Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2412.03840 · The European Physical Journal C

Bell-CHSH inequality and unitary transformations in quantum field theory

D. O. R. Azevedo, F. M. Guedes, M. S. Guimarães, I. Roditi, Silvio Paolo sorella, Alves Vieira

Published Mar 6, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $$1+1$$ 1 + 1 Minkowski space-time is scrutinized by relying on the Tomita–Takesaki modular theory. The example of the bounded Hermitian operator $$sign(\varphi (f))$$ s i g n ( φ ( f ) ) , where $$\varphi (f)$$ φ ( 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.

DOI
10.1140/epjc/s10052-026-15445-1

Related papers

On a class of bounded Hermitian operators for the Bell-CHSH inequality in Quantum Field Theory

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

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 $1+1$ Minkowski spacetime is scrutinized.

arXiv:2506.00504•May 31, 2025•0 citations

Modular theory and the Bell-CHSH inequality in relativistic scalar quantum field theory

J. G. A. Caribé, M. S. Guimarães, I. Roditi +1 more

Abstract 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 $$1+1$$ 1 + 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.

The European Physical Journal C•Jul 18, 2026•0 citations

Modular Theory and the Bell-CHSH inequality in relativistic scalar Quantum Field Theory

J. G. A. Caribé, M. S. Guimaraes, I. Roditi +1 more

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 $1+1$ 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.

hep-thCross-listedarXiv:2603.25873•Mar 30, 2026•0 citations

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

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.

Physical review. D/Physical review. D.•Sep 24, 2025•4 citations

Cat states and violation of the Bell-CHSH inequality in relativistic Quantum Field Theory

M. S. Guimaraes, I. Roditi, S. P. Sorella

A cat state localized in the right Rindler wedge is employed to study the violation of the Bell-CHSH inequality in a relativistic scalar free Quantum Field Theory. By means of the bounded Hermitian operator $sign(\varphi(f))$, where $\varphi(f)$ stands for the smeared scalar field, it turns out that the Bell-CHSH correlator can be evaluated in closed analytic form in terms of the imaginary error function. Being the superposition of two coherent states, cat states allow for the existence of interference terms which give rise to a violation of the Bell-CHSH inequality. As such, the present setup can be considered as an explicit realization of the results obtained by Summers-Werner.

hep-thCross-listedarXiv:2601.05216•Physical review. D/Physical review. D.•Jan 9, 2026•1 citation

Bell-CHSH inequality and unitary operators

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

Unitary operators are employed to investigate the violation of the Bell-CHSH inequality. The ensuing modifications affecting both classical and quantum bounds are elucidated. The relevance of a particular class of unitary operators whose expectation values are real is pointed out. For these operators, the classical and quantum bounds remain unaltered, being given, respectively, by 2 and 2 2 . As examples, we discuss the explicit realization of phase space Bell-CHSH inequality violation and the Weyl unitary operators for a real scalar field in relativistic Quantum Field Theory.

arXiv:2403.15276•Nuclear Physics B•Oct 16, 2024•6 citations
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+11+11+1 1 + 1 Minkowski space-time is scrutinized by relying on the Tomita–Takesaki modular theory. The example of the bounded Hermitian operator sign(φ(f))sign(\varphi (f))sign(φ(f)) s i g n ( φ ( f ) ) , where φ(f)\varphi (f)φ(f) φ ( 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.
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.
Abstract 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 1 + 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.
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.
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.
A cat state localized in the right Rindler wedge is employed to study the violation of the Bell-CHSH inequality in a relativistic scalar free Quantum Field Theory. By means of the bounded Hermitian operator sign(φ(f))sign(\varphi(f))sign(φ(f)), where φ(f)\varphi(f)φ(f) stands for the smeared scalar field, it turns out that the Bell-CHSH correlator can be evaluated in closed analytic form in terms of the imaginary error function. Being the superposition of two coherent states, cat states allow for the existence of interference terms which give rise to a violation of the Bell-CHSH inequality. As such, the present setup can be considered as an explicit realization of the results obtained by Summers-Werner.
Unitary operators are employed to investigate the violation of the Bell-CHSH inequality. The ensuing modifications affecting both classical and quantum bounds are elucidated. The relevance of a particular class of unitary operators whose expectation values are real is pointed out. For these operators, the classical and quantum bounds remain unaltered, being given, respectively, by 2 and 2 2 . As examples, we discuss the explicit realization of phase space Bell-CHSH inequality violation and the Weyl unitary operators for a real scalar field in relativistic Quantum Field Theory.