On computing quantum waves exactly from classical action

Quantum mechanics works. Nobody disputes that. The predictions are exact, the experiments confirm them, and the formalism has held for a century. What nobody agrees on is what it means. Lohmiller and Slotine have a concrete answer to that question — not a philosophical position, but a mathematical theorem: you can construct the exact Schrödinger wave function from nothing but classical least action and classical probability density. No approximation. No extra postulates. No sum over infinite non-classical paths. Let that sit for a moment, because it changes the shape of the interpretive debate considerably. The three major pictures of quantum mechanics have always made the same predictions while disagreeing on what is actually happening. The Copenhagen view says the wave function is a calculational tool — there are no particle trajectories until measurement forces a collapse, and collapse is simply a primitive axiom you accept. The Einstein-Bohm picture insists trajectories are real: particles follow determined paths guided by a quantum potential, and the statistical appearance of quantum mechanics reflects a distribution over hidden initial conditions. Feynman's path integral sits somewhere in between — it keeps the wave, abandons trajectories, but replaces the Schrödinger propagator with an integral over an infinity of wildly non-classical zigzag paths, each weighted by the phase of its action. All three reproduce the same numbers. None of them obviously explains the others. What Lohmiller and Slotine show is that a fourth construction produces identical wave functions and probability distributions — and that this fourth construction is, in a precise sense, the most economical of the four. It uses only a finite or countable set of classical extremal paths, the ones that actually satisfy the Hamilton-Jacobi variational condition, weighted by classical probability densities propagated along those paths. And it is exact. The central object in their construction is what they call a multi-valued classical action. For most quantum problems, the Hamilton-Jacobi equation has not one but several distinct least-action solutions that reach the same final point at a given time. These multiple branches arise for two reasons: multiplicity in initial conditions — a spread of initial positions or momenta already generates a family of extremal rays — and branch points, where the topology or geometry of the problem forces the creation of new branches. Branch points are generated by a multiply connected manifold, as in the double-slit experiment; by spatial inequality constraints, as in a particle in a box; or by a singularity in the potential, as in the Coulomb problem. Theorem 2.4 in the paper packages this precisely: the action is J-valued, where J indexes every classical branch arising from all initial conditions and all branch points encountered during the evolution. The quantum wave emerges from Theorem 3.2. For each classical branch, you form a complex amplitude: the phase is the classical action divided by ħ, and the magnitude is the square root of the classical probability density transported along that branch by the classical continuity equation. Sum over all branches and you have the Schrödinger wave. In words: ψ at position x and time t equals the sum over j of the square root of ρ_j times e to the i φ_j over ħ. This is not a WKB approximation or a Van Vleck semiclassical estimate — those approaches add Maslov phases and determinants as corrections. Theorem 3.2 is exact. Computing ρ_j along each extremal path means solving the classical continuity equation on that branch: you follow an infinitesimal packet of classical probability through the determined classical flow generated by φ_j, tracking how that packet expands or contracts as it reaches the measurement point. The transported density sets the amplitude weight for that branch. The entire construction is classical mechanics applied branch by branch. The canonical examples make this vivid. In the double-slit experiment, the two holes are the branch points. Every pixel on the detection screen is reached by exactly two classical straight-line paths, one from each slit. The classical action for branch j is φ_j over ħ equals p₀ r_j over ħ, where r_j is the geometric distance from slit j to the screen point. The classical density behind each slit falls as r_j to the minus two. Summing the two branch contributions — the square root of r_j squared times e to the i p₀ r_j over ħ — gives the standard Fraunhofer two-slit wave, exactly. Feynman's infinite zigzag sum reduces to two classical paths. The Aharonov-Bohm extension adds a magnetic potential in the region behind the wall with curl A equal to zero: the classical paths are unchanged, but each action term acquires the additional phase from the line integral of A along that path, reproducing the Aharonov-Bohm phase shift without modifying any trajectory. The particle in a box works because elastic reflections at the rigid walls generate new branches with each reflection. The quantization condition emerges from Lemma 3.4: for the density to remain consistent across the periodic multipath structure, you need two Lp over ħ equal to two π k for integer k, which gives p equals π k ħ over L and the familiar discrete energy sequence proportional to k squared over mass times L squared. The square-root classical density on each branch is constant at the value of the square root of two over L, and the normalized standing-wave eigenfunctions come out directly. For the harmonic oscillator, the classical Laplacian acting on the action gives N ω cot of ω t, and the resulting classical density takes the form of M ω over two i π ħ times sin of ω t, raised to the power N over two. Feeding this density and the classical harmonic action into Theorem 3.2 produces the eigenfunctions exactly — Hermite-polynomial-weighted Gaussians — with eigenvalues E_k equals ħ ω times (k plus N over two). No process noise is added to the classical path, in contrast to Feynman's stochastic discretization. The hydrogen atom is the most striking case. The Coulomb singularity at the origin is the branch point. Lohmiller and Slotine use two-valued quaternion coordinates to handle it, which generates two families of counter-rotating Kepler orbits as the classical least-action branches. Periodicity of those orbits introduces additional branches, and Lemma 3.4 quantizes the Kepler rotation rate: ω equals G over ħ k for integer k. The full branch index set is the initial quaternion coordinate in R to the power of four, times a spin index up or down, times k in the positive integers. Combining the classical densities along these quantized Kepler multipaths reproduces the three-dimensional Coulomb eigenfunctions. The paper gives explicit densities: for the one S state, ρ equals one over the square root of π times e to the minus r over two; for the two P state, ρ involves x₂ squared times e to the minus r over two; and the three-dimensional combinations involve Hermite polynomial factors. These are not approximations to the standard hydrogenic orbitals — they are the standard hydrogenic orbitals, derived from classical Kepler orbit structure. Now the interpretive question the listener came here to have answered: what does this framework say about measurement, collapse, and entanglement? Wave collapse, in this framework, is Lemma 3.3. A measurement is modeled as a unitary map from configuration coordinates to measurement coordinates. When the measurement yields a specific result y_k, the classical square-root density in measurement coordinates collapses to a Dirac impulse at y_k. The pre-measurement wave, built as the sum over J branches, maps through U into an eigenwave concentrated at y_k, multiplied by an overall phase sum from the classical actions. Normalization removes that phase. In plain terms: collapse happens because the classical density becomes singular in the measurement basis — not because of a separate postulate about observation disturbing the system, but as a mechanical consequence of how the classical density behaves under a unitary projection. The wave doesn't mysteriously jump; the classical density carried by the multipath structure sharpens to a point. One important caveat the authors are explicit about: Born's rule — the probability of measuring y_k equals the squared amplitude at that eigenvalue — remains a postulate in their framework. The construction produces the correct density over x and t, but the interpretive step connecting density to probability is not derived. Entanglement follows from a similar logic. For P classically decoupled spinning particles, the total Hamiltonian and total action are simply sums — H equals the sum over p of H_p, and φ equals the sum over p of φ_p — and the total wave is a tensor product of individual spinors. Lohmiller and Slotine construct the Einstein-Podolsky-Rosen singlet state explicitly: one over the square root of two times ψ-up tensor ψ-down minus ψ-down tensor ψ-up, with each ensemble carrying probability one-half and a common unknown initial spin direction. Measurements at two spatially separated detectors are geometric projections of the initial classical eigenspinors onto the local filter directions. The resulting correlation matches the quantum formula confirmed by experiment. The paper's treatment of Bell's theorem deserves attention. Bell's inequality is derived from a binary detector model that quantizes the Bloch sphere to plus or minus one at each relative angle. Lohmiller and Slotine show that this binary quantization is classically valid only at relative Euler angles equal to integer multiples of two π. Replace the binary detector with a full classical spinor detector — a continuous spinor representation on the Bloch sphere — and plug it into Bell's hidden-parameter integral, and you recover the quantum correlation formula rather than Bell's inequality. The authors argue that the classical correlation of two spinning particles, when the detector is described by full classical spinors rather than binary outputs, is exactly the quantum formula. Whether this resolves or sidesteps the deeper issue Bell's theorem raises is a question they leave open, but the operational claim is precise. The contrast with Copenhagen and Bohm now has clean edges. Copenhagen treats collapse as a primitive axiom and offers no account of why it happens or when. The pilot-wave Bohmian picture gives real trajectories but requires a quantum potential as an additional ingredient — a nonlocal guiding field that has no classical precursor and must be postulated separately. The Lohmiller-Slotine framework derives both collapse and the entanglement correlations from the classical multipath structure and classical density evolution alone, without introducing a quantum potential or a new measurement axiom. The relativistic extension shows the framework is not a non-relativistic trick. Replacing the scalar action with a relativistic action on R to the power of four — proper-time parametrization, covariant Hamilton-Jacobi equation — yields the Klein-Gordon equation. The classical density uses the proper four-dimensional volume element, so the resulting quantum density matrix remains Lorentz-invariant. For spin-half particles, the scalar action is replaced by a two by two pure-imaginary quaternion action, and Lemma 3.1 maps this to the Pauli equation. A four by four pure-imaginary quaternion action maps to a Dirac spinor via the same lemma, with Pauli and Dirac matrices appearing from the algebraic decoupling of the quaternionic substitution. Setting the rest mass to zero gives the relativistic eikonal equation — geometric ray optics — and because the relevant metric M prime is the metric tensor satisfying the Einstein field equations, the geodesic limit connects the construction to general relativity. Setting the wave to a real four by one vector potential recovers Maxwell's equations, both homogeneous and, with a current density term, non-homogeneous. Every major wave equation in relativistic quantum theory falls out of the same theorems with appropriate substitutions. On computational cost, the comparison with Feynman path integrals is direct and concrete. Feynman's kernel integrates over an infinite, time-sliced family of stochastic paths in function space — that is the formal object you approximate when you do lattice field theory or numerical path integration. The Lohmiller-Slotine construction requires only the J-valued set of classical extremal branches from Theorem 2.4, plus the classical densities computed by the continuity equation along each branch. The double-slit problem takes two classical paths. The harmonic oscillator reproduces Feynman's Gaussian integrals without discretization or process noise. For hydrogen, the alternative in the literature is the Duru-Kleinert time-reparametrized path integral, a technically demanding construction; Lohmiller and Slotine obtain the same Coulomb eigenwaves from the structured set of quantized Kepler orbits indexed by k and spin. They describe their approach as using "only a minimal subset of classical paths" and add explicitly that "there is no process noise added to the classical path in contrast to the Feynman path integral." Fewer paths, exact result, no stochastic regularization. One connection the listener may have expected but should know is absent: the paper contains no information-geometry content. There is no Fisher information metric, no Cramér-Rao bound, no metric on the space of actions or densities. The geometry that appears is the classical Riemannian structure of the configuration manifold and the quaternionic geometry of spinor space — but information geometry as a framework is not developed or invoked. If you came hoping for a bridge to the Amari-Chentsov theory or to geometric quantum mechanics via statistical manifolds, this paper does not provide it. Lohmiller and Slotine close by naming what their result actually establishes: the fundamental quantum postulates — the existence of a wave function, its propagation by the Schrödinger equation, and wave collapse at measurement — are derived from the classical multipath theorem, not assumed. Born's rule remains a postulate. The construction extends to Klein-Gordon, Pauli, and Dirac dynamics and is a multi-particle result. They flag several concrete research directions: deriving complex actions for general nonlinear potentials where perturbation theory has been the only tool; applying the framework to hybrid quantum-classical systems; obtaining exact quantum simulations from classical systems whose extremal paths match; and using the differentiability of classical paths in machine-learning approaches to computational quantum chemistry. The quantum-classical boundary, in this account, is not a wall separating two incompatible descriptions of nature. It is a fold in the action structure — a place where a single classical least-action field becomes multi-valued, and where coherently summing those values with their classical densities produces interference, quantization, and entanglement. That is a sharper and more actionable claim than any of the three interpretive pictures it sits alongside. This lecture was created by ennepō. Go to https://ennepo.ai to Discover, Create and Follow the latest research in your field. Read when you can. Listen when you want to.