Dynamics of algebras in quantum unstable systems

TL;DR

This paper develops a non-unitary dynamical evolution operator within rigged Hilbert space formalism to describe unstable quantum processes, showing how non-commutative observables become commutative over time.

Contribution

It introduces a novel non-unitary evolution framework for unstable quantum systems using rigged Hilbert space, capturing the transition from non-abelian to abelian observable algebras.

Findings

Non-unitary evolution describes decay and decoherence processes.

Observable algebras become commutative at large times.

Examples illustrate the transition from non-abelian to abelian observables.

Abstract

We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is non-unitary. This allows to describe observables that are initially non-commutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally present some relevant examples.