State Vector Reduction as a Shadow of a Noncommutative Dynamics

TL;DR

This paper develops a noncommutative geometric model unifying general relativity and quantum mechanics, showing how noncommutative dynamics relate to state vector reduction and standard quantum evolution.

Contribution

It introduces a noncommutative geometric framework where state and probability concepts coincide, linking quantum dynamics with space-time limits.

Findings

Noncommutative dynamics described by one-parameter groups of random operators

In the quantum gravitational approximation, dynamics yield standard unitary evolution

In the space-time limit, the model leads to state vector reduction

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.