Majorana bilinears, compact modular seeds and the approach to the Tsirelson bound
Abstract
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.
Related
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- An Explicit Counterexample to Tsirelson's Problem via a Linear System Game
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- Fermions and the Tsirelson Problem in Real Quantum Theory
- The seeds and silhouettes of optimal gravitational shielding
BibTeX
@misc{guimaraes2026majorana,
title = {{Majorana bilinears, compact modular seeds and the approach to the Tsirelson bound}},
author = {M. S. Guimaraes and I. Roditi and S. P. Sorella},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04729},
eprint = {2610.04729},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04729},
}
RIS
TY - UNPB
TI - Majorana bilinears, compact modular seeds and the approach to the Tsirelson bound
AU - M. S. Guimaraes
AU - I. Roditi
AU - S. P. Sorella
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04729
UR - https://arxiv.org/abs/2610.04729
AB - 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.
ER -