On computing quantum waves exactly from classical action

This paper presents an intriguing mathematical framework for computing quantum wave functions from classical multi-valued actions and densities. The internal logic is consistent - the authors clearly define how constraints or singularities lead to multi-valued actions, and how these combine with classical densities to construct wave functions. The mathematical structure follows logically from constrained Hamilton-Jacobi theory through to wave construction. However, the mathematical validity is significantly undermined by the absence of key derivations. Most critically, the paper never proves its central claim that Ψ_j = √ρ_j exp(iΦ_j/ℏ) satisfies the Schrödinger equation. This is not a minor omission - it is the fundamental result on which the entire construction depends. Similarly, the density evolution equation is presented without justification. While the examples provide plausibility by recovering known results, they do not constitute rigorous mathematical proof. The paper reads more as a research program outline than a complete mathematical demonstration.

Mathematically, the submission proposes a clear ansatz but does not establish the exactness it claims. The central issue is that the paper starts from a classical Hamilton–Jacobi equation and classical density transport, then treats the resulting phase and density as if they solve the exact quantum polar decomposition. In general they do not: exact Schrödinger evolution requires the quantum-potential correction. Because that missing step is the load-bearing bridge between the classical construction and the quantum claims, the main theorem is unsupported in the form presented. Internally, the manuscript also shifts a core definition: the same Φ_j is first a classical least-action field and later effectively the exact quantum phase, without proving equivalence. The examples may illustrate semiclassical intuition, but as summarized they do not mathematically justify exact recovery of box eigenstates, hydrogen orbitals, entanglement correlations, or relativistic spinor equations. The work therefore contains an interesting classical-branch framework, but its central mathematical claim of exact equivalence is not demonstrated and appears inconsistent with the standard exact amplitude-phase decomposition.

The paper proposes that exact Schrödinger wave functions can be constructed from purely classical multi-valued actions Φⱼ and classical densities ρⱼ via Ψⱼ = √ρⱼ exp(iΦⱼ/ℏ), summed over branches. This is the Madelung ansatz with a branch-sum prescription. The mathematical framework is internally consistent at the definitional level, but the central theorem (Theorem 2) — that this construction is *exact* and not merely WKB — is asserted without derivation in the submitted text. The standard Madelung decomposition produces a quantum-potential correction to the Hamilton-Jacobi equation, and reconciling the claim of exactness with this well-known fact is the load-bearing missing step. Without it, the construction reduces to a WKB-plus-branch-summation method, which is useful but not the exact, Feynman-replacing result claimed. Several examples raise additional concerns. The double-slit formula given is the Fraunhofer far-field amplitude (an approximation), inconsistent with claims of exactness. The hydrogen-atom claim that classical Kepler multi-valued actions reproduce the standard eigenfunctions is asserted but not shown, and is in obvious tension with the structure of s-state wave functions. The EPR treatment is qualitative and does not engage with Bell's theorem's constraints on local hidden-variable-like reformulations. Mathematical validity is therefore capped at 2 due to the unverified central derivation; internal consistency is 3 because while definitions are used uniformly, the exactness claim conflicts with the approximate character of several worked examples.

The submission’s mathematical core is the claim that exact Schrödinger solutions can be built from (multi-valued) classical Hamilton–Jacobi actions Φ_j and associated classical densities ρ_j via Ψ = Σ_j √ρ_j e^{iΦ_j/ħ}. As stated in the excerpt, this is not mathematically established: substituting the polar form into Schrödinger normally produces a modified Hamilton–Jacobi equation containing a quantum-potential term depending on ρ, whereas the paper uses the purely classical Hamilton–Jacobi PDE. Without a demonstrated mechanism by which the proposed density evolution cancels or reproduces that quantum term, “exactness” is unsupported. Additionally, the work relies on several compressed steps that are central (Theorem 2’s proof; branch-point matching; singular potential handling) but not shown here. Those gaps directly affect the main conclusion that a discrete set of classical extremals replaces path integrals exactly. Within the excerpt, the framework is suggestive and uses standard mathematical ingredients, but the crucial equivalence to full quantum dynamics is not rigorously demonstrated.

This paper presents an intriguing approach to computing quantum waves from classical action, but suffers from a critical gap in its theoretical foundation. While the authors provide their central construction formula and demonstrate it across multiple examples, they do not derive this formula from first principles or show why it should satisfy the Schrödinger equation. The jump from classical Hamilton-Jacobi equations to the quantum wave construction (Ψⱼ = √ρⱼ · exp(iΦⱼ/ℏ)) is presented without mathematical justification. Additionally, key variables in the ensemble formulation are introduced without proper definition. The examples are well-chosen and illustrative, but cannot substitute for the missing theoretical derivation. The work reads more like a proposal or conjecture than a complete mathematical proof of the claimed exact equivalence.

This paper is ambitious and organized around a concrete central claim: that exact quantum wave functions can be reconstructed from multi-valued classical least-action branches plus associated classical densities. As a completeness matter, the submission succeeds in presenting a coherent program and identifying the main mathematical ingredients it intends to use. It also names specific target systems where the method is supposed to apply. However, the support for the main claim is structurally incomplete in the provided content. The key theorem is asserted without the full derivation needed to show that the proposed ansatz solves the Schrödinger equation exactly rather than approximately. Several core quantities are insufficiently defined, and the examples and extensions are presented more as declarations of success than as fully worked demonstrations. On completeness alone, this reads as a promising outline or compressed summary of a larger argument, but not as a self-sufficiently supported presentation of the paper's central result.

The paper proposes an intriguing and potentially powerful method for deriving quantum wave functions directly from classical multi-valued action and density, demonstrating its application through several well-known quantum problems. This approach offers a conceptual bridge between classical and quantum physics and could simplify calculations compared to Feynman path integrals. However, a significant gap in completeness exists in the absence of a mathematical derivation or proof for the paper's central Theorem 2. The assertion that the proposed wave function exactly solves the Schrödinger equation, without a detailed mathematical justification of this equivalence or derivation from the classical premises, undermines the rigor of the core argument. Additionally, a key variable governing classical density evolution remains undefined. While the examples are illustrative, the foundational theoretical justification for the method's exactness is not fully developed.

This submission is scientifically interesting because it advances a clear, ambitious thesis: that exact quantum wavefunctions can be reconstructed from multi-valued classical least-action solutions and associated classical densities, potentially replacing path-integral summation over arbitrary paths with a discrete set of extremal ones. As a conceptual synthesis, that is novel and, if made fully rigorous, would be important both computationally and foundationally. The paper is also commendably paradigm-independent in spirit, offering an alternative interpretation of standard quantum phenomena rather than merely rephrasing orthodoxy. The main weaknesses are not lack of ambition but lack of operational closure in the presented text. The work does not clearly articulate falsifying empirical tests that distinguish it from standard quantum mechanics, and several headline claims—especially exactness in general, treatment of entanglement/collapse, and extension to relativistic spinor equations—appear stronger than what is visibly demonstrated in the provided summary. Overall, this looks like a novel and potentially significant framework paper whose scientific value depends critically on whether the full manuscript truly supplies the missing constructive detail and exact proofs for the broadest claims.

This is a mathematically-oriented paper proposing that exact Schrödinger (and Klein-Gordon, Dirac) wave functions can be reconstructed from a discrete sum over multi-valued classical action branches weighted by classical densities, Ψ = Σⱼ √ρⱼ exp(iΦⱼ/ℏ). The ansatz is the familiar polar/WKB form, and the proposed novelty lies in the claim that summing over a discrete branch set yields exact (not semiclassical) results across a wide range of canonical problems. As scientific communication, the work is reasonably organized but suffers from a gap between its strong abstract claims and the demonstrations actually presented in the submitted text — the hydrogen spectrum, particle-in-a-box quantization, and EPR correlations are asserted to follow but not derived in adequate detail, and the relationship to existing exact-semiclassical results (van Vleck propagator for quadratic Lagrangians, Maslov theory, Gutzwiller) is not clearly delineated. From a falsifiability standpoint, the paper is empirically inert by design: it aims to reproduce standard QM, so no observation distinguishes it from textbook quantum mechanics. Its testability is therefore mathematical (does the construction actually yield exact solutions?) rather than physical. The work could become considerably stronger with (a) explicit derivation of one quantization spectrum (e.g., hydrogen) from the branch sum, (b) clear comparison to and improvement over Maslov/van Vleck constructions, and (c) identification of any regime — even computational — where the discrete-path method gives different or more efficient results than path integrals. As submitted, the contribution reads as a reformulation with computational appeal but limited demonstrated novelty and no empirical differentiation.

The non-relativistic Schrödinger wavefunction for the double-slit experiment can be computed exactly from the two classical extremal action branches (one per slit) and their classical densities, reproducing the standard Fraunhofer interference pattern.

Falsifiable if: If reconstructing the wavefunction from the two classical extremal paths and associated densities fails to reproduce experimentally observed interference fringes (intensity distribution) to within experimental uncertainty, this claim is falsified.

Hydrogen atom eigenfunctions (energy eigenstates and spectral predictions) can be obtained exactly from the multi-valued classical action induced by the Coulomb singularity (Kepler orbits) combined with the classical density along those action branches.

Falsifiable if: If the reconstructed eigenfunctions do not produce the observed hydrogen energy levels, wavefunction shapes, or spectral line positions/strengths (within experimental and theoretical precision), the claim is falsified.

Feynman path integrals for typical quantum systems reduce to a discrete finite (or countable) sum over classical extremal action paths; contributions from non-extremal zig-zag paths are not required to recover exact quantum results.

Falsifiable if: If for some quantum system the exact quantum amplitude requires contributions that cannot be represented by any sum over classical extremal actions (i.e., discrepancies remain that vanish only when non-extremal path contributions are included), the reduction claim is falsified.

Entanglement correlations (e.g., EPR correlations) can be represented exactly as sums of classical individual-particle actions mapping to tensor products of spinors, with measurement collapse corresponding to transitions between action branches at branch points.

Falsifiable if: If quantum correlation statistics (e.g., Bell-inequality-violating statistics) predicted by standard quantum mechanics cannot be reproduced by the classical-action-sum construction for entangled states, or if the branch-transition model contradicts observed measurement statistics, the claim is falsified.

The same density-weighted multi-valued action construction extends exactly to relativistic wave equations (Klein–Gordon and Dirac) producing their solutions from relativistic classical actions and densities.

Falsifiable if: If applying the construction to relativistic systems yields solutions that disagree with known solutions of the Klein–Gordon or Dirac equations (or with experimentally validated relativistic quantum predictions), the extension claim is falsified.