Projective representation of the Galilei group for classical and quantum-classical systems

TL;DR

This paper demonstrates that classical mechanics can have a projective Galilei group representation with mass as the central charge, extending to some quantum-classical hybrid systems, contrasting with the non-projective classical case.

Contribution

It introduces a new projective representation of the Galilei group in classical mechanics, where the mass acts as the central charge, and extends this to hybrid quantum-classical systems.

Findings

Classical mechanics admits a projective Galilei group representation with mass as central charge.

The classical non-projective representation has zero central charge, unlike the quantum case.

Extension of the projective representation to certain quantum-classical hybrid systems.

Abstract

A physically relevant unitary irreducible non-projective representation of the Galilei group is possible in the Koopman-von Neumann formulation of classical mechanics. This classical representation is characterized by the vanishing of the central charge of the Galilei algebra. This is in contrast to the quantum case where the mass plays the role of the central charge. Here we show, by direct construction, that classical mechanics also allows for a projective representation of the Galilei group where the mass is the central charge of the algebra. We extend the result to certain kind of quantum-classical hybrid systems.