Non-linear Liouville and Shr\"odinger equations in phase space

TL;DR

This paper develops a phase space framework for classical and quantum systems using non-linear Liouville and Schr"odinger equations, establishing conditions for their formulation and exploring their physical implications.

Contribution

It introduces a unified phase space approach for non-linear classical and quantum dynamics, deriving new non-linear equations and connecting them with the Wigner formalism.

Findings

Derived a non-linear Liouville operator for classical systems.

Formulated a non-linear Schr"odinger equation in phase space.

Established a link between the formalism and Wigner's quantum mechanics.

Abstract

Unitary representations of the Galilei group are studied in phase space, in order to describe classical and quantum systems. Conditions to write in general form the generator of time translation and Lagrangians in phase space are then established. In the classical case, Galilean invariance provides conditions for writing the Liouville operator and Lagrangian for non-linear systems. We analyze, as an example, a generalized kinetic equation where the collision term is local and non-linear. The quantum counter-part of such unitary representations are developed by using the Moyal (or star) product. Then a non-linear Schr\"odinger equation in phase space is derived and analyzed. In this case, an association with the Wigner formalism is established, which provides a physical interpretation for the formalism.