Quantum tunneling, global phases and the limits of classical action reconstructions

TL;DR

This paper critically examines a recent method claiming exact wave function reconstruction from classical actions, revealing its limitations in quantum tunneling and phase phenomena where complex or quantum potentials are necessary.

Contribution

The study demonstrates that classical action-based reconstruction fails in forbidden regions and for global phase effects, emphasizing the need for complex actions or quantum potentials.

Findings

Reconstruction breaks down in classically forbidden regions.

Wave functions require complex actions or quantum potentials.

Global phase phenomena impose constraints beyond local classical trajectories.

Abstract

It was proposed recently that the Schr\"odinger wave function can be reconstructed exactly from a discrete superposition of classical action branches weighted by associated classical densities, without semiclassical approximations. We examine this construction for quantum tunneling through finite potential barriers and for quantum phase phenomena. Although formally consistent when the Hamilton-Jacobi equation admits globally defined real branches, the construction breaks down in classically forbidden regions where no real classical action exists. Using rectangular and Coulomb barrier tunneling in alpha decay and nuclear fusion, we show that the wave function requires either a non-vanishing quantum potential or complex-valued action. The growing barrier component fixed by global boundary conditions is essential for transmission and cannot arise from local real classical trajectories alone. Berry phase, flux quantization, Josephson tunneling, and dc SQUID interference likewise impose global phase constraints absent from local classical action transport.