arXiv:2603.25873

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

J. G. A. Caribé, M. S. Guimaraes, I. Roditi, S. P. Sorella

hep-thmath-phquant-phSubmitted Mar 26, 2026• Announced Mar 30, 2026• 0 citations• Revised Jun 29, 2026 (v2)

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 $1+1$ 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.

Comments: 24 pages, 2 figures. Updated version that matches the one acepted by EPJC

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arXiv:2603.25873

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

J. G. A. Caribé, M. S. Guimaraes, I. Roditi, S. P. Sorella

hep-thmath-phquant-phSubmitted Mar 26, 2026• Announced Mar 30, 2026• 0 citations• Revised Jun 29, 2026 (v2)

arXiv HTML PDF

Abstract

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.

Comments: 24 pages, 2 figures. Updated version that matches the one acepted by EPJC

Related papers

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.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 CJul 18, 20260 citations

Modular wedge localization, Majorana fields and the Tsirelson limit of the Bell-CHSH inequality

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

The massive Majorana field in $1+1$ dimension is employed to investigate the violation of the Bell-CHSH inequality in relativistic Quantum Field Theory. We give an explicit rapidity-space realization of the Summers-Werner modular-localization construction and reduce the vacuum Bell-CHSH correlator to a single spectral weight $h^2(ω)$ for the modular operator. The resulting analytic families approach the Tsirelson bound in the vacuum state as their spectral weight concentrates near $ω\approx0$ , corresponding to the eigenvalue $λ^2 \approx 1$ of the modular operator.The massive Majorana field in $1+1$ dimension is employed to investigate the violation of the Bell-CHSH inequality in relativistic Quantum Field Theory. We give an explicit rapidity-space realization of the Summers-Werner modular-localization construction and reduce the vacuum Bell-CHSH correlator to a single spectral weight $h^2(ω)$ for the modular operator. The resulting analytic families approach the Tsirelson bound in the vacuum state as their spectral weight concentrates near $ω\approx0$, corresponding to the eigenvalue $λ^2 \approx 1$ of the modular operator.

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Majorana bilinears, compact modular seeds and the approach to the Tsirelson bound

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

In this work we study the Bell-CHSH inequality in the vacuum of the free massive Majorana field in $1+1$ dimensions, taking as the dichotomic observables of the two parties even bilinears of smeared fields localized in the right and in the left Rindler wedges. The test functions are generated from seeds in the modular-frequency variable through the Tomita-Takesaki operator of the right wedge and, on the side of the left wedge, through its adjoint multiplied by $i$ , which accounts for twisted duality. For seeds supported at positive modular frequency the one-particle overlaps between the two parties do not depend on the modular weights, and we obtain the Bell parameter in closed form. When the seeds are phase rotations of a single compact bump, the bilinears generate a copy of the Pauli algebra in each wedge, and we compute exactly the correlation matrix of the vacuum restricted to the two copies. The optimal settings violate the classical bound up to spectral widths more than twice as large as those reached with planar settings. We also show that the Tsirelson bound is never attained in the wedge vacuum for any choice of dichotomic observables, and that approaching it forces the modular spectral weight of the Bell vectors to concentrate at the spectral point $λ=1$ of the modular operator. This establishes in a precise way how the even sector of the free Majorana field realizes the Summers-Werner mechanism.In this work we study the Bell-CHSH inequality in the vacuum of the free massive Majorana field in $1+1$ dimensions, taking as the dichotomic observables of the two parties even bilinears of smeared fields localized in the right and in the left Rindler wedges. The test functions are generated from seeds in the modular-frequency variable through the Tomita-Takesaki operator of the right wedge and, on the side of the left wedge, through its adjoint multiplied by $i$, which accounts for twisted duality. For seeds supported at positive modular frequency the one-particle overlaps between the two parties do not depend on the modular weights, and we obtain the Bell parameter in closed form. When the seeds are phase rotations of a single compact bump, the bilinears generate a copy of the Pauli algebra in each wedge, and we compute exactly the correlation matrix of the vacuum restricted to the two copies. The optimal settings violate the classical bound up to spectral widths more than twice as large as those reached with planar settings. We also show that the Tsirelson bound is never attained in the wedge vacuum for any choice of dichotomic observables, and that approaching it forces the modular spectral weight of the Bell vectors to concentrate at the spectral point $λ=1$ of the modular operator. This establishes in a precise way how the even sector of the free Majorana field realizes the Summers-Werner mechanism.

hep-thCross-listedarXiv:2610.04729Oct 6, 20260 citations

More on Majorana fields, modular localization and near-saturation of the Tsirelson bound

J. R. Apfata, J. G. A. Caribé, Silvio Paolo sorella

The Bell-CHSH inequality provides a powerful probe of quantum correlations in relativistic quantum field theory. We study its violation in the vacuum of the free massless Majorana field in 1+1 dimensions using modular wedge localization. By introducing a rapidity variable, we obtain a particularly simple representation of the one-particle Hilbert space as a direct sum of two scalar components, in which the spinor mixing present in previous formulations is eliminated. This allows us to give an explicit algorithm for constructing four bounded, Hermitian, dichotomic operators whose Bell-CHSH expectation value can be made arbitrarily close to the Tsirelson bound ( 2 2 ). We derive the resulting Bell-CHSH functional in terms of the modular spectral weight and provide a semi-analytic family of test functions that realizes this near-saturation explicitly.The Bell-CHSH inequality provides a powerful probe of quantum correlations in relativistic quantum field theory. We study its violation in the vacuum of the free massless Majorana field in 1+1 dimensions using modular wedge localization. By introducing a rapidity variable, we obtain a particularly simple representation of the one-particle Hilbert space as a direct sum of two scalar components, in which the spinor mixing present in previous formulations is eliminated. This allows us to give an explicit algorithm for constructing four bounded, Hermitian, dichotomic operators whose Bell-CHSH expectation value can be made arbitrarily close to the Tsirelson bound ( 2 2 ). We derive the resulting Bell-CHSH functional in terms of the modular spectral weight and provide a semi-analytic family of test functions that realizes this near-saturation explicitly.

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Bell-CHSH inequality and unitary transformations in quantum field theory

D. O. R. Azevedo, F. M. Guedes, M. S. Guimarães +3 more

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.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.

arXiv:2412.03840The European Physical Journal CMar 6, 20260 citations

1+1

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 massive Majorana field in

1+1

dimension is employed to investigate the violation of the Bell-CHSH inequality in relativistic Quantum Field Theory. We give an explicit rapidity-space realization of the Summers-Werner modular-localization construction and reduce the vacuum Bell-CHSH correlator to a single spectral weight

h^2(ω)

for the modular operator. The resulting analytic families approach the Tsirelson bound in the vacuum state as their spectral weight concentrates near

ω\approx0

, corresponding to the eigenvalue

λ^2 \approx 1

of the modular operator.

1+1

Minkowski spacetime is scrutinized.

In this work we study the Bell-CHSH inequality in the vacuum of the free massive Majorana field in

1+1

dimensions, taking as the dichotomic observables of the two parties even bilinears of smeared fields localized in the right and in the left Rindler wedges. The test functions are generated from seeds in the modular-frequency variable through the Tomita-Takesaki operator of the right wedge and, on the side of the left wedge, through its adjoint multiplied by

i

, which accounts for twisted duality. For seeds supported at positive modular frequency the one-particle overlaps between the two parties do not depend on the modular weights, and we obtain the Bell parameter in closed form. When the seeds are phase rotations of a single compact bump, the bilinears generate a copy of the Pauli algebra in each wedge, and we compute exactly the correlation matrix of the vacuum restricted to the two copies. The optimal settings violate the classical bound up to spectral widths more than twice as large as those reached with planar settings. We also show that the Tsirelson bound is never attained in the wedge vacuum for any choice of dichotomic observables, and that approaching it forces the modular spectral weight of the Bell vectors to concentrate at the spectral point

λ=1

of the modular operator. This establishes in a precise way how the even sector of the free Majorana field realizes the Summers-Werner mechanism.

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.