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Picard--Lefschetz theory is applied to path integrals of quantum mechanics,in order to compute real-time dynamics directly. After discussing basicproperties of real-time path integrals on Lefschetz thimbles, we demonstrateits computational method in a concrete way by solving three simple examples ofquantum mechanics. It is applied to quantum mechanics of a double-wellpotential, and quantum tunneling is discussed. We identify all of the complexsaddle points of the classical action, and their properties are discussed indetail. However a big theoretical difficulty turns out to appear in rewritingthe original path integral into a sum of path integrals on Lefschetz thimbles.We discuss generality of that problem and mention its importance. Real-timetunneling processes are shown to be described by those complex saddle points,and thus semi-classical description of real-time quantum tunneling becomespossible on solid ground if we could solve that problem.

Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling