Correspondence Principle as Equivalence of Categories

TL;DR

This paper explores the categorical relationship between quantum and classical mechanics, proposing that the correspondence principle can be viewed as a functor between categories, but finds limitations in certain models like the Ising spin glass.

Contribution

It introduces a categorical framework for quantum-classical correspondence and demonstrates the limitations of this approach using the Ising model.

Findings

Constructivist approach to quantum-classical mapping faces limitations

Functorial correspondence cannot be fully established in the Ising spin glass model

Implications for the emergence of classicality from quantum mechanics

Abstract

If quantum mechanics were to be applicable to macroscopic objects, classical mechanics would have to be a limiting case of quantum mechanics. Then the category Set that packages classical mechanics has to be in some sense a 'limiting case' of the category Hilb packaging quantum mechanics. Following from this assumption, quantum-classical correspondence can be considered as a mapping of the category Hilb to the category Set, i.e., a functor from Hilb to Set, taking place in the macroscopic limit. As a procedure, which takes us from an object of the category Hilb (i.e., a Hilbert space) in the macroscopic limit to an object of the category Set (i.e., a set of values that describe the configuration of a system), this functor must take a finite number of steps in order to make the equivalence of Hilb and Set verifiable. However, as it is shown in the present paper, such a constructivist requirement cannot be met in at least one case of an Ising model of a spin glass. This could mean that it is impossible to demonstrate the emergence of classicality totally from the formalism of standard quantum mechanics.