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

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