arXiv (quant-ph)
2026-06-15
DOI: arXiv:2605.20443
arXiv:2605.20443v3 Announce Type: replace
Abstract: The recent posting arXiv:2605.02621 [14], commenting on the article rspa.2025.0413 [7], argues that the proof of Lemma 3.1 in [7] is missing the spatial derivative of the density, which would lead to a Bohm-like quantum potential. This technical note shows why the propagated density is independent of space in the Feynman propagator construction of Lemma 3.1. This is done by extending the proof of Lemma 3.1 explicitly with Bohm-like quantum potential terms along the stationary action paths, and then showing that these terms are exactly zero. In [7], this property can also be verified directly on most examples (double slit, Aharonov-Bohm, potential well, harmonic oscillator, tunneling, EPR, QED), as well as in the derivations of the Pauli, Dirac, and Maxwell equations. For more general nonlinear actions, a time rescaling may be required to guarantee this space independence along stationary paths. In the hydrogen atom example, this time rescaling can be computed in closed form.
In contrast to the general wave of the Madelung solution [9] Lemma 3.1 of [7] is defined first for a propagator, and a general wave is then constructed in a second step. Recall that a propagator is a specific quantum wave, which is initialized at $t=0$ with a Dirac impulse at a given initial position or momentum. In turn, a general wave is constructed in a second step by superposing a distribution of initial conditions using the propagator.
This key difference is why the Bohm-like quantum potential terms disappear in the construction [7] (specifically, in the first step) while the Bohm potential in the Madelung analysis does not.
This fundamental difference is also consistent with the fact that the wave construction in [7] extends naturally to relativistic contexts, while Bohmian non-locality notoriously prevents such extensions.
Keywords - Response to arXiv:2605.02621, in relation to rspa.2025.0413