Title: Real-time 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 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. - Highlights: • Real-time 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 double-well potential. • Difficulty for practical applications is revealed, and we discuss its generality. • Quantum tunneling is shown to be closelymore » related to complex classical solutions.« less

Department of Physics, The University of Tokyo, Tokyo 113-0033 (Japan)

(Japan)

Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914 (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