Comments on the Bell-Clauser-Horne-Shimony-Holt inequality

Abstract: We discuss the relationship between the Bogoliubov transformations, squeezed states, entanglement and maximum violation of the Bell-CHSH inequality. In particular, we point out that the construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in a simple and general way, covering a large variety of models, ranging from Quantum Mechanism to relativistic Quantum Field Theories. Various examples are employed to illustrate the above mentioned framework. We start by considering a pair of entangled spin 1 particles and a squeezed oscillator in Quantum Mechanics, moving then to the relativistic complex quantum scalar field and to the analysis of the vacuum state in Minkowski space-time in terms of the left and right Rindler modes. In the latter case, the Bell-CHSH inequality turns out to be parametrized by the Unruh temperature.