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Quantum tunnelling, real-time dynamics and Picard-Lefschetz thimbles

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


Zong-Gang Mou

Anders Tranberg


We follow up the work, where in light of the Picard-Lefschetz thimble approach, we split up the real-time path integral into two parts: the initial density matrix part which can be represented via an ensemble of initial conditions, and the dynamic part of the path integral which corresponds to the integration over field variables at all later times. This turns the path integral into a two-stage problem where, for each initial condition, there exits one and only one critical point and hence a single thimble in the complex space, whose existence and uniqueness are guaranteed by the characteristics of the initial value problem. In this paper, we test the method for a fully quantum mechanical phenomenon, quantum tunnelling in quantum mechanics. We compare the method to solving the Schrödinger equation numerically, and to the classical-statistical approximation, which emerges naturally in a well-defined limit. We find that the Picard-Lefschetz result matches the expectation from quantum mechanics and that, for this application, the classical-statistical approximation does not.


Mou, Z., Saffin, P. M., & Tranberg, A. (2019). Quantum tunnelling, real-time dynamics and Picard-Lefschetz thimbles. Journal of High Energy Physics, 2019(11),

Journal Article Type Article
Acceptance Date Nov 12, 2019
Online Publication Date Nov 25, 2019
Publication Date 2019-11
Deposit Date Jul 29, 2020
Publicly Available Date Jul 31, 2020
Journal Journal of High Energy Physics
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 2019
Issue 11
Article Number 135
Keywords Nuclear and High Energy Physics
Public URL
Publisher URL


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