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

We point out that the violation of the Bell-CHSH inequality in Quantum Mechanics exhibits a simple understanding when the entangled spin singlet states are thought as the vacuum states of suitable Hamiltonians. The construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in an elementary way. The inequality acquires a form in which its violation can be traced back to the vacuum properties, a feature which enables us to make a bridge among a large class of models, whose vacuum state can be described by a Bogoliubov transformation as, for example: superfluids, superconductors and Quantum Field Theories. The examples of a pair of entangled spin 1 particles in Quantum Mechanics and of the scalar field in relativistic Quantum Field Theory are discussed. In the latter case, we rely on the relation expressing the Minkowski vacuum in terms of left and right Rindler modes. As such, the Bell-CHSH inequality turns out to be parametrized by the Unruh temperature.