Finite Weyl polynomials and the approach to Tsirelson’s bound in relativistic scalar quantum field theory
- 1UERJ-Universidade do Estado do Rio de Janeiro
- 2CBPF-Centro Brasileiro de Pesquisas Físicas
Abstract
We construct four bounded Hermitian operators, each one a finite polynomial in the unitary Weyl operators, for the Bell-CHSH inequality in a free massive scalar field in 1+1 1 + 1 dimensions. The operators are localized in complementary wedges. Odd Weyl harmonics provide two exactly anticommuting axis observables in each wedge, while normalizable packets with compact support in the spectrum of the boost generator give exact modular inner products at nonvanishing bandwidth. The resulting Bell-CHSH correlator is a finite double sum. An operator with six Weyl terms per axis already gives 2.14885. Using normalized Fejér approximants of sgn(cos (x)) sgn ( cos ( x ) ) , we show that the supremum over the finite-polynomial family equals 2√2 2 2 , although no finite member attains it; a degree-511 example gives 2.80027. We also point out that, in a centered quasifree state, a Bell-CHSH test whose four final settings are bounded functions of individual quadratures admits one common Gaussian representation and therefore stays below 2. The finite-Weyl construction avoids this restriction because Bob’s final settings mix two noncommuting axis observables.
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BibTeX
@article{caribe2026finite,
title = {{Finite Weyl polynomials and the approach to Tsirelson’s bound in relativistic scalar quantum field theory}},
author = {J. G. A. Caribé and M. S. Guimaraes and I. Roditi and S. P. Sorella},
journal = {The European Physical Journal C},
year = {2026},
month = oct,
doi = {10.1140/epjc/s10052-026-16508-z},
url = {https://link.springer.com/article/10.1140/epjc/s10052-026-16508-z},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Finite Weyl polynomials and the approach to Tsirelson’s bound in relativistic scalar quantum field theory
AU - J. G. A. Caribé
AU - M. S. Guimaraes
AU - I. Roditi
AU - S. P. Sorella
PY - 2026
DA - 2026/10/23
JO - The European Physical Journal C
DO - 10.1140/epjc/s10052-026-16508-z
UR - https://link.springer.com/article/10.1140/epjc/s10052-026-16508-z
AB - We construct four bounded Hermitian operators, each one a finite polynomial in the unitary Weyl operators, for the Bell-CHSH inequality in a free massive scalar field in 1+1 1 + 1 dimensions. The operators are localized in complementary wedges. Odd Weyl harmonics provide two exactly anticommuting axis observables in each wedge, while normalizable packets with compact support in the spectrum of the boost generator give exact modular inner products at nonvanishing bandwidth. The resulting Bell-CHSH correlator is a finite double sum. An operator with six Weyl terms per axis already gives 2.14885. Using normalized Fejér approximants of sgn(cos (x)) sgn ( cos ( x ) ) , we show that the supremum over the finite-polynomial family equals 2√2 2 2 , although no finite member attains it; a degree-511 example gives 2.80027. We also point out that, in a centered quasifree state, a Bell-CHSH test whose four final settings are bounded functions of individual quadratures admits one common Gaussian representation and therefore stays below 2. The finite-Weyl construction avoids this restriction because Bob’s final settings mix two noncommuting axis observables.
SN - 1434-6044
SN - 1434-6052
ER -