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

TL;DR

This paper applies Picard--Lefschetz theory to real-time quantum path integrals, demonstrating a computational approach for quantum tunneling and discussing the theoretical challenges involved.

Contribution

It introduces a method to compute real-time quantum dynamics using Lefschetz thimbles, with detailed analysis of complex saddle points in tunneling phenomena.

Findings

Successfully applied to simple quantum mechanics examples

Identified complex saddle points relevant to tunneling

Discussed theoretical difficulties in rewriting path integrals

Abstract

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.