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On the emergence of preferred structures in quantum theory

Antoine Soulas, Guilherme Franzmann, Andrea Di Biagio

TL;DR

The paper analyzes whether a Hamiltonian can uniquely determine a tensor product structure (TPS) in Hilbert space, addressing competing results by Cotler et al. and Stoica. It develops a relational, unitary-invariant framework grounded in invariant theory to assess how structures emerge and demonstrates that the pair (H, |ψ⟩) can uniquely select a TPS up to global unitary equivalence. By distinguishing relational vs absolute uniqueness and introducing determined kinds and invariant properties, the authors reconcile the previous theorems and provide a constructive path to TPS uniqueness. The work advances Hilbert-space foundationalism and has implications for emergent locality and quantum reference frames, while noting current finite-dimensional limitations and avenues for extension to quantum gravity contexts.

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.

On the emergence of preferred structures in quantum theory

TL;DR

The paper analyzes whether a Hamiltonian can uniquely determine a tensor product structure (TPS) in Hilbert space, addressing competing results by Cotler et al. and Stoica. It develops a relational, unitary-invariant framework grounded in invariant theory to assess how structures emerge and demonstrates that the pair (H, |ψ⟩) can uniquely select a TPS up to global unitary equivalence. By distinguishing relational vs absolute uniqueness and introducing determined kinds and invariant properties, the authors reconcile the previous theorems and provide a constructive path to TPS uniqueness. The work advances Hilbert-space foundationalism and has implications for emergent locality and quantum reference frames, while noting current finite-dimensional limitations and avenues for extension to quantum gravity contexts.

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.

Paper Structure

This paper contains 15 sections, 12 theorems, 24 equations.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Can the Hamiltonian uniquely determine a TPS?
  4. Cotler et al.'s theorem
  5. Stoica's theorem
  6. $K$-locality is unitary-invariant
  7. $\hat{H}$ has more symmetries than $\mathcal{T}$
  8. The time evolution symmetry
  9. A continuous infinity of distinct TPSs
  10. How structures see each other
  11. Which notion of uniqueness to keep?
  12. Insights from invariant theory
  13. Can $(\hat{H}, \ket{\psi})$ uniquely determine a TPS?
  14. Conclusion
  15. Proof of lemma \ref{['lemma_tensor']}

Key Result

Theorem 2.6

Assume $d_1 = \dots = d_n \equiv d$, and suppose the existence of a single Hamiltonian on $\mathcal{H}$ admitting a TPS ${\mathcal{H}}\xspace \rightarrow \bigotimes_{i=1}^n \mathcal{H}_i$ that makes it $K$-local but without any $K$-duals. Then, if $K$ is sufficiently small, almost all $K$-local Hami

Theorems & Definitions (29)

  • Definition 2.1: Tensor product structure
  • Definition 2.2: $K$-locality
  • Remark 2.3
  • Definition 2.4: Global unitary equivalence
  • Definition 2.5: $K$-duality
  • Theorem 2.6: Cotler et al.
  • Theorem 2.7: Stoica
  • Lemma 2.8
  • Lemma 2.9
  • Proposition 2.10
  • ...and 19 more