# Evolution of quantum observables: from non-commutativity to commutativity

## TL;DR

This paper explores how quantum observables transition from non-commutative to commutative algebras over infinite time, using models involving Gamow vectors and scattering theory to illustrate the process.

## Contribution

It introduces two mathematical models demonstrating the transition from quantum to classical observables in the infinite time limit.

## Key findings

- Non-commutative algebra becomes commutative over infinite time.
- Models involve operators on Gamow vectors and scattering theory formalism.
- Transition explained through algebraic structures in quantum resonances.

## Abstract

A fundamental aspect of the quantum-to-classical limit is the transition from a non-commutative algebra of observables to commutative one. However, this transition is not possible if we only consider unitary evolutions. One way to describe this transition is to consider the Gamow vectors, which introduce exponential decays in the evolution. In this paper, we give two mathematical models in which this transition happens in the infite time limit. In the first one, we consider operators acting on the space of the Gamow vectors, which represent quantum resonances. In the second one, we use an algebraic formalism from scattering theory. We construct a non-commuting algebra which commutes in the infinite time limit.