  We present an investigation of the $CHSH$ inequality within a relativistic
quantum field theory model built up with a pair of free massive scalar fields
$(\varphi_A, \varphi_B)$ where, as it is customary, the indices $(A,B)$ refer
to Alice and Bob, respectively. A set of bounded Hermitian operators is
introduced by making use of the Weyl operators. A $CHSH$ type correlator is
constructed and evaluated in the Fock vacuum by means of the canonical
quantization. Although the observed violation of the $CHSH$ inequality turns
out to be rather small as compared to Tsirelson's bound of Quantum Mechanics,
the model can be employed for the study of Bell's inequalities in the more
physical case of gauge theories such as: the Higgs models, for which local
$BRST$ invariant operators describing both the massive gauge boson as well as
the Higgs particle have been devised. These operators can be naturally
exponentiated, leading to $BRST$ invariant type of Weyl operators useful to
analyze Bell's inequalities within an invariant $BRST$ environment.
