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

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


Peter Millington

Zong-Gang Mou

Anders Tranberg


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.


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.

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
Keywords Nuclear and High Energy Physics
Public URL
Publisher URL


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