On the problem of emergence of classical spacetime: The quantum-mechanical approach

The Riemannian manifold structure of the classical (i.e. Einsteinian) spacetime is derived from the structure of an abstract infinite-dimensional separable Hilbert space S. For this S is first realized as a Hilbert space H of functions of abstract parameters. The spaceH is associated with the space of states of a macroscopic testparticle in the universe. The spatial localization of state of the particle through its interaction with the environment is associated with the selection of a submanifoldM of realization H. The submanifold M is then identified with the classical space (i.e. a space-like hypersurface in spacetime). The mathematical formalism is developed which allows recovering of the usual Riemannian geometry on the classical space and, more generally, on space and time from the Hilbert structure on S. The specific functional realizations of S are capable of generating spacetimes of different geometry and topology. Variation of the length-type action functional on S is shown to produce both the equation of geodesics on M for macroscopic particles and the Schrödinger equation for microscopic particles.