Special Features of the Upper Half-Plane in Semiclassical Theory

We study conditions that the semiquantised Riemannian metric $g_Q$ and Levi-Civita connection $\nabla_Q$ have the same form as classical one in semiclassical theory. Concrete examples of semiclassical theory have been computed by Majid et al. for upper half-planes, hemispheres and complex projective spaces. However, only in the example of the upper half-planes the semiquantised metric and connection have the same form as classical one. In this study, we first investigate the conditions that the Riemannian metric has the same form as classical one when its components are replaced by general functions. Next, based on these conditions we compute the generalised Ricci form and investigate the conditions that a quantised connection is a quantum Levi-Civita connection. Finally, we semiquantise the upper half-plane with Poincar\'e metric generalised by a parameter $t$. This paper is based on my Master thesis.