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

Two separate studies are done in this thesis:

1. The Wigner 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 the Wigner 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 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) the quantity ins. Though we expect the boundary contribution to be similar to bound, the computation is left for future work.