Statistics on Lefschetz thimbles: Bell/Leggett-Garg inequalities and the classical-statistical approximation

TL;DR

This paper explores how complex probability distributions in quantum field theory, inspired by Lefschetz thimble theory, relate to Bell and Leggett-Garg inequalities, highlighting differences between classical-statistical and quantum theories.

Contribution

It demonstrates that the classical-statistical approximation cannot violate temporal Bell inequalities, unlike full quantum theory, and explains the underlying reasons for this discrepancy.

Findings

Classical-statistical approximation does not violate temporal Bell inequalities.

Full quantum theory can violate Bell inequalities, unlike the classical approximation.

Key differences between spatial and temporal Bell inequalities are identified.

Abstract

Inspired by Lefschetz thimble theory, we treat Quantum Field Theory as a statistical theory with a complex Probability Distribution Function (PDF). Such complex-valued PDFs permit the violation of Bell-type inequalities, which cannot be violated by a real-valued, non-negative PDF. In this paper, we consider the Classical-Statistical approximation in the context of Bell-type inequalities, viz. the familiar (spatial) Bell inequalities and the temporal Leggett-Garg inequalities. We show that the Classical-Statistical approximation does not violate temporal Bell-type inequalities, even though it is in some sense exact for a free theory, whereas the full quantum theory does. We explain the origin of this discrepancy, and point out the key difference between the spatial and temporal Bell-type inequalities. We comment on the import of this work for applications of the Classical-Statistical approximation.