Recent work shows that the Schroedinger equation can be solved exactly based only on classical least action. The computation is based on solving a HamiltonJacobi equation for the action computing the classical density accordingly along all stationary action paths and finally constructing the exact wave function based on these classical quantities alone. The method requires that the action Laplacian or more generally the Laplacian of the propagated density be purely time-varying along stationary action paths. In the case of arbitrary nonlinear potentials this condition can still be verified without loss of generality by using a time rescaling. As the complexity of the wave computation is thus shifted to that of the action and the possible time rescaling this paper proposes a new analytical approach both to compute the action itself for a general nonlinear Hamilton-Jacobi pde and to concurrently construct a time rescaling as needed In contrast to solving the Schroedinger equation directly this computation of action and density extends naturally to systems with nonlinear potentials or position dependent inertia tensors. Hence in principle it can replace the approximations of quantum perturbation theory. For general nonlinear potentials, extending a result of Duru and Kleinert the time rescaling in a given metric is shown to correspond to a change of variables in the computed action density and wave with each eigenwave computed using that change of variables. We show that the approach makes it straightforward to construct the quantum wave for basic cases where no exact solution has been yet derived with the time rescaling unifying the computation. The approach first illustrated for the known three dimensional hyperbolic potential waves of the hydrogen atom is then used to compute the quantum waves for a quartic oscillator for which so far no direct exact solutions are known.
Recent work shows that the Schroedinger equation can be solved exactly based only on classical least action. The computation is based on solving a HamiltonJacobi equation for the action computing the classical density accordingly along all stationary action paths and finally constructing the exact wave function based on these classical quantities alone. The method requires that the action Laplacian or more generally the Laplacian of the propagated density be purely time-varying along stationary action paths. In the case of arbitrary nonlinear potentials this condition can still be verified without loss of generality by using a time rescaling. As the complexity of the wave computation is thus shifted to that of the action and the possible time rescaling this paper proposes a new analytical approach both to compute the action itself for a general nonlinear Hamilton-Jacobi pde and to concurrently construct a time rescaling as needed In contrast to solving the Schroedinger equation directly this computation of action and density extends naturally to systems with nonlinear potentials or position dependent inertia tensors. Hence in principle it can replace the approximations of quantum perturbation theory. For general nonlinear potentials, extending a result of Duru and Kleinert the time rescaling in a given metric is shown to correspond to a change of variables in the computed action density and wave with each eigenwave computed using that change of variables. We show that the approach makes it straightforward to construct the quantum wave for basic cases where no exact solution has been yet derived with the time rescaling unifying the computation. The approach first illustrated for the known three dimensional hyperbolic potential waves of the hydrogen atom is then used to compute the quantum waves for a quartic oscillator for which so far no direct exact solutions are known.
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 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.
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].
It is shown how using the classical Hamilton-Jacobi equation one can arrive at the time-dependent wave equation. Although the former equation was originally used by E.Schroedinger to get the wave equation, we propose a different approach. In the first place, we do not use the principle of least action and, in addition, we arrive at the time-dependent equation, while Schroedinger (in his first seminal paper) used the least action principle and obtained the stationary wave equation. The proposed approach works for any classical Hamilton-Jacobi equation. In addition, by introducing information loss into the Hamilton-Jacobi equation we derive in an elementary fashion the wave equations (ranging from the Shroedinger to Klein-Gordon, to Dirac equations). We also apply this technique to a relativistic particle in the gravitational field and obtain the respective wave equation. All this supports 't Hooft's proposal about a possibility of arriving at quantum description from a classical continuum in the presence of information loss.
The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schroedinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton's equations also give the Schroedinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order derivatives. For a time-independent system the Principle of Least Action reduces to the energy variational principle. For time-dependent systems the Principle of Least Action gives time-dependent approximations. Using a Hartree product trial wave function for a time-dependent many-boson system, we apply the Principle of Least Action to obtain the Gross-Pitaevskii equation that describes a Bose-Einstein condensate.
First, we show that there exists in classical mechanics three actions corresponding to different boundary conditions: two well-known actions, the Euler-Lagrange classical action S_cl(x,t;x_0), which links the initial position x_0 and its position x at time t, the Hamilton-Jacobi action S(x,t), which links a family of particles of initial action S_0(x) to their various positions x at time t, and a new action, the deterministic action S(x,t;x_0,v_0), which links a particle in initial position x_0 and initial velocity v_0 to its position x at time t. We study, in the semi-classical approximation, the convergence of the quantum density and the quantum action, solutions to the Madelung equations, when the Planck constant h tends to 0. We find two different solutions which depend on the initial density. In the first case, where the initial quantum density is a classical density, the quantum density and the quantum action converge to a classical action and a classical density which satisfy the statistical Hamilton-Jacobi equations. These are the equations of a set of classical particles whose initial positions are known only by the initial density. In the second case where initial density converges to a Dirac density, the density converges to the Dirac function and the quantum action converges to a deterministic action. Therefore we introduce into classical mechanics non-discerned particles, which satisfy the statistical Hamilton-Jacobi-equations and explain the Gibbs paradox, and discerned particles, which satisfy the deterministic Hamilton-Jacobi equations. Finally, we propose an interpretation of the Schrodinger wave function that depends on the initial conditions (preparation). This double interpretation seems to be the interpretation of Louis de Broglie's "double solution" idea.