Randomness in Classical Mechanics and Quantum Mechanics

TL;DR

This paper challenges the classical deterministic view by proposing that classical mechanics inherently contains fundamental randomness, similar to quantum mechanics, and introduces a functional formulation emphasizing irreversibility and observer ensembles.

Contribution

It introduces a functional approach to classical mechanics that incorporates irreducible randomness and reinterprets quantum mechanics with observer ensembles.

Findings

Classical uncertainty relations show positive, non-zero errors in measurement.

Solutions to the Liouville equation exhibit delocalization, explaining irreversibility.

Newtonian trajectories are approximate, with computed corrections.

Abstract

The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is always positive (non zero). A "functional" formulation of classical mechanics was suggested. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. An interpretation of quantum mechanics is attempted in which both classical and quantum mechanics contain fundamental randomness. Instead of an ensemble of events one introduces an ensemble of observers.