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Bell Nonlocality as a Covariance Obstruction in Locally Covariant Quantum Field Theory

Joseph Samper Finberg

TL;DR

This work identifies a fundamental covariance obstruction in locally covariant quantum field theory: no covariant hidden-variable model can both reproduce quantum statistics and satisfy Bell factorization for CHSH-violating states. Framed as a category-theoretic no-go theorem, the result shows that diffeomorphism covariance and the demand for past-based screening of correlations are mutually incompatible in LCQFT. The obstruction is given a non-abelian cohomological interpretation and persists under spacetime embeddings, clarifying that Bell nonlocality in relativistic quantum theory reflects the impossibility of covariant classical causal explanations rather than a breakdown of relativistic causality. The findings have broad implications for interpretations of quantum theory, the problem of time, and approaches to quantum gravity, suggesting that covariant spacetime structure may emerge from entanglement rather than from covariant classical beables.

Abstract

Locally covariant algebraic quantum field theory (LCQFT) satisfies Einstein causality through microcausality and operational no-signalling, yet Bell-type correlations persist in entangled field states across spacelike regions. We demonstrate that this apparent tension reflects a fundamental covariance obstruction: no assignment of classical past variables can simultaneously be covariant under spacetime embeddings, screen off quantum correlations, and reproduce AQFT statistics. This obstruction is distinct from dynamical nonlocality or signalling violations. We formalize this as a no-go theorem in the category-theoretic framework of LCQFT, showing that Bell's notion of local causality -- requiring factorization conditioned on a common past -- is structurally incompatible with diffeomorphism covariance. The failure of Bell locality thus reflects not a breakdown of relativistic causality but the impossibility of embedding quantum correlations into a classical causal framework without introducing preferred foliations or non-covariant beables. This clarifies the conceptual status of nonlocality in relativistic quantum theory.

Bell Nonlocality as a Covariance Obstruction in Locally Covariant Quantum Field Theory

TL;DR

This work identifies a fundamental covariance obstruction in locally covariant quantum field theory: no covariant hidden-variable model can both reproduce quantum statistics and satisfy Bell factorization for CHSH-violating states. Framed as a category-theoretic no-go theorem, the result shows that diffeomorphism covariance and the demand for past-based screening of correlations are mutually incompatible in LCQFT. The obstruction is given a non-abelian cohomological interpretation and persists under spacetime embeddings, clarifying that Bell nonlocality in relativistic quantum theory reflects the impossibility of covariant classical causal explanations rather than a breakdown of relativistic causality. The findings have broad implications for interpretations of quantum theory, the problem of time, and approaches to quantum gravity, suggesting that covariant spacetime structure may emerge from entanglement rather than from covariant classical beables.

Abstract

Locally covariant algebraic quantum field theory (LCQFT) satisfies Einstein causality through microcausality and operational no-signalling, yet Bell-type correlations persist in entangled field states across spacelike regions. We demonstrate that this apparent tension reflects a fundamental covariance obstruction: no assignment of classical past variables can simultaneously be covariant under spacetime embeddings, screen off quantum correlations, and reproduce AQFT statistics. This obstruction is distinct from dynamical nonlocality or signalling violations. We formalize this as a no-go theorem in the category-theoretic framework of LCQFT, showing that Bell's notion of local causality -- requiring factorization conditioned on a common past -- is structurally incompatible with diffeomorphism covariance. The failure of Bell locality thus reflects not a breakdown of relativistic causality but the impossibility of embedding quantum correlations into a classical causal framework without introducing preferred foliations or non-covariant beables. This clarifies the conceptual status of nonlocality in relativistic quantum theory.

Paper Structure

This paper contains 39 sections, 4 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Three Notions of Locality
  3. Operational No-Signalling
  4. AQFT Microcausality
  5. Bell Local Causality
  6. Logical Independence
  7. Locally Covariant AQFT Framework
  8. Category of Spacetimes
  9. Category of Algebras
  10. Locally Covariant Quantum Field Theory
  11. Einstein Causality in LCQFT
  12. States and Natural Transformations
  13. Why Covariance Matters for Bell Locality
  14. Bell Violations in AQFT
  15. Entanglement Across Spacelike Regions
  16. ...and 24 more sections

Key Result

Theorem 1

Let $\mathfrak{A}: \mathsf{Loc} \to \mathsf{Alg}$ be a locally covariant quantum field theory, and let $\omega$ be a covariant state. If $\omega$ violates the CHSH inequality for some choice of observables in spacelike-separated regions, then no covariant hidden variable model satisfying Bell factor

Figures (2)

  • Figure 1: Logical relationships between locality notions. Solid arrows denote implication; dashed arrows with $\times$ indicate no implication. Quantum mechanics satisfies (✓) microcausality and no-signalling while violating ($\times$) Bell locality.
  • Figure 2: The covariance obstruction illustrated. Measurement regions $\mathcal{O}_A$ and $\mathcal{O}_B$ are spacelike separated. Hidden variables $\lambda$ screening off correlations must be defined on a Cauchy surface in their common past $J^-(\mathcal{O}_A \cup \mathcal{O}_B)$. In the rest frame, this is $\Sigma$ (solid); in a boosted frame related by $\chi$, it is $\Sigma'$ (dashed). Covariance demands $\lambda = \lambda'$, but the surfaces intersect different spacetime regions. No foliation-independent assignment can satisfy both Bell factorization and covariance.

Theorems & Definitions (21)

  • Definition 1: No-Signalling
  • Definition 2: Microcausality
  • Definition 3: Bell Local Causality
  • Definition 4: LCQFT Functor
  • Definition 5: Covariant State
  • Definition 6: Category of Cauchy Data
  • Definition 7: Hidden Variable Model
  • Definition 8: Bell Factorization
  • Definition 9: Covariance
  • Theorem 1: Covariance Obstruction
  • ...and 11 more