Coherent loop states and their Application in Geometric Quantisation
In the first part, we will study coherent loop states on \(S^2\), with application to the representation theory of \(\text{SU}(2)\). We will show that they recover the usual basis of angular momentum eigenstates used in Physics, and give a self- contained proof of the asymptotics of their inner product. As an application, we will use these states to derive Littlejohn and Yu’s geometric formula for the asymptotics of the Wigner matrix elements.
In the second part, we will consider coherent loop states on a general Riemann surface \(M\). We will show that for a regular polarization of \(M\), the second derivatives of the Bergman kernel on the diagonal of \(M\) can be completed precisely in terms of the Kahler form of \(M\). Therefore, the asymptotics of the inner product of coherent loop states can be computed using the complex phase principle. This gives an alternative proof, for regular polarised Riemann surfaces, of a result of Borthwick, Paul and Uribe.