Quantum tunneling with a Lorentzian path integral

          @misc{ scivideos_PIRSA:17060019,
            doi = {10.48660/17060019},
            url = {https://pirsa.org/17060019},
            author = {Sberna, Laura},
            keywords = {Other Physics},
            language = {en},
            title = {Quantum tunneling with a Lorentzian path integral},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2017},
            month = {jun},
            note = {PIRSA:17060019 see, \url{https://scivideos.org/index.php/pirsa/17060019}}
          }
          

Abstract

We describe the tunneling of a quantum mechanical particle with a Lorentzian (realtime) path integral. The analysis is made concrete by application to the inverted harmonic oscillator potential, where the path integral is known exactly. We apply Picard-Lefschetz theory to the time integral of the Feynmann propagator at fixed energy, and show that the Euclidean integration contour is obtained as a Lefschetz thimble, or a sum of them, in a suitable limit. Picard-Lefschetz theory is used to make the integral manifestly convergent and is also essential for the saddle point or semiclassical approximation. The very simple example of the inverted harmonic oscillator presents many interesting mathematical features, such as the Stokes phenomenon and multiple relevant complex saddles. We also attempt to construct a more realistic picture of the tunneling process, by allowing for spreading in energy and duration.