A categorical view of Bell's inequalities in quantum field theory

TL;DR

This paper extends Bell's inequalities to algebraic and locally covariant quantum field theories, demonstrating their preservation under spacetime embeddings using a functorial state space framework.

Contribution

It generalizes Bell's inequalities within LCQFT using a functorial approach, showing their invariance under algebraic morphisms and spacetime embeddings.

Findings

Bell's inequalities are preserved under algebraic morphisms.

The functorial state space framework ensures invariance under isometric spacetime embeddings.

The approach applies to all admissible quadruples in the subcategory.

Abstract

We propose a generalization of the description of Bell's inequalities in algebraic quantum field theory (AQFT) to the context of locally covariant quantum field theory (LCQFT). We use the functorial formulation of the state space as proposed in the seminal work \cite{BFV03} to show that for the suitable subcategory which consists of all possible admissible quadruples, yields a category of states over such observables which satisfy the Clauser-Horne-Shimony-Holt (CHSH) inequality. Such inequality is trivially preserved under algebraic morphisms and functoriality of the state space asserts us that it is preserved under isometric embeddings between the globally hyperbolic spacetimes where the algebras are defined and which constitute the locally covariant QFT functor.