In contrast to his earlier posting arXiv260502621 [6] commenting on the article rspa20250413 [5] and answered in arXiv260520443 [3] the same author of the new posting arXiv260605197 [7] does not seem to dispute any longer the following points i Assuming in [5] that along each extremal action path the action Laplacian or more generally the propagated density is space independent implies that the Bohm like potential terms are exactly zero a condition which is directly verified in all examples of [5] as well as in its relativistic Dirac and Maxwell extensions ii Just as in Feynmans well known results the standard polynomials in the treatment in [5] of the harmonic oscillator appear naturally from the Taylor expansion of the kernel and thus there is no circularity. He does not appear to dispute either that Bohm like terms along individual stationary action paths are indeed very different from the standard Bohm Madelung potential on the overall wave. This is illustrated e.g. in the detailed double slit example in [3] whose Bohm like terms are exactly zero while the usual Bohm Madelung potential can be very large. Also in that example the action behind the slits is a conic function so that exact Feynman or Van Vleck computations based on quadratic actions do not apply. His new posting [7] therefore concentrates on the claim in [3] that the space independence assumption on the Laplacian of the action if not directly verified can be achieved through time rescaling which is a fair discussion topic We all agree that the Schrodinger equation has to be fulfilled in the original x t coordinates. To this effect as in the hydrogen atom example of the original paper [5] we use a general result of Duru and Kleinert which the author of [6] and [7] may be unaware of. There independence of the constructed eigenwave on the rescaled time is key to the exact backmapping to the original x t coordinates.
Journal reference
Proceedings of the Royal Society A, 482(2336):20250413, 2026
Appendix2$.$pdf in this posting is a response to arXiv:2606.05197 , in relation to rspa.2025.0413. The main paper iis unchanged from the previous (v3) version
In contrast to his earlier posting arXiv260502621 [6] commenting on the article rspa20250413 [5] and answered in arXiv260520443 [3] the same author of the new posting arXiv260605197 [7] does not seem to dispute any longer the following points i Assuming in [5] that along each extremal action path the action Laplacian or more generally the propagated density is space independent implies that the Bohm like potential terms are exactly zero a condition which is directly verified in all examples of [5] as well as in its relativistic Dirac and Maxwell extensions ii Just as in Feynmans well known results the standard polynomials in the treatment in [5] of the harmonic oscillator appear naturally from the Taylor expansion of the kernel and thus there is no circularity. He does not appear to dispute either that Bohm like terms along individual stationary action paths are indeed very different from the standard Bohm Madelung potential on the overall wave. This is illustrated e.g. in the detailed double slit example in [3] whose Bohm like terms are exactly zero while the usual Bohm Madelung potential can be very large. Also in that example the action behind the slits is a conic function so that exact Feynman or Van Vleck computations based on quadratic actions do not apply. His new posting [7] therefore concentrates on the claim in [3] that the space independence assumption on the Laplacian of the action if not directly verified can be achieved through time rescaling which is a fair discussion topic We all agree that the Schrodinger equation has to be fulfilled in the original x t coordinates. To this effect as in the hydrogen atom example of the original paper [5] we use a general result of Duru and Kleinert which the author of [6] and [7] may be unaware of. There independence of the constructed eigenwave on the rescaled time is key to the exact backmapping to the original x t coordinates.
In contrast to his earlier posting arXiv:2605.02621 [5] commenting on the articlerspa.2025.0413 [4] and answered in arXiv:2605.20443 [3], the same author of the newposting arXiv:2606.05197 [6] does not seem to dispute any longer that, as detailedin [3], (i) Assuming in [4] that, along each extremal action path, the action Laplacianor more generally the propagated density is space-independent, implies that theBohm-like potential terms are exactly zero – a condition which is directly verified inall examples of [4] (double slit, Aharonov-Bohm, potential well, harmonic oscillator,tunneling, EPR, QED) as well as in its relativistic Dirac and Maxwell extensions, withthe exception of the hydrogen atom example, and (ii) Just as in Feynman’s well-knownresults, the standard polynomials in the treatment in [4] of the harmonic oscillator andthe hydrogen atom appear naturally from the Taylor expansion of the kernel, and thusthere is no ”circularity”.He does not appear to dispute either that Bohm-like terms along individual stationaryaction paths are indeed very different from the standard Bohm-Madelung potentialon the overall wave. This is illustrated, e.g., in the detailed double slit example in [3].which can be solved very simply using the construction in [4], and whose Bohm-liketerms are exactly zero while the usual Bohm-Madelung potential can be very large.Also, in that example the action behind the slits is a conic function, so that exactFeynman or Van Vleck computations based on quadratic actions do not apply.His new posting [6] therefore concentrates on the claim in [3] that the spaceindependenceassumption on the Laplacian of the action, if not directly verified, canbe achieved through time-rescaling, which is a fair discussion topic. We all agree thatthe Schr¨odinger equation has to be fulfilled in the original x, t coordinates. To thiseffect, as in the hydrogen atom example of the original paper [4], we use a general resultof Duru and Kleinert, which the author of [5] and [6] may be unaware of. There,independence of the constructed eigenwave on the rescaled time is key to the exactbackmapping to the original x, t coordinates.Keywords − Response to arXiv:2606.05197, in relation to rspa.2025.0413 ”On computingquantum waves exactly from classical action”, Feynman path integrals, Duru-Kleinert time rescaling, Bohm-Madelung potential.
The recent arXiv posting arXiv:2605.20443 by Lohmiller and Slotine attempts to address the omission of the Bohm quantum potential in their proposed exact equivalence between classical action and the Schrödinger equation. They introduce a position-dependent time transformation to argue that the spatial derivatives of the probability density amplitude vanish for the Feynman kernel. A rigorous mathematical examination of this transformation reveals a violation of the multivariable chain rule. The spatial derivatives do not vanish in the physical reference frame. The mathematical framework presented by the authors remains identical to the well-established semiclassical Van Vleck propagator, which is exact exclusively for quadratic potentials.
A recent article by Lohmiller \& Slotine (Proc.\ R.\ Soc.\ A \textbf{482}: 20250413) claims that the Schrödinger equation can be solved exactly using only classical least action and classical fluid density, asserting that this formulation avoids semiclassical approximations. We show that their mathematical derivation contains a foundational error. By neglecting the spatial derivatives of the probability density amplitude, the authors inadvertently omit the quantum potential -- the term originally identified by Madelung and later emphasised by Bohm. Consequently, their proposed equivalence is not exact but rather constitutes the standard semiclassical approximation. We further demonstrate that each of the paper's illustrative examples either belongs to a class where the quantum potential vanishes identically due to the geometry of the problem, or recovers the correct quantum result by importing quantum eigenfunctions through the initial conditions, thereby concealing the error.
Recent work shows that the Schrödinger equation can be solved exactly based only on classical least action. The computation is based on first computing the multi-valued action solutions of the Hamilton-Jacobi equation, computing the classical propagated density along all stationary action multipaths, and finally constructing the exact kernel and wave function based on these classical multi-valued quantities alone. The method requires that the propagated classical square root density along each stationary action path is harmonic, as is the case for basic examples such as e.g., the double-slit, quantum tunneling, or the Einstein-Podolsky-Rosen experiment. For more complex cases, residual Bohm-like quantum potentials along each stationary action path can be eliminated through a time-rescaling.
This paper proposes an extended version of the above construction which avoids the need for time-rescaling even in the most general cases. This is achieved by introducing in the Hamilton-Jacobi equation a real viscosity term, which classically simply corresponds to replacing a lumped parameter model with a continuum model including a volumetric stress tensor. This keeps the earlier construction unchanged, except that the real stationary action multipaths are now real stationary viscous action multipaths defined with respect to an associated modified Lagrangian. As in the earlier version, the construction extends straightforwardly to the special and general relativistic contexts, thus describing in a common language the quantum, classical, and relativistic cases.
Note that the global Bohm potential term in the lumped Hamilton-Jacobi equation associated with a Madelung transform does not have a classical equivalent, while the volumetric stress tensor term in our viscous Hamilton-Jacobi equation directly represents the real paths of a classical deformed particle field.
In a recent publication, Goldfarb et al. have proposed a new trajectory scheme to solve quantum problems: Bohmian mechanics with complex action BOMCA . According to these authors, their contribution represents “a novel formulation of Bohmian mechanics.” In our opinion, rather than a new formulation of Bohmian mechanics, the proposal of Goldfarb et al. consists of a novel quantum computational scheme, which is based on solving the complex trajectory equations resulting from an already known complex version of Bohmian dynamics. On the other hand, Goldfarb et al. also claimed that their numerical scheme describes “fully local complex quantum trajectories,” with the reward of a significantly higher degree of localization than in the conventional Bohmian mechanics by lowering the magnitude of the quantum potential. At least from a theoretical and interpretative point of view, we consider that it is very important to clarify what “fully local quantum trajectories” means. Bohmian mechanics is a nonlocal theory because standard quantum mechanics is a nonlocal theory. This nonlocality manifests via the presence of a quantum potential, and therefore the corresponding quantum trajectories will display features that reflect such nonlocality. Something similar should also occur for any trajectory-based approach that intends to be equivalent to standard quantum mechanics. In this Comment we would like to provide a general discussion on these two aspects arising from the work of Goldfarb et al., which can be of much interest to the Chemical Physics community. In particular, we would like to stress the special interest relying on how the term locality should be employed. The approach of Goldfarb et al. starts with the timedependent quantum Hamilton-Jacobi equation. This equation was considered by Leacock and Padgett in 1983 within the context of the quantum transformation theory as a way to obtain bound-state energy levels of quantum systems with no need to calculate the corresponding eigenfunctions. According to Leacock and Padgett, the quantum Hamilton-Jacobi equation can be either stated as a postulate as done by these authors or derived from the Schrodinger equation through a simple connection formula. In the second case, the quantum action corresponds to the complex phase of the wave function when the latter is expressed as a pure complex exponential function. This relationship allows one to pass straightforwardly from the quantum Hamilton-Jacobi formalism to the standard one in terms of the time-dependent Schrodinger and vice versa . Although both the Ansatz and the formulation used by Leacock and Padgett are similar to those considered in the standard quantum WKB semiclassical approximation, as pointed out by these authors, their approach conceptually differs from the WKB one. Within Bohmian mechanics, energy eigenstates present an inconvenience: they give rise to zero velocity fields. Hence, when particles are described by this type of states, they will remain steady motionless . To overcome this problem, Floyd and Faraggi and Matone developed timeindependent quantum Hamilton-Jacobi-like formulations starting from real bipolar Ansatze, though they did not claim full equivalence with standard quantum mechanics regarding their predictions. Later, John proposed a timedependent complex quantum trajectory formalism based on the same connection formula mentioned by Leacock and Padgett to study the dynamics associated with some simple analytical cases, such as the harmonic oscillator or the step barrier. This “modified de Broglie–Bohm approach to quantum mechanics,” as termed by John, is formally equivalent to the BOMCA formulation of Goldfarb et al. Nevertheless, we would like to specify that, due to the fundamental theoretical nature of John’s contribution, a separate calculation of the wave function is required in order to obtain the corresponding trajectories John makes use of the “guiding wave” idea from Bohmian mechanics . Meanwhile, in the version of Goldfarb et al. the modified de Broglie–Bohm approach is formulated in terms of a closed set of equations in the spirit of quantum hydrodynamical formalisms , which avoids the calculation of the wave functions, thus providing a new, practical use to such an approach. Furthermore, within the approach of Goldfarb et al., the expression for the quantum force happens to be typically much smaller than in standard Bohmian mechanics. On the other hand, we would also like to note that a related formulation has also been used by Chou and Wyatt, though they developed a different, novel numerical scheme to tackle the problem of bound-energy levels and scattering in one dimension. When trying to understand the quantum-classical correspondence from the perspective of a WKB-like formulation of Bohmian mechanics WKB-BM , Sanz and co-workers reached the complex quantum HamiltonJacobi equation the same year as John, but independently. The WKB-BM formulation constitutes an excellent, compreTHE JOURNAL OF CHEMICAL PHYSICS 127, 197101 2007
A recent attempt by Lohmiller and Slotine to derive the Schrödinger wave function exactly from classical action and classical density was shown by Vattay to silently drop the Madelung–Bohm quantum potential Q = −(ℏ²/2M) ∇²√ρ/√ρ, producing the standard semiclassical (WKB) approximation rather than an exact result. Vattay's mathematical correction is unambiguous, but it leaves the quantum potential unexplained: Q is treated as an irreducible mathematical fact about quantum mechanics with no underlying mechanism. We show that within Quantum Wavespace Theory (QWST) — a framework based on a finite propagation speed C and a finite saturation density P₀ — the quantum potential acquires an explicit physical origin. Starting from the QWST Lagrangian and applying the Madelung decomposition, we show how Q emerges from the gradient-energy term (C²/2)|∇Ψ|² of the wavespace pattern. In the small-amplitude limit, QWST is mathematically equivalent to standard quantum mechanics, with Q identified as the gradient energy of the wavespace pattern written in polar coordinates. When the saturation constraint is reinstated, the Madelung–Bohm equation acquires a calculable nonlinear modification governed by V_sat(|∇Ψ|; P₀) in the QWST Lagrangian. The correction is parametrically negligible in atomic physics (in agreement with all current measurements) and becomes order-unity within nucleon cores, where standard linear quantum mechanics is known to be insufficient. The saturation scale P₀ entering this correction is consistent with the order-of-magnitude proton internal pressure reported by Burkert et al., and the framework places this scale near ~10³⁵ Pa through a cosine-envelope integration of the proton mass.