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

Millington, Peter; Mou, Zong-Gang; Saffin, Paul M.; Tranberg, Anders

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.

Citation

Millington, P., Mou, Z., Saffin, P. M., & Tranberg, A. (2021). Statistics on Lefschetz thimbles: Bell/Leggett-Garg inequalities and the classical-statistical approximation. Journal of High Energy Physics, 2021(3), Article 77. https://doi.org/10.1007/jhep03%282021%29077

• Journal Article Type: Article
• Acceptance Date: Jan 22, 2021
• Online Publication Date: Mar 8, 2021
• Publication Date: 2021-03
• Deposit Date: Mar 9, 2021
• Publicly Available Date: Mar 25, 2021
• Journal: Journal of High Energy Physics
• Electronic ISSN: 1029-8479
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 2021
• Issue: 3
• Article Number: 77
• DOI: https://doi.org/10.1007/jhep03%282021%29077
• Keywords: Nuclear and High Energy Physics
• Public URL: https://nottingham-repository.worktribe.com/output/5382297
• Publisher URL: https://link.springer.com/article/10.1007/JHEP03(2021)077