Hybrid Koopman C*-formalism and the hybrid quantum-classical master equation

TL;DR

This paper introduces a formalism for hybrid quantum-classical dynamical systems using Koopman C*-algebra, defining automorphisms and hybrid states to unify classical and quantum dynamics.

Contribution

It develops a novel hybrid formalism based on Koopman C*-algebra, classifies hybrid dynamical systems, and characterizes hybrid Hamiltonians.

Findings

Defined hybrid states as density matrices on a GNS-derived Hilbert space

Classified all unitary hybrid dynamical systems

Identified possible hybrid Hamiltonian operators

Abstract

Based on Koopman formalism for classical statistical mechanics, we propose a formalism to define hybrid quantum-classical dynamical systems by defining (outer) automorphisms of the C*-algebra of hybrid operators and realizing them as linear transformations on the space of hybrid states. These hybrid states are represented as density matrices on the Hilbert space obtained from the hybrid C*-algebra by the GNS formalism. We also classify all possible dynamical systems which are unitary and obtain the possible hybrid Hamiltonian operators.