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

• Main Authors: Peter Millington, Zong-Gang Mou, Paul M. Saffin, Anders Tranberg
• Format: Article
• Language: English
• Published: SpringerOpen 2021-03-01
• Series: Journal of High Energy Physics
• Subjects: Field Theories in Lower Dimensions; Nonperturbative Effects; Lattice Quantum Field Theory
• Online Access: https://doi.org/10.1007/JHEP03(2021)077

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.