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The interaction between Unruh-De Witt spin $1/2$ detectors and a real scalarfield is scrutinized by making use of the Tomita-Takesaki modular theory asapplied to the Von Neumann algebra of the Weyl operators. The use of themodular theory enables to evaluate in an exact way the trace over the quantumfield degrees of freedom. The resulting density matrix is employed to the studyof the Bell-CHSH correlator. It turns out that, as a consequence of theinteraction with the quantum field, the violation of the Bell-CHSH inequalityexhibits a decreasing as compared to the case in which the scalar field isabsent.

Unruh-De Witt detectors, Bell-CHSH inequality and Tomita-Takesaki theory