arXiv:2605.06224

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

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

hep-thmath-phquant-phSubmitted May 7, 2026• Announced May 8, 2026• 0 citations• Revised Jul 29, 2026 (v2)

Abstract

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.

Comments: 13 pages, two figures, refs. added

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

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Induced Bell-CHSH observables in the Fewster-Verch measurement scheme

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

quant-pharXiv:2610.33773Oct 27, 20260 citations

arXiv:2605.06224

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

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

hep-thmath-phquant-phSubmitted May 7, 2026• Announced May 8, 2026• 0 citations• Revised Jul 29, 2026 (v2)

arXiv HTML PDF

Abstract

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.

Comments: 13 pages, two figures, refs. added

Related papers

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.

arXiv:2607.07949Nuclear Physics BOct 8, 20260 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.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:2603.25873Mar 30, 20260 citations

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

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

Bell-CHSH Violation for a Massless Majorana Field in a Double Cone from Modular Localization

S. P. Sorella

We investigate Bell-CHSH correlations for a chiral massless Majorana field localized in a double cone. The conformal modular flow of the associated light-ray interval is converted into translations by a rapidity-like coordinate, and the one-particle scalar product is derived in modular momentum space. Particular attention is paid to the fact that this scalar product carries a Fermi-Dirac weight. Requiring the modular conjugation to be antiunitary with respect to that weighted scalar product fixes the Fourier-space realization of the modular objects. The resulting Tomita-Takesaki operator and its adjoint generate Alice and twisted-dual Bob directions for which all four crossed CAR orthogonality relations follow automatically from modular localization. The Bell problem then reduces to a simple one-function functional. For a Gaussian modular-momentum profile $h_σ(k)=\exp[-k^2/(2σ^2)]$ the Bell parameter is larger than the classical bound for a broad range of widths and approaches the Tsirelson value analytically, $\lim_{σ\to 0^+}{\it B}(σ)=2\sqrt2$ Thus near-maximal Bell violation is obtained from an explicit sequence of smooth rapidly decreasing profiles concentrated around zero modular momentum, corresponding to the modular spectral value $λ=1$ .We investigate Bell-CHSH correlations for a chiral massless Majorana field localized in a double cone. The conformal modular flow of the associated light-ray interval is converted into translations by a rapidity-like coordinate, and the one-particle scalar product is derived in modular momentum space. Particular attention is paid to the fact that this scalar product carries a Fermi-Dirac weight. Requiring the modular conjugation to be antiunitary with respect to that weighted scalar product fixes the Fourier-space realization of the modular objects. The resulting Tomita-Takesaki operator and its adjoint generate Alice and twisted-dual Bob directions for which all four crossed CAR orthogonality relations follow automatically from modular localization. The Bell problem then reduces to a simple one-function functional. For a Gaussian modular-momentum profile $h_σ(k)=\exp[-k^2/(2σ^2)]$ the Bell parameter is larger than the classical bound for a broad range of widths and approaches the Tsirelson value analytically, \[ \lim_{σ\to 0^+}{\it B}(σ)=2\sqrt2 \] Thus near-maximal Bell violation is obtained from an explicit sequence of smooth rapidly decreasing profiles concentrated around zero modular momentum, corresponding to the modular spectral value $λ=1$.

hep-thCross-listedarXiv:2608.23753Aug 26, 20260 citations

Induced Bell-CHSH observables in the Fewster-Verch measurement scheme

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

The Summers-Werner theorem shows that the vacuum of a relativistic quantum field theory violates the Bell-CHSH inequality maximally on complementary wedges, but it is an existence statement: it exhibits neither the observables nor an apparatus that measures them. We give an explicit operator-level realization within the Fewster-Verch measurement framework, obtaining dichotomic Bell observables of a free Majorana field in $1+1$ dimensions as observables induced by local couplings to Majorana probes. The induced system smearing factorizes exactly as $f_h=k\,b_{λ,h}$ , so its support is controlled by the compactly supported coupling profile $k$ ; dichotomy is imposed by calibrating and normalizing the induced smearings rather than required of the probe observable; and a probe-only normal ordering yields the product of Alice's and Bob's induced bilinears, which the non-multiplicative induced map alone does not provide. Under explicit attainability hypotheses the induced observables saturate the Tsirelson bound in a state prepared from the vacuum. The construction extends to coupled scalar fields.The Summers-Werner theorem shows that the vacuum of a relativistic quantum field theory violates the Bell-CHSH inequality maximally on complementary wedges, but it is an existence statement: it exhibits neither the observables nor an apparatus that measures them. We give an explicit operator-level realization within the Fewster-Verch measurement framework, obtaining dichotomic Bell observables of a free Majorana field in $1+1$ dimensions as observables induced by local couplings to Majorana probes. The induced system smearing factorizes exactly as $f_h=k\,b_{λ,h}$, so its support is controlled by the compactly supported coupling profile $k$; dichotomy is imposed by calibrating and normalizing the induced smearings rather than required of the probe observable; and a probe-only normal ordering yields the product of Alice's and Bob's induced bilinears, which the non-multiplicative induced map alone does not provide. Under explicit attainability hypotheses the induced observables saturate the Tsirelson bound in a state prepared from the vacuum. The construction extends to coupled scalar fields.

quant-pharXiv:2610.33773Oct 27, 20260 citations

The massive Majorana field in

1+1

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

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.

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.

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.

h_σ(k)=\exp[-k^2/(2σ^2)]

the Bell parameter is larger than the classical bound for a broad range of widths and approaches the Tsirelson value analytically,

\lim_{σ\to 0^+}{\it B}(σ)=2\sqrt2

Thus near-maximal Bell violation is obtained from an explicit sequence of smooth rapidly decreasing profiles concentrated around zero modular momentum, corresponding to the modular spectral value

λ=1

1+1

dimensions as observables induced by local couplings to Majorana probes. The induced system smearing factorizes exactly as

f_h=k\,b_{λ,h}

, so its support is controlled by the compactly supported coupling profile

k

; dichotomy is imposed by calibrating and normalizing the induced smearings rather than required of the probe observable; and a probe-only normal ordering yields the product of Alice's and Bob's induced bilinears, which the non-multiplicative induced map alone does not provide. Under explicit attainability hypotheses the induced observables saturate the Tsirelson bound in a state prepared from the vacuum. The construction extends to coupled scalar fields.