The real-time Feynman path integral for step potentials

Complex (semi-)classical paths, or instantons, form an integral part of our understanding of quantum physics. Whereas real classical paths describe classically allowed transitions in the real-time Feynman path integral, classically forbidden evolution is captured by complex semi-classical paths or instantons. In this paper, we uncover the rich, intricate nature of complex semi-classical paths and interference in the Feynman propagator of a non-relativistic quantum particle in both a smooth Woods-Saxon and a discontinuous Heaviside step potential. We demonstrate that the complex semi-classical paths are connected to caustics and may cease to exist as naive solutions to the boundary value problem when the semi-classical path encounters singularities of the potential. We generalise complex semi-classical paths to equivalence classes. Using this generalisation, we track the contribution of complex semi-classical paths beyond these singularity crossings and identify the instanton responsible for the quantum reflection. Whereas most complex contributions to the path integral are small, we demonstrate that in some cases the contribution of the complex semi-classical path is unsuppressed and persists into the semi-classical limit. Finally, we develop methods to detect the presence of complex semi-classical paths from propagation amplitudes. The structure of complex semi-classical paths and methods developed here generalises to a large set of problems in real-time quantum physics.