Computing Quantum Waves Exactly from Classical Action: Lohmiller & Slotine's Bridge Between Classical and Quantum Mechanics

Forum topic · ✨步子哥 · 2026-04-26

Summary

A detailed Chinese-language forum post on zhichai.net reviews the 2025 paper 'On computing quantum waves exactly from classical action' by Winfried Lohmiller and Jean-Jacques Slotine of MIT's Nonlinear Systems Laboratory, published in Proceedings of the Royal Society A (482(2336), 20250413). The paper proves that the Schrödinger equation can be derived exactly, for any finite value of Planck's constant, from multi-valued classical action fields satisfying the Hamilton-Jacobi equation, combined with classical density fields satisfying a continuity equation, without semiclassical approximations or infinite path integrals. The wavefunction is assembled as a sum of terms of the form sqrt(rho_j)·exp(i Phi_j / hbar), where each term corresponds to a classical extremal path. The post explains the two main theorems, the four physical origins of multi-valued action (spatial constraints, Hamiltonian singularities, closed configuration manifolds, initial distributions), and exact reproductions of the double-slit experiment, particle in a box, harmonic oscillator, and hydrogen atom. It also covers the authors' control-theory background, the paper's 21-month arXiv iteration, comparison with Feynman path integrals and Bohmian mechanics, EPR/Bell implications, and remaining open questions in interpretation and many-body scalability.

Introduction

This forum post reviews the 2025 paper 'On computing quantum waves exactly from classical action' by Winfried Lohmiller and Jean-Jacques Slotine (MIT Nonlinear Systems Laboratory), published in *Proceedings of the Royal Society A*, 482(2336), 20250413. The central claim: the Schrödinger equation can be constructed exactly from classical action — no quasi-classical approximations, no infinite path-integral time slicing.

Key points

Theorem 1 (multi-valued classical fields): On a constrained manifold G^N, a multi-valued local minimal-action field Φ_j and a classical density field ρ_j naturally coexist. Φ_j satisfies the Hamilton-Jacobi equation ∂Φ_j/∂t + H(x, ∇Φ_j) = 0, while ρ_j satisfies the continuity equation ∂ρ_j/∂t + ∇·(ρ_j v_j) = 0, with velocities defined via the M-gradient of the action and a metric-weighted Laplacian (M-Laplacian) capturing curvature effects.
Theorem 2 (exact wavefunction construction): Ψ = ∑_j √ρ_j · exp(i Φ_j / ℏ) maps classical extremal paths and densities onto a full quantum wavefunction, exactly for arbitrary finite ℏ — not an approximation. No Bohmian quantum potential Q is required.
Four sources of multi-valuedness: spatial inequality constraints (double slit), Hamiltonian singularities (Coulomb potential at r=0), closed configuration manifolds (e.g., spin on S^3), and extended initial distributions. Branch points (Δ_M Φ_j → ±∞) create or remove branches, offering a classical picture of measurement collapse.
Exact reproductions: the double-slit pattern from just two paths (vs. Feynman's infinite zig-zags), particle-in-a-box quantization E_n = n²π²ℏ²/(2mL²), the harmonic oscillator with Hermite polynomials via Poisson summation and the Mehler kernel, and the hydrogen atom's energy levels E_k = −MG²/(2ℏ²k²) via Coulomb singularity branching and SO(4) symmetry.
Relation to prior work: the result closes a loop begun with Schrödinger (1926) and Madelung (1926), and extends past Van Vleck propagators, WKB, de Broglie-Bohm theory, Gutzwiller trace formulas, and Berry phase. Unlike WKB/Gutzwiller asymptotics, the construction is exact; unlike Bohmian mechanics, classical trajectories are unmodified and may cross.
Relation to Feynman path integrals: complementary rather than competing. L&S avoid the curse of dimensionality in constrained geometries, with the bottleneck shifted to enumerating extremal paths; path integrals remain dominant for QFT free fields.

Authors and publication history

Both authors come from control theory rather than physics: Slotine founded contraction theory; Lohmiller specializes in differential-geometric treatment of Hamilton-Jacobi-Bellman equations. The paper evolved through 11 arXiv versions (arXiv:2405.06328, from May 2024) over 21 months before Royal Society publication, gaining Maxwell equation derivations, EPR hidden-variable analysis, and derivations of four quantum postulates along the way. It was first presented at the Helgoland 2025 centenary conference.

EPR and Bell theorem

The paper interprets EPR entanglement using quaternionic spinors as hidden variables, matching the singlet correlation −n₁ᵀn₂ exactly. Bell inequality violations arise because spinor detectors (rather than scalars) are measured. However, the framework does not truly evade Bell: it requires abandoning locality or measurement independence, and initial probability input remains necessary (as Peter Morgan has noted).

Assessment

The post frames the work as a computational breakthrough rather than a replacement for quantum ontology: quantum behavior 'grows' from the non-uniqueness of classical paths. Practical value is seen in teaching (engineers can reach quantum mechanics via classical intuition) and in constrained low-body systems, with open challenges in many-body scalability and QFT loop renormalization. Media coverage (MIT press release, April 2026; Interesting Engineering) was enthusiastic but major journals offered no dedicated commentary.

References

1. Lohmiller, W. & Slotine, J.-J. (2025). On computing quantum waves exactly from classical action. *Proceedings of the Royal Society A*, 482(2336), 20250413. 2. Schrödinger, E. (1926). Quantisierung als Eigenwertproblem. *Annalen der Physik*, 384(4), 361-376. 3. Madelung, E. (1926). Quantentheorie in hydrodynamischer Form. *Zeitschrift für Physik*, 40(3-4), 322-326. 4. Feynman, R. P. (1948). Space-time approach to non-relativistic quantum mechanics. *Reviews of Modern Physics*, 20(2), 367-387. 5. Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of "hidden" variables. *Physical Review*, 85(2), 166-179.