---
abstract: 'Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations.'
author:
- 'Alessandro Sergi[^1]'
- 'Gabriel Hanna[^2]'
- 'Roberto Grimaudo[^3]'
- 'Antonino Messina[^4]'
title: 'Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths'
---

Introduction
============

A growing community of physicists is interested in both monitoring and controlling the time evolution of small numbers of quantum degrees of freedom (DOF) that are embedded in noisy and uncontrollable environments [@caldeira; @weiss; @bp]. A specific case of such a system is encountered when the environment is classical-like in nature. This situation is one of fundamental importance because, ultimately, we and our experimental tools behave classically, at least from a coarse-grained perspective. In recent years, we have 