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

TL;DR

The paper develops a real-time path-integral framework based on Picard--Lefschetz theory, deforming oscillatory integrals onto Lefschetz thimbles to achieve convergence and enabling per-saddle semiclassical analysis. It provides explicit constructions and results for simple quantum systems (free particle on a line, free particle on a circle, and the harmonic oscillator) to illustrate how thimbles implement quantum dynamics and Maslov-type phase factors. The central application to a double-well potential reveals a rich set of complex saddle points controlling real-time tunneling, but also highlights a major unresolved challenge: determining the intersection numbers $n_\sigma$ that weight each thimble in the original path integral. The work argues that real-time tunneling must be described by highly oscillatory complex saddles with potentially infinitely many contributing sectors, offering a path toward a solid semiclassical understanding while acknowledging substantial mathematical and computational hurdles.

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.

Figures (9)

• Figure 1: Possible behaviors of Lefschetz thimbles for non-real classical solutions in real-time path integrals. Red dashed lines show upward flows, and blue solid ones show downward flows. (a) Upward flows from $z_{\sigma}$ do not intersect $\mathcal{Y}$. (b) $z_{\sigma}$ and its complex conjugate $z_{\overline{\sigma}}$ are connected by an upward/downward flow. (c) There exists a real classical solution $x_{\mathrm{cl}}$ with $\mathrm{Im}\;\mathcal{I}[x_{\mathrm{cl}}]=\mathrm{Im}\;\mathcal{I}[z_{\sigma}]$, and those critical points are connected by downward/upward flows.
• Figure 2: List of classical solutions (\ref{['eq:dw04']}) with the boundary condition $x_i=-1$ and $x_f=1$: Left and right panels plot the list of parameters in complex $k^2$ and $p$ planes, respectively. These plots of parameters $k^2$ and $p$ are obtained by numerically solving the formula (\ref{['eq:relation_classical_solutions_integers02']}).
• Figure 3: Typical behaviors of real classical solutions. The left and right panels show those of $p>1$ and $p<1$, respectively.
• Figure 4: Typical behaviors of complex classical solutions. $(n,m)$ refers an element of $\Sigma$ constructed in the next subsection.
• Figure 5: $[t_i+C, t_f+C]$ as a line segment embedded in $\mathbb{C}$. The classical solution labeled by $(n,m)\in \Sigma$ goes straight from $\omega_2$ to $\omega_2+2n\omega_1+2m\omega_3$ during the time $t_f-t_i$ without intersecting the lattice $\Lambda_k$.