Investigations on the Wigner derivative and on an integral formula for the quantum 6j symbols

Ranaivomanana, Valimbavaka Hosana (2022-04)

Thesis (PhD)--Stellenbosch University, 2022.

Thesis

ENGLISH SUMMARY: wo separate studies are done in this thesis: 1. TheWigner derivative is the partial derivative of dihedral angle with respect to opposite edge length in a tetrahedron, all other edge lengths remaining fixed. We compute the inverse Wigner derivative for spherical tetrahedra, namely the partial derivative of edge length with respect to opposite dihedral angle, all other dihedral angles remaining fixed. We show that the inverse Wigner derivative is actually equal to theWigner derivative. 2. We investigate a conjectural integral formula for the quantum 6j symbols suggested by Bruce Bartlett. For that we consider the asymptotics of the integral and compare it with the known formula for the asymptotics of the quantum 6j symbols due to Taylor and Woodward. Taylor and Woodward’s formula can be rewritten as a sum of two quantities: ins and bound. The asymptotics of the integral splits into an interior and boundary contribution. We successfully compute the interior contribution using the stationary phase method. The result is indeed quite similar to although not exactly the same as ins. Though we expect the boundary contribution to be similar to bound, the computation is left for future work.

AFRIKAANS OPSOMMING: Twee afsonderlike studies word in hiedie tesis gedoen: 1. Die Wigner-afgeleide is die parsiële afgeleide van ’n tweevlakshoek met betrekking tot die teenoorgestelde kandlengte in ’n tetraëder, terwyl alle ander kandlengtes onveranderd bly. Ons bereken die inverse Wigner-afgeleide vir sferiese tetraëders, naamlik die parsiële afgeleide van die kandlengte met betrekking tot teenoortaande tweevlakshoek, terwyl alle ander tweevlakshoeke konstant bly. Ons wys dat die inverse Wigner-afgeleide inderdaad gelyk is aan die Wigner-afgeleide. 2. Ons ondersoek ’n beweerde integralformule vir die kwantum 6j simbole, wat deur Bruce Bartlett as moontlikheid voorgestel is. Daarvoor oorweeg ons die asimptotika van die integraal en vergelyk dit met die bekende formule van die kqantum 6j simbole as gevolg van Taylor enWoodward. Taylor enWoodward se formule kan herskryf word as ’n som van twee hoeveelhede: ins en bound. Die asimptotika van die integraal verdeel in ’n binne- en grensbydrae. Ons het die interne bydrae suksesvol met behulp van die stilstaande fase metode bereken. Die resultaat is inderdaad baie soortgelyk aan hoewel nie presies dieselfde as ins nie. Alhoewel ons verwag dat die grensbydrae soortgelyk aan bound sal wees, word die berekening gelaat vir toekomstige werk.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/124919
This item appears in the following collections: