Approximately Macroscopically Unique States and Quantum Mechanics

TL;DR

This paper demonstrates the existence of Approximately Macroscopically Unique states in quantum systems with small commutators, highlighting their persistence even as Planck's constant approaches zero, and contrasting quantum and classical systems.

Contribution

It introduces the concept of AMU states for unbounded operators and shows their existence in position and momentum systems with small Planck constant, emphasizing quantum-classical differences.

Findings

AMU states exist for quantum systems with small commutators.

Standard quantum systems differ significantly from classical systems even as |7| 7 0.

AMU states are present in position and momentum systems when |7| is sufficiently small.

Abstract

We show that Mumford's Approximately Macroscopically Unique (AMU) states exist for quantum systems consisting of unbounded self-adjoint operators when the commutators are small. In particular, AMU states always exist in position and momentum systems when the Planck constant $|\hbar|$ is sufficiently small. However, we show that these standard quantum mechanical systems are far away from classical mechanical (commutative) systems even when $|\hbar|\to 0.$