Universal Local symmetries and non-superposition in classical mechanics

TL;DR

This paper reveals that extending the Hilbert space formulation of classical mechanics introduces extra observables that are not invariant under universal local symmetries, leading to the impossibility of superposition in classical states.

Contribution

It demonstrates that many additional observables in the KvN formulation are non-invariant under local symmetries, which necessitates their removal and prevents superposition in classical mechanics.

Findings

Extra observables are non-invariant under local symmetries.

Removing these observables enforces classical superselection rules.

Superposition is impossible in the extended KvN framework.

Abstract

In the Hilbert space formulation of classical mechanics (CM), pioneered by Koopman and von Neumann (KvN), there are potentially more observables that in the standard approach to CM. In this paper we show that actually many of those extra observables are not invariant under a set of universal local symmetries which appear once the KvN is extended to include the evolution of differential forms. Because of their non-invariance, those extra observables have to be removed. This removal makes the superposition of states in KvN, and as a consequence also in CM, impossible.