Quantum mechanical model in gravity theory

We consider a model of a real massive scalar field defined as homogeneous on a $d$-dimensional sphere such that the sphere radius, time scale, and scalar field are related by the equations of the general theory of relativity. We quantize this system with three degrees of freedom, define the observables, and find dynamical mean values of observables in the regime where the scalar field mass is much less than the Planck mass.