Non-commutativity from coarse grained classical probabilities

TL;DR

This paper demonstrates how non-commutative quantum behavior, including position, momentum, and spin, can emerge from classical statistical ensembles through coarse graining at the Planck scale, challenging the classical-quantum boundary.

Contribution

It introduces a classical statistical framework that reproduces quantum non-commutativity and discreteness via coarse graining, providing an alternative perspective on quantum phenomena.

Findings

Non-commutative product structure for position and momentum derived from classical ensembles.

Explicit example showing emergence of quantum spin within classical statistical context.

Coarse graining leads to incomplete statistics, explaining quantum correlations.

Abstract

Non-commutative quantum physics at the atom scale can arise from coarse graining of a classical statistical ensemble at the Planck scale. Position and momentum of an isolated particle are classical observables which remain computable in terms of the coarse grained information. However, the commuting classical product of position and momentum observables is no longer defined in the coarse grained system, which is therefore described by incomplete statistics. The microphysical classical statistical ensemble at the Planck scale admits an alternative non-commuting product structure for position and momentum observables which is compatible with the coarse graining. Measurement correlations for isolated atoms are based on this non-commutative product structure. We present an explicit example for these ideas. It also realizes the discreteness of the spin observable within a microphysical classical statistical ensemble.