2014/06/30 by Yuya Tanizaki, Takayuki Koike · 5 citations
Mathematics · Physics and Astronomy · #Feynman diagram #Path integral formulation #Quantum #Quantum Mechanics and Applications #Quantum chaos and dynamical systems #Quantum dissipation #Quantum dynamics #Quantum tunnelling #Relation between Schrödinger's equation and the path integral formulation of quantum mechanics #Saddle point #Simple (philosophy) #advanced mathematical theories #hep-lat #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1016/j.aop.2014.09.003
published as Annals of Physics 351 (2014) 250-274 · 24 pages, 9 figures; (v2) Discussion in Sec.4.3 is improved, Figs. 8 and 9 added, typos corrected, references added; (v3) typos corrected
openalex publication_date 2014/09/16 · arxiv created 2014/09/18 · arxiv updated 2014/09/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Picard--Lefschetz theory is applied to path integrals of quantum mechanics, in order to compute real-time dynamics directly. After discussing basic properties of real-time path integrals on Lefschetz thimbles, we demonstrate its computational method in a concrete way by solving three simple examples of quantum mechanics. It is applied to quantum mechanics of a double-well potential, and quantum tunneling is discussed. We identify all of the complex saddle points of the classical action, and their properties are discussed in detail. However a big theoretical difficulty turns out to appear in rewriting the original path integral into a sum of path integrals on Lefschetz thimbles. We discuss generality of that problem and mention its importance. Real-time tunneling processes are shown to be described by those complex saddle points, and thus semi-classical description of real-time quantum tunneling becomes possible on solid ground if we could solve that problem.