Quantum effects in a noncommutative Friedman world model

Abstract

We present a noncommutative version of the closed Friedman world model and show how its classical space–time geometry can be expressed in terms of typically quantum mathematical structures, namely in terms of an operator algebra M0 on a family of Hilbert spaces. The operator algebra M0 can be completed to the von Neumann algebra M, but the geometry cannot be prolonged from M0 to M. This mathematical fact is a stumbling block in creating full quantum gravity theory. Two effects appearing in this model, generation of matter and probabilistic properties of singularities, are also discussed.