Remarks on the Clauser-Horne-Shimony-Holt inequality in relativistic quantum field theory

TL;DR

This paper investigates the CHSH inequality within a relativistic quantum field theory model using free scalar fields, revealing small violations and suggesting applications to gauge theories like the Higgs model.

Contribution

It introduces a CHSH-type correlator in a relativistic QFT setting and explores its violation, providing a foundation for studying Bell inequalities in gauge-invariant frameworks.

Findings

Small violation of CHSH inequality compared to Tsirelson's bound.

Model applicable to gauge theories such as the Higgs model.

Constructs BRST invariant operators for Bell inequality analysis.

Abstract

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.