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.
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.
Comment on arXiv:2605.02621 [23] and arXiv:2606.05197 [24]:
In contrast to his earlier comment [23] on rspa.2025.0413 [12] and answered in arXiv:2605.20443 [11], the same author of [24] does not seem to dispute any longer that the use in [12] of tools from Feynman's [5] namely
- the kernel approach, which computes a wave starting at a given initial position or momentum, and is a key element in [12] to ensure that the Bohm-like quantum potential vanishes. This is directly verified in all examples of [12] as well as in the Pauli, Dirac and Maxwell derivations from classical action. This approach is very different from the standard Bohm-Madelung potential on the overall wave, as illustrated in detail in [11] for the two-slit experiment.
- for the harmonic oscillator, the kernel computation directly from a Taylor series expansion, which does not involve any circularity.
His second post [24] now
- criticizes the use in [12], [11] of Duru and Kleinert's time-independent eigenwave equivalence of the original and the time-scaled Schrödinger equations [4], confusing this standard result with a multivariable chain rule in the time-dependent kernel computation from action in [12].
- claims that "The mathematical framework presented by the authors remains identical to the well-established semiclassical Van Vleck propagator [sic]" which requires that the "Hamiltonian is at most quadratic in position and momentum [sic]". This confuses a quadratic action in the vector case with a harmonic propagated density based on the Laplacian of the action (with harmonicity always fulfilled e.g., for holomorphic complex functions [10]. In addition, it ignores the extension of a real position to a complex or quaternion position or spin, the key result on multi-valued action branches (which enables a classical derivation of quantum randomness) and the quantization of classical action in [12].
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.
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.
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.
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.
By re-expressing the generalized Schrodinger equation in its coordinate representation in terms of amplitude and phase of the state vector, a quantum-mechanical change of a system can be made to correspond to an ensemble of classical motions of the system to which is added some internal potential. This ensemble may be taken as the statistical. This enables an alternative formulation of quantum mechanics with which is associated that classical picture. Such a formulation leads, however, if developed straightforwardly according to the picture, to a theoretical scheme quite different from that of ordinary quantum mechanics concerning the problems other than the equation of motion. Furthermore, this formulation is not applicable to Fermions when spin, or exclusion principle is taken into account. Bohm's recent renewed form of the statistical interpretation of quantum mechanics referring to this formulation is criticized in several points. On this formulation we can construct a singular ensemble which is shown to correspond to the transformation kernel of the wave function to infinitesimally later instant, and thus a connection between this formulation and that of Feynman's is taken out. Hydrodynamical picture formally equivalent to the statistical picture in one-body problems is also considered, and from such analogy certain formal generalization of Schrodinger equation is suggested.