Coherent loop states and their applications in geometric quantization

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dc.contributor.advisorBartlett, Bruceen_ZA
dc.contributor.authorNzaganya, Nzaganya Edsonen_ZA
dc.contributor.otherStellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.en_ZA
dc.date.accessioned2024-02-26T18:42:48Z
dc.date.accessioned2024-04-26T12:28:56Z
dc.date.available2024-02-26T18:42:48Z
dc.date.available2024-04-26T12:28:56Z
dc.date.issued2024-03
dc.descriptionThesis (PhD)--Stellenbosch University, 2024. en_ZA
dc.description.abstractENGLISH ABSTRACT: In the first part of this study, we study coherent loop states (also known as Bohr‑ Sommerfeld states) on 𝑆², with application to the representation theory of 𝑆𝑈(2). These states offer a precise bridge between the classical and quantum descriptions of angular momentum. We show that they recover the usual basis of angular mo‑ mentum eigenstates used in physics, and give a self‑contained proof of the asymp‑ totics of their inner products. As an application, we use these states to derive Little‑ john and Yu’s geometric formula for the asymptotics of the Wigner matrix elements. In the second part of this thesis, we consider coherent loop states on a general Riemann surface 𝑀. We show that for quasi‑regular polarizations of 𝑀, the second derivatives of the Bergman kernel on the diagonal of 𝑀 can be computed precisely in terms of the Kähler form of 𝑀. Therefore, the asymptotics of the inner product of coherent loop states can be computed using the complex stationary phase principle. This gives an alternative proof, for quasi‑regular polarized Riemann surfaces, of a variant of a result of Borthwick, Paul and Uribe.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING: In die eerste deel van hierdie studie ondersoek ons koherente lusstate (ook bekend as Bohr‑Sommerfeld‑state) op 𝑆², met die toepassing op die matrikselemente van onontbindbare representasies van 𝑆𝑈(2). Hierdie state bied ’n presiese brug tus‑ sen die klassieke en kwantum beskrywings van hoekmomentum. Ons toon aan dat hulle die gewone basis van hoekmomentum‑eigenstate herwin wat in fisika gebruik word, en gee ’n selfinhoudelike bewys van die asymptote van hul binneprodukte. As ’n toepassing gebruik ons hierdie state om Littlejohn en Yu se meetkundige for‑ mule vir die asymptote van die Wigner‑matrikselemente af te lei. In die tweede deel van hierdie tesis, beskou ons koherente lusstate op ’n alge‑ mene Riemann‑oppervlak 𝑀. Ons toon aan dat vir reguliere polarisasies van 𝑀, die tweede afgeleides van die Bergmankernel op die diagonaal van 𝑀 presies bere‑ ken kan word in terme van die Kähler‑vorm van 𝑀. Daarom kan die asymptote van die binneproduk van koherente lusstate bereken word deur gebruik te maak van die komplekse stasionêre fase‑beginsel. Dit gee ’n alternatiewe bewys, vir amper‑ gereelde gepolariseerde Riemann‑oppervlaktes, van ’n resultaat van Borthwick, Paul, en Uribe.af_ZA
dc.description.versionDoctorateen_ZA
dc.format.extentxii, 95 pages : illustrations (some color)en_ZA
dc.identifier.urihttps://scholar.sun.ac.za/handle/10019.1/130297
dc.language.isoen_ZAen_ZA
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch Universityen_ZA
dc.rights.holderStellenbosch Universityen_ZA
dc.subject.lcshCoherent states -- Mathematical modelsen_ZA
dc.subject.lcshRiemann surfacesen_ZA
dc.subject.lcshGeometric quantizationen_ZA
dc.subject.lcshAngular momentumen_ZA
dc.subject.lcshWigner small d matricesen_ZA
dc.subject.nameUCTDen_ZA
dc.titleCoherent loop states and their applications in geometric quantizationen_ZA
dc.typeThesisen_ZA
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