Noncommutative Dynamics of Random Operators

TL;DR

This paper explores a noncommutative algebraic framework unifying general relativity and quantum mechanics, where dynamics are described via state-dependent automorphisms and relate to free probability, revealing a deep connection between quantum evolution and noncommutative probability.

Contribution

It introduces a novel noncommutative algebraic approach to quantum dynamics on a transformation groupoid, linking automorphism groups with free probability and unifying dynamics with probabilistic structures.

Findings

Dynamics are governed by automorphisms derived from the Tomita-Takesaki theorem.

Under certain conditions, the usual quantum evolution is recovered.

The framework unifies quantum mechanics and probability through noncommutative algebra.

Abstract

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $\Gamma = E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $\Gamma $. We show that every $a \in {\cal A}$ defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, \phi)$, where $\phi $ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $\phi $ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, \phi)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $\phi $ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.