Cat states and violation of the Bell-Clauser-Horne-Shimony-Holt inequality in relativistic quantum field theory

A cat state localized in the right Rindler wedge is employed to study the violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality in a relativistic scalar-free quantum field theory. By means of the bounded Hermitian operator <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>sign</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>φ</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>f</a:mi> <a:mo stretchy="false">)</a:mo> <a:mo stretchy="false">)</a:mo> </a:mrow> </a:math> , where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>φ</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>f</g:mi> <g:mo stretchy="false">)</g:mo> </g:math> stands for the smeared scalar field, it turns out that the Bell-CHSH correlator can be evaluated in closed analytic form in terms of the imaginary error function. Being the superposition of two coherent states, cat states allow for the existence of interference terms, which give rise to a violation of the Bell-CHSH inequality. As such, the present setup can be considered as an explicit realization of the results obtained by Summers-Werner.