Journal
ANNALS OF PHYSICS
Volume 351, Issue -, Pages 250-274Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.aop.2014.09.003
Keywords
Real-time dynamics; Path integral; Picard-Lefschetz theory; Lefschetz thimble; Quantum tunneling
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Funding
- RIKEN iTHES project
- Program for Leading Graduate Schools, MEXT, Japan
- [25-6615]
- [25-2869]
- Grants-in-Aid for Scientific Research [13J06615] Funding Source: KAKEN
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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 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. (C) 2014 Elsevier Inc. All rights reserved.
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