Emergent Classicality via Commuting Position and Momentum Operators

TL;DR

This paper explores how classical behavior emerges from quantum mechanics by constructing commuting operators close to position and momentum, addressing the non-commutativity issue through phase space decoherence.

Contribution

It introduces a method to approximate commuting position and momentum operators using decohered states, revisiting von Neumann's idea to explain classicality emergence.

Findings

Constructed commuting operators close to quantum position and momentum.

Avoided Balian-Low theorem limitations by focusing on decohered states.

Provided a framework for understanding classicality emergence in quantum systems.

Abstract

Any account of the emergence of classicality from quantum theory must address the fact that the quantum operators representing positions and momenta do not commute, whereas their classical counterparts suffer no such restrictions. To address this, we revive an old idea of von Neumann, and seek a pair of commuting operators $X,P$ which are, in a specific sense, "close" to the canonical non-commuting position and momentum operators, $x,p$. The construction of such operators is related to the problem of finding complete sets of orthonormal phase space localized states, a problem severely limited by the Balian-Low theorem. Here these limitations are avoided by restricting attention to situations in which the density matrix is reasonably decohered (i.e., spread out in phase space).