Path integrals on complex contours that terminate in prescribed Stokes sectors yield spectral formulas for resonant energies, explaining why the instanton bounce calculation and real-time decay rates agree.
Lorentzian path integral for quantum tunneling and WKB approximation for wave-function
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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 Schr\"{o}dinger 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.
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Path integral analysis of Schr\"odinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states
Path integrals on complex contours that terminate in prescribed Stokes sectors yield spectral formulas for resonant energies, explaining why the instanton bounce calculation and real-time decay rates agree.