Interpreting Bohm quantum potentials in Computing quantum waves exactly from classical action

The recent arXiv posting [13], commenting on Lemma 3.1 of the paper [7], argues that the proof is missing the spatial derivative of the density, which would lead to a Bohm quantum potential. This technical note shows why the propagated density is never space-dependent in the Feynman propagator construction of Lemma 3.1. This is done by extending the proof of Lemma 3.1 explicitly with the Bohm quantum potential, and then showing that it vanishes. The continuity p.d.e. and the Hamilton-Jacobi p.d.e., extended by the Bohm potential, are undisputed. In constrast 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 Feynman 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. We will see that this key difference is why the Bohm quantum potential disappears in our construction [7] but does not in the Madelung solution, while both consider a general wave.