General linear dynamics - quantum, classical or hybrid

TL;DR

This paper introduces a unified linear framework for classical, quantum, and hybrid dynamics using a path integral approach, highlighting quantum mechanics' unique features within this generalization.

Contribution

It proposes a new linear ensemble theory that unifies classical, quantum, and hybrid dynamics, providing insights into their differences and similarities.

Findings

Quantum mechanics is distinguished from other linear generalizations.

The framework encompasses classical and quantum dynamics as special cases.

A simple two-dimensional model illustrates the unique aspects of quantum mechanics.

Abstract

We describe our recent proposal of a path integral formulation of classical Hamiltonian dynamics. Which leads us here to a new attempt at hybrid dynamics, which concerns the direct coupling of classical and quantum mechanical degrees of freedom. This is of practical as well as of foundational interest and no fully satisfactory solution of this problem has been established to date. Related aspects will be observed in a general linear ensemble theory, which comprises classical and quantum dynamics in the form of Liouville and von Neumann equations, respectively, as special cases. Considering the simplest object characterized by a two-dimensional state-space, we illustrate how quantum mechanics is special in several respects among possible linear generalizations.