10T16:18:04Z"],"dc:date.issued":["2019-08-06"],"dc:description":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."],"dc:description.abstract":["In noncommutative geometry, the metric information of a noncommutative space is encoded in the data of a spectral triple $(mathcal{A}, mathcal{H},D)$, where $D$ plays the role of the Dirac operator acting on the Hilbert space of spinors. Ideas of spectral geometry can then be used to define suitable notions such as volume, scalar curvature, and Ricci curvature. In particular, one can construct the Ricci curvature from the asymptotic expansion of the heat trace $textrm{Tr}(e^{-tD^2})$. In Chapter 2, we will compute the Ricci curvature of a curved noncommutative three torus. The computation is done for both conformal and a non-conformal perturbation of the flat metric. By applying Connes' pseudodifferential calculus for the noncommutative tori, we explicitly compute the second density of the heat trace expansion for the perturbed Laplacians on both functions and $1-$forms. On the other hand, in noncommutative geometry one also wants to get a good notion of an action functional which depends only on the spectrum of $D$, called spectral action functional. It is known that such a functional can be expressed as $textrm{Tr(f(D))}$ for some function $f$. In chapter 3, we show that the von Neumann entropy, average energy, and negative free energy of the Gibbs state of the second quantized Dirac operator $dGamma D$ has a spectral action functional interpretation of the original Dirac operator $D$. To be able to carry on the computations, we have to incorporate the chemical potential $mu$. All those spectral action coefficients can be given in terms of the modified Bessel functions."],"dc:identifier.uri":["https://hdl.handle.net/20.500.14721/29183"],"dc:language.iso":["en_ca"],"dc:publisher":["The University of Western Ontario"],"dc:subject":["Noncommutative Geometry","Spectral Triples","Second Quantization","Spectral Geometry","Differential Geometry","Modified Bessel Functions","Chemical Potential","Entropy","Ricci Curvature","Scalar Curvature"],"dc:title":["Ricci Curvature of Noncommutative Three Tori, Entropy, and Second Quantization"],"dc:type":["thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["Ph D"]},"updated_at":"2026-07-27T21:56:18Z"}