A cat state localized in the right Rindler wedge is employed to study the violation of the Bell-CHSH inequality in a relativistic scalar free Quantum Field Theory. By means of the bounded Hermitian operator
sign(φ(f)), where
φ(f) stands for the smeared scalar field, it turns out that the Bell-CHSH correlator can be evaluated in closed analytic form in terms of the imaginary error function. Being the superposition of two coherent states, cat states allow for the existence of interference terms which give rise to a violation of the Bell-CHSH inequality. As such, the present setup can be considered as an explicit realization of the results obtained by Summers-Werner.
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(φ(f)) s i g n ( φ ( f ) ) , where
φ(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.
We devise a general setup to investigate the violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality in the vacuum state in the context of quantum field theory. We test the method with massless spinor fields in (
1+1)-dimensional Minkowski space-time. Alice's and Bob's test functions are explicitly constructed, first by employing Haar wavelets which are then bumpified into proper test functions via a smoothening procedure relying on the Planck-taper window function. Relativistic causality is implemented by requiring the support of Alice's and Bob's test functions to be located in the left and right Rindler wedges, respectively. Violations of the Bell-CHSH inequality as close as desired to Tsirelson's bound are reported. We briefly comment on the extra portal, compared to earlier works; this opens the opportunity to scrutinize Bell-CHSH inequalities with generic, interacting quantum field theories.
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
h2(ω) for the modular operator. The resulting analytic families approach the Tsirelson bound in the vacuum state as their spectral weight concentrates near
ω≈0, corresponding to the eigenvalue
λ2≈1 of the modular operator.