State Vector Reduction as a Shadow of a Noncommutative Dynamics

Abstract

A model, based on a noncommutative geometry, unifying general relativity with quantum mechanics, is further develped. It is shown that the dynamics in this model can be described in terms of one-parameter groups of random operators. It is striking that the noncommutative counterparts of the concept of state and that of probability measure coincide. We also demonstrate that the equation describing noncommutative dynamics in the quantum gravitational approximation gives the standard unitary evolution of observables, and in the "space-time limit" it leads to the state vector reduction. The cases of the spin and position operators are discussed in details.