Quantum observables for probabilistic classical particles
Christof Wetterich
Abstract
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.
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