On the emergence of preferred structures in quantum theory

TL;DR

This paper investigates how minimal quantum ingredients like the Hamiltonian and state can determine preferred structures such as tensor product decompositions, clarifying previous conflicting theorems and proposing a formalism for emergence in quantum theory.

Contribution

It clarifies and resolves conflicting theorems about Hamiltonian-induced tensor product structures and introduces a formalism for emergent structures based on unitary-invariant properties.

Findings

A Hamiltonian and a state can uniquely determine a preferred tensor product structure.

The paper resolves misunderstandings about existing theorems by Cotler et al. and Stoica.

A new mathematical approach for emergent structures in quantum theory is proposed.

Abstract

We assess the possibilities offered by Hilbert space fundamentalism, an attitude towards quantum physics according to which all physical structures (e.g. subsystems, locality, spacetime, preferred observables) should emerge from minimal quantum ingredients (typically a Hilbert space, Hamiltonian, and state). As a case study, we first mainly focus on the specific question of whether the Hamiltonian can uniquely determine a tensor product structure, a crucial challenge in the growing field of quantum mereology. The present paper reviews, clarifies, and critically examines two apparently conflicting theorems by Cotler et al. and Stoica. We resolve the tension, show how the former has been widely misinterpreted and why the latter is correct only in some weaker version. We then propose a correct mathematical way to address the general problem of preferred structures in quantum theory, relative to the characterization of emergent objects by unitary-invariant properties. Finally, we apply this formalism in the particular case we started with, and show that a Hamiltonian and a state are enough structure to uniquely select a preferred tensor product structure.