Classical Representations of Quantum Mechanics Related to Statistically Complete Observables

TL;DR

This paper reformulates quantum mechanics using classical probability measures and functions, emphasizing the role of statistically complete observables and exploring the connection between quantum and classical dynamics.

Contribution

It introduces a unified classical framework for quantum mechanics based on statistically complete observables and duality principles, bridging quantum and classical descriptions.

Findings

Quantum states can be represented as probability measures on phase space.

Joint position-momentum observables are analyzed within this classical framework.

Quantum dynamics are related to classical Liouville dynamics.

Abstract

We present a reformulation of quantum mechanics in terms of probability measures and functions on a general classical sample space and in particular in terms of probability densities and functions on phase space. The basis of our proceeding is the existence of so-called statistically complete observables and the duality between the state spaces and the spaces of the observables, the latter holding in the quantum as well as in the classical case. In the phase-space context, we further discuss joint position-momentum observables, Hilbert spaces of infinitely differentiable functions on phase space, and dequantizations. Finally, the relation of quantum dynamics to the classical Liouville dynamics is investigated.