The observables of a dissipative quantum system

TL;DR

This paper introduces a new time-dependent product for observables in dissipative quantum systems, capturing dissipation effects and leading to a contracted algebra at long times, supported by explicit examples.

Contribution

It proposes a novel time-dependent product for observables that models dissipation effects and results in a contracted algebra in the long-time limit.

Findings

The new product accounts for dissipation effects on observables.

In the long-time limit, the algebra contracts to a simpler form.

Explicit examples validate the theoretical framework.

Abstract

A time-dependent product is introduced between the observables of a dissipative quantum system, that accounts for the effects of dissipation on observables and commutators. In the $t \to \infty$ limit this yields a contracted algebra. The general ideas are corroborated by a few explicit examples.