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

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 $h^2(\omega )$ for the modular operator. The resulting analytic families approach the Tsirelson bound in the vacuum state as their spectral weight concentrates near $\omega \approx 0$, corresponding to the eigenvalue ${\lambda }^2\approx 1$ of the modular operator.