Quantization of Lagrangian Descriptors

TL;DR

This paper introduces a quantum version of Lagrangian descriptors using path integrals, revealing how quantum fluctuations broaden invariant manifolds and enable tunneling, thus providing a geometric framework for quantum phase space transport.

Contribution

It formulates quantum Lagrangian descriptors within the path integral framework, linking classical transport structures to quantum effects and tunneling phenomena.

Findings

Path integral sampling shows manifold broadening under quantum fluctuations.

Quantum LDs reveal barrier penetration and delocalization.

Framework applicable to field theory and beyond classical regimes.

Abstract

We formulate Lagrangian descriptors (LDs) in the path integral framework. Averaging the classical LD over fluctuations about extremal trajectories defines a quantum LD that incorporates quantum effects. Invariant manifolds, which sharply organize classical transport, become finite-width phase space structures under quantum fluctuations, and their overlap provides a geometric mechanism consistent with tunneling as fluctuation-induced delocalization of transport barriers. We demonstrate this approach for the Hamiltonian saddle, where path integral sampling reveals manifold broadening and barrier penetration. This establishes a geometric framework for studying phase space transport and tunneling beyond the classical regime, while also providing a natural route toward the application of LDs to field theory.