Realtime Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling
Picard–Lefschetz theory is applied to path integrals of quantum mechanics, in order to compute realtime dynamics directly. After discussing basic properties of realtime 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 doublewell 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. Realtime tunneling processes are shown to be described by those complex saddle points, and thus semiclassical description of realtime quantum tunneling becomes possible on solid ground if we could solve that problem.  Highlights: • Realtime path integral is studied based on Picard–Lefschetz theory. • Lucid demonstration is given through simple examples of quantum mechanics. • This technique is applied to quantum mechanics of the doublewell potential. • Difficulty for practical applications is revealed, and we discuss its generality. • Quantum tunneling is shown to be closelymore »
 Authors:

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 Department of Physics, The University of Tokyo, Tokyo 1130033 (Japan)
 (Japan)
 Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 1538914 (Japan)
 Publication Date:
 OSTI Identifier:
 22403469
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Annals of Physics (New York); Journal Volume: 351; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FEYNMAN PATH INTEGRAL; MATHEMATICAL SOLUTIONS; POTENTIALS; QUANTUM MECHANICS; QUANTUM WELLS; TUNNEL EFFECT