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

TL;DR

This paper bridges two longstanding approaches to quantum tunneling: the instanton method and the resonant-state (Gamow–Siegert) formalism. It shows that resonant states arise from a generalized eigenvalue problem formulated in the complex plane with boundary conditions encoded by Stokes wedges, and that a corresponding path integral can be decomposed into Lefschetz thimbles to reproduce the instanton result. The key insight is that Callan–Coleman’s contour prescription is naturally realized by choosing resonant contours, explaining why only half of the bounce contributes to the decay rate and clarifying the real-time vs Euclidean connection. The work focuses on a 1D setup and outlines routes to extend the framework to higher dimensions and field theory, where a fully first-principles resonant-state formulation remains an open problem.

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.

Figures (5)

• Figure 1: Prototypical potential $V(z)$ for which one studies the survival probability of an initial state $\Psi_{T=0}(z)$, supported inside the FV region at the onset of the decay.
• Figure 2: Typical eigenvalue problem encountered in ordinary quantum mechanics, with the eigenfunctions defined in $L^2(\mathbb{R})$.
• Figure 3: Generic boundary conditions imposed on the ODE \ref{['eq:TimeIndependentSchrödingerEquation']}, leading to a well-defined eigenvalue problem. Introducing an essentially arbitrary contour $\varGamma$ terminating in the desired regions of subdominance, the eigenvalue problem can be projected onto $\mathbb{R}$, allowing us to handle it with mostly standard quantum-mechanical tools.
• Figure 4: Example of a real, unbounded cubic potential permitting Gamow--Siegert boundary conditions via an appropriate choice of Stokes wedges bordering the real axis in the spatial directions allowing for quantum tunneling.
• Figure 5: (Left) Formal potential deformation required to naturally encode the desired Gamow--Siegert boundary conditions demanded to access resonant states. (Center) Emerging resonant-state eigenvalue problem defined in the modified, unbounded potential ${V^{\hbox{(deformed)}}(z)}$. (Right) Simplified thimble decomposition in the complexified function space $\mathcal{C}^\mathbb{C}([0,T])$, schematically reducing the infinite-dimensional function space to a single field direction along which the FV and bounce trajectory are connected by the steepest-descent gradient flow. Accordingly, the saddle-point functions $z_{\hbox{FV}}(t)$ and $z_{\hbox{bounce}}(t)$ are represented as single points in the function space. Since the functional integration contour ${\mathcal{C}([0,T],\varGamma^{\hbox{(resonant)}}\:\!)}$ is deformed into the upper complex half-"plane", the path integral \ref{['eq:Ground_State_Energy_Resonant_FullySpelledOut']} picks up half the bounce contribution, as previously indicated by Callan & Coleman CallanColemanFateOfFalseVac2. As the system lies on a Stokes line, the residual contributions from the bounce thimble in the lower complex half-plane are canceled by its overlap with the FV thimble. To render all thimbles fully visible, this aforementioned overlap has been lifted.