Bell-CHSH inequality and unitary transformations in Quantum Field Theory

TL;DR

The paper addresses Bell-CHSH nonlocality in relativistic quantum field theory and develops a modular framework based on Tomita-Takesaki theory to define localized, bounded observables. It then shows that unitary deformations of these observables, exemplified by sign-operators built from smeared fields, can induce CHSH violations in the vacuum, with explicit 1+1D scalar and Proca dual models. In the undeformed case the vacuum correlator stays below the bound $2$, but suitably chosen deformations yield genuine violations, e.g. |⟨C⟩| ≈ 2*0.0101, while the Tsirelson bound is $2*sqrt(2)$. The Proca-field analysis in 1+1 dimensions reduces to the scalar case via duality, confirming the method’s consistency and pointing to extensions to higher dimensions and interacting theories for stronger nonlocal correlations.

Abstract

Unitary transformations are employed to enhance the violations of the Bell-CHSH inequality in relativistic Quantum Field Theory. The case of the scalar field in $1+1$ Minkowski space-time is scrutinized by relying on the Tomita-Takesaki modular theory. The example of the bounded Hermitian operator $sign(\varphi(f))$, where $\varphi(f)$ stands for the smeared scalar field, is worked out. It is shown that unitary deformations enable for violations of the Bell-CHSH inequality. The setup is generalized to the Proca vector field by means of its equivalence with the scalar theory.

Figures (1)

• Figure 1: Behavior of the Bell-CHSH correlator $\langle {\cal C} \rangle$ as a function of the parameters $\lambda$. Although $\lambda$ belongs to the spectral interval $[0,1]$, the plot has been extended over a larger interval to show that $\langle {\cal C} \rangle$ is bounded by 2.