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Coherent loop states and their applications in geometric quantization
Thesis (PhD)--Stellenbosch University, 2024.
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| Format: | Thesis |
| Language: | en_ZA en_ZA |
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Stellenbosch : Stellenbosch University
2024
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| _version_ | 1867613968067985408 |
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| access_status_str | Open Access |
| author | Nzaganya, Nzaganya Edson |
| author2 | Bartlett, Bruce |
| author_browse | Bartlett, Bruce Nzaganya, Nzaganya Edson |
| author_facet | Bartlett, Bruce Nzaganya, Nzaganya Edson |
| author_sort | Nzaganya, Nzaganya Edson |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2024. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/130297 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA en_ZA |
| last_indexed | 2026-06-10T12:44:34.165Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/130297 Coherent loop states and their applications in geometric quantization Nzaganya, Nzaganya Edson Bartlett, Bruce Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Coherent states -- Mathematical models Riemann surfaces Geometric quantization Angular momentum Wigner small d matrices UCTD Thesis (PhD)--Stellenbosch University, 2024. ENGLISH 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. AFRIKAANSE 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. Doctorate 2024-02-26T18:42:48Z 2024-04-26T12:28:56Z 2024-02-26T18:42:48Z 2024-04-26T12:28:56Z 2024-03 Thesis https://scholar.sun.ac.za/handle/10019.1/130297 en_ZA en_ZA Stellenbosch University xii, 95 pages : illustrations (some color) application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Coherent states -- Mathematical models Riemann surfaces Geometric quantization Angular momentum Wigner small d matrices UCTD Nzaganya, Nzaganya Edson Coherent loop states and their applications in geometric quantization |
| title | Coherent loop states and their applications in geometric quantization |
| title_full | Coherent loop states and their applications in geometric quantization |
| title_fullStr | Coherent loop states and their applications in geometric quantization |
| title_full_unstemmed | Coherent loop states and their applications in geometric quantization |
| title_short | Coherent loop states and their applications in geometric quantization |
| title_sort | coherent loop states and their applications in geometric quantization |
| topic | Coherent states -- Mathematical models Riemann surfaces Geometric quantization Angular momentum Wigner small d matrices UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/130297 |
| work_keys_str_mv | AT nzaganyanzaganyaedson coherentloopstatesandtheirapplicationsingeometricquantization |