Quantum observables for probabilistic classical particles

The classical observables of position and momentum are not well adapted to particles in a microphysical situation where typical probability distributions are characterized by a substantial dispersion.

We propose the use of more robust quantum observables for probabilistic classical particles.

The quantum observables are statistical observables which do not take fixed values for a given classical position and momentum.

Solutions of the Liouville equation are discussed in the quantum formalism for classical statistics.

Statistical observables are represented by non-commuting operators.

No classical correlation function is defined for these observables and Bell's inequalities do not apply.

We demonstrate for a general potential how a quantum system emerges from classical statistics.

For the particular cases of a harmonic potential and a Coulomb potential we investigate subsystems which describe all features of a quantum particle.

This covers the discrete energy spectrum of the hydrogen atom and quantum harmonic oscillator.

We discuss the interference for the double-slit experiment.

Conserved statistical observables may also be relevant for the probabilistic dynamics of dust or planets.