Instantons meet resonances: Unifying two seemingly distinct approaches to quantum tunneling

TL;DR

This paper unifies the resonant-state and instanton approaches to quantum tunneling, showing their equivalence through a contour-deformation prescription that corresponds to resonant state boundary conditions.

Contribution

It demonstrates the equivalence of the resonant-state and instanton methods in quantum tunneling, clarifying their conceptual relationship.

Findings

Resonant states and instanton methods are mathematically equivalent.

Contour deformation in path integrals corresponds to resonant state boundary conditions.

Provides a unified physical interpretation of tunneling decay rates.

Abstract

In the study of quantum-mechanical tunneling processes, numerous approaches have been developed to determine the decay rate of states initially confined within a metastable potential region. Virtually all analytical treatments, however, fall into one of two superficially unrelated conceptual frameworks: the resonant-state approach and the instanton method. Whereas the concept of resonant states and their associated decay widths is grounded in physical reasoning by capturing the regime of uniform probability decay, the instanton method lacks a comparably clear physical interpretation. We demonstrate the equivalence of the two approaches, revealing that the contour-deformation prescription in the functional integral put forward by Callan and Coleman directly corresponds to the outgoing Gamow--Siegert boundary conditions defining resonant states.