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Physical review. D/Physical review. D.

Class of bounded Hermitian operators for the Bell-Clauser-Horne-Shimony-Holt inequality in quantum field theory

M. S. Guimarães, I. Roditi, Silvio Paolo sorella

Published Sep 24, 2025• 4 citations
Publisher

Abstract

The violation of the Bell-Clauser-Horne-Shimony-Holt inequality in a relativistic scalar quantum field theory is analyzed by means of a set of bounded Hermitian operators constructed out of the unitary Weyl operators. These operators allow for both analytic and numerical approaches. While the former relies on the modular theory of Tomita-Takesaki, the latter is devised through an explicit construction of the test functions needed for the localization of the aforementioned operators. The case of causal tangent diamonds in $1+1$ Minkowski spacetime is scrutinized.

DOI
10.1103/x939-vzq7

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The violation of the Bell-Clauser-Horne-Shimony-Holt inequality in a relativistic scalar quantum field theory is analyzed by means of a set of bounded Hermitian operators constructed out of the unitary Weyl operators. These operators allow for both analytic and numerical approaches. While the former relies on the modular theory of Tomita-Takesaki, the latter is devised through an explicit construction of the test functions needed for the localization of the aforementioned operators. The case of causal tangent diamonds in 1+11+11+1 Minkowski spacetime is scrutinized.
The violation of the Bell-CHSH inequality in a relativistic scalar Quantum Field Theory is analysed by means of a set of bounded Hermitian operators constructed out of the unitary Weyl operators. These operators allow for both analytic and numerical approaches. While the former relies on the modular theory of Tomita-Takesaki, the latter is devised through an explicit construction of the test functions needed for the localization of the aforementioned operators. The case of causal tangent diamonds in 1+11+11+1 Minkowski spacetime is scrutinized.
The Bell-CHSH (Clauser-Horne-Shimony-Holt) inequality in the vacuum state of a relativistic scalar quantum field is analyzed. Using Weyl operators built with smeared fields localized in the Rindler wedges, the Bell-CHSH inequality is expressed in terms of the Lorentz invariant inner products of test functions. A numerical framework for these inner products is devised. Causality is also explicitly checked by a numerical evaluation of the Pauli-Jordan function. Violations of the Bell-CHSH inequality are reported for different values of the particle mass parameter. Published by the American Physical Society 2024
We present an investigation of the Clauser-Horne-Shimony-Holt (CHSH) inequality within a relativistic quantum field theory model built up with a pair of free massive scalar fields $({\ensuremath{\varphi}}_{A},{\ensuremath{\varphi}}_{B})$ where, as is customary, the indices (A,B)(A,B)(A,B) refer to Alice and Bob, respectively. A set of bounded Hermitian operators is introduced by making use of the Weyl operators. A CHSH-type correlator is constructed and evaluated in the Fock vacuum by means of canonical quantization. Although the observed violation of the CHSH inequality turns out to be rather small as compared to Tsirelson's bound of quantum mechanics, the model can be employed for the study of Bell's inequalities in the more physical case of gauge theories such as the Higgs models, for which local Becchi-Rouet-Stora-Tyutin (BRST) invariant operators describing both the massive gauge boson as well as the Higgs particle have been devised. These operators can be naturally exponentiated, leading to BRST invariant type of Weyl operators useful to analyze Bell's inequalities within an invariant BRST environment.
The Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality in the vacuum state of a relativistic scalar quantum field is revisited by making use of the Hilbert space $\mathcal{H}\ensuremath{\bigotimes}{\mathcal{H}}_{AB}$, where H\mathcal{H}H and HAB{\mathcal{H}}_{AB}HAB​ stand, respectively, for the Hilbert space of the scalar field and of a generic bipartite quantum mechanical system. The construction of Hermitian, field-dependent, dichotomic operators is devised as well as the Bell-CHSH inequality. Working out the ABABAB part of the inequality, the resulting Bell-CHSH correlation function for the quantum field naturally emerges from unitary Weyl operators. Furthermore, introducing a Jaynes-Cummings type Hamiltonian accounting for the interaction between the scalar field and a pair of qubits, the quantum corrections to the Bell-CHSH inequality in the vacuum state of the scalar field are evaluated till the second order in perturbation theory.
The Tomita-Takesaki modular theory is employed to discuss the Bell-CHSH inequality in wedge regions. By using the Bisognano-Wichmann results, the construction of a set of wedge localized vectors in the one-particle Hilbert space of a relativistic massive scalar field in 1+11+11+1 dimensions is devised to establish whether violations of the Bell-CHSH inequality might occur for different choices of Bell's operators. In particular, the construction of the wedge localized vectors employed in the seminal work by Summers-Werner is scrutinized and applied to Weyl and other operators. We also outline a possible path towards the saturation of Tsirelson's bound.