On the Feynman path integral formulation of the Bell-Clauser-Horne-Shimony-Holt inequality in Quantum Field Theory

TL;DR

This paper explores formulating the Bell-CHSH inequality within Quantum Field Theory using the Feynman path integral approach, demonstrating compatibility with time ordering and recovering canonical results.

Contribution

It introduces a novel path integral formulation of the Bell-CHSH inequality in Quantum Field Theory, extending its applicability and linking it to canonical quantization methods.

Findings

Path integral formulation reproduces canonical results

Compatibility with time ordering in QFT established

Framework for analyzing Bell inequalities in QFT developed

Abstract

By employing a free scalar Quantum Field Theory model previously introduced \cite{Peruzzo:2022pwv}, we attempt at formulating the Bell-CHSH inequality within the Feynman path integral. This possibility relies on the observation that the Bell-CHSH inequality exhibits a natural extension to Quantum Field Theory in such a way that it is compatible with the time ordering $T$. By treating the Feynman propagator as a distribution and by introducing a suitable localizing set of compact support smooth test functions, we work out the path integral setup for the Bell-CHSH inequality, recovering the same results of the canonical quantization.