Quantum tunneling from paths in complex time

TL;DR

This paper introduces a novel method for analyzing quantum tunneling by examining complex classical solutions and their negative modes in the complex time plane, unifying previous approaches and extending applicability.

Contribution

It develops a unified framework using complex solutions and negative mode analysis to identify dominant tunneling paths, applicable even with singularities or absence of Euclidean solutions.

Findings

Negative modes determine dominant tunneling paths

Method unifies and generalizes previous tunneling treatments

Applicable to cases with singularities or no Euclidean solutions

Abstract

We study quantum mechanical tunneling using complex solutions of the classical field equations. Simple visualization techniques allow us to unify and generalize previous treatments, and straightforwardly show the connection to the standard approach using Euclidean instanton solutions. We demonstrate that the negative modes of solutions along various contours in the complex time plane reveal which paths give the leading contribution to tunneling and which do not, and we provide a criterion for identifying the negative modes. Central to our approach is the solution of the background and perturbation equations not only along a single path, but over an extended region of the complex time plane. Our approach allows for a fully continuous and coherent treatment of classical evolution interspersed by quantum tunneling events, and is applicable in situations where singularities are present and also where Euclidean solutions might not exist.