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

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