SUNScholar - Browsing by Author "Nzaganya, Nzaganya Edson"

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Coherent loop states and their applications in geometric quantization
(Stellenbosch : Stellenbosch University, 2024-03) Nzaganya, Nzaganya Edson; Bartlett, Bruce; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
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.