Lorentzian path integral for quantum tunneling and WKB approximation for wave-function

Abstract

Recently, the Lorentzian path integral formulation using the Picard-Lefschetz theory has attracted much attention in quantum cosmology. In this paper, we analyze the tunneling amplitude in quantum mechanics by using the Lorentzian Picard-Lefschetz formulation and compare it with the WKB analysis of the conventional Schrödinger equation. We show that the Picard-Lefschetz Lorentzian formulation is consistent with the WKB approximation for wave-function and the Euclidean path integral formulation utilizing the solutions of the Euclidean constraint equation. We also consider some problems of this Lorentzian Picard-Lefschetz formulation and discuss a simpler semiclassical approximation of the Lorentzian path integral without integrating the lapse function.

14 pages, 14 figures; fixed typos, added figures, updated discussion

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• Truly Lorentzian quantum cosmology
• Jackiw-Teitelboim Gravity and Lorentzian Quantum Cosmology
• Vacuum Decay and Euclidean Lattice Monte Carlo
• Remote Hawking-Moss instanton and the Lorentzian path integral