Abstract
It is well known that quantum tunneling can be described by instantons in the imaginary-time path integral formalism. However, its description in the real-time path integral formalism has been elusive. Here we establish a statement that quantum tunneling can be characterized in general by the contribution of complex saddle points, which can be identified by using the Picard-Lefschetz theory. We demonstrate this explicitly by performing Monte Carlo simulations of simple quantum mechanical systems, overcoming the sign problem by the generalized Lefschetz thimble method. We confirm numerically that the contribution of complex saddle points manifests itself in a complex “weak value” of the Hermitian coordinate operator \( \hat{x} \) evaluated at time t, which is a physical quantity that can be measured by experiments in principle. We also discuss the transition to classical dynamics based on our picture.
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Acknowledgments
We would like to thank Yuhma Asano and Masafumi Fukuma for valuable discussions. The computations were carried out on the PC clusters in KEK Computing Research Center and KEK Theory Center. K.S. is supported by the Grant-in-Aid for JSPS Research Fellow, No. 20J00079. A. Y. is supported by a Grant-in-Aid for Transformative Research Areas “The Natural Laws of Extreme Universe — A New Paradigm for Spacetime and Matter from Quantum Information” (KAKENHI Grant No. JP21H05191) from JSPS of Japan.
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Nishimura, J., Sakai, K. & Yosprakob, A. A new picture of quantum tunneling in the real-time path integral from Lefschetz thimble calculations. J. High Energ. Phys. 2023, 110 (2023). https://doi.org/10.1007/JHEP09(2023)110
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DOI: https://doi.org/10.1007/JHEP09(2023)110