We solve numerically exactly a simple toy model to quantum general relativity or more properly to path integral on a curved space. We consider the thermal equilibrium of a quantum many body problem on the sphere, the surface of constant positive curvature. We use path integral Monte Carlo to measure the kinetic energy, the internal energy and the static structure of a bosons, fermions and anyons fluid at low temperatures on the sphere. For bosons we also measure the superfluid fraction and compare its behavior at the critical temperature with the universal jump predicted by Nelson and Kosterlitz in flat space in the thermodynamic limit at the superfluid phase transition. For fermions and anyons it is necessary to use the restricted path integral recipe in order to overcome the sign problem. Even if this recipe is exact for the non interacting fluid it reduces to just an approximation for an interacting system. And we make the example of the electron gas at low temperature. Snapshots of the many body path configuration during the evolution of the computer experiment show that the ``speed'' of the single particle path near the poles slows down as a consequence of the ``hairy ball theorem'' of Poincaré. The influence of curvature on the thermodynamic and structural properties of the many body fluid is also studied.
In a previous paper, a quantum-mechanical formulation involving both mean fields and independent-particle fields in many-fermion systems was proposed using the path-integral technique. Then the semiclassical calculation of the energy spectra was performed, and the quantization rule was derived by applying a stationary phase approximation on the path integral. In this paper, a boson-fermion expansion is derived from our formulation using Dirac quantization. As an illustration, the Lipkin model is utilized.