  The Bell-CHSH inequality in the vacuum state of a relativistic scalar quantum
field is revisited by making use of the Hilbert space ${\cal H} \otimes {\cal
H}_{AB}$, where ${\cal H}$ and ${\cal H}_{AB}$ stand, respectively, for the
Hilbert space of the scalar field and of a generic bipartite quantum mechanical
system. The construction of Hermitian, field-dependent, dichotomic operators is
devised as well as the Bell-CHSH inequality. Working out the $AB$ part of the
inequality, the resulting Bell-CHSH correlation function for the quantum field
naturally emerges from unitary Weyl operators. Furthermore, introducing a
Jaynes-Cummings type Hamiltonian accounting for the interaction between the
scalar field and a pair of qubits, the quantum corrections to the Bell-CHSH
inequality in the vacuum state of the scalar field are evaluated till the
second order in perturbation theory.
