A complex-linear reformulation of Hamilton--Jacobi theory and the emergence of quantum structure

TL;DR

This work introduces a minimal complex embedding of the Hamilton-Jacobi (HJ) ensemble into a single complex field $\psi = R e^{i S/\kappa}$, yielding a linear Schrödinger-type evolution that unifies classical and quantum dynamics. The two mild structural requirements—first-order time evolution and absence of nonlinear gradient terms—force a unique embedding, leading to the HJS system: a deformed HJ equation with a quantum-like potential $Q[R] = -\frac{\kappa^{2}}{2m}\frac{\nabla^{2}R}{R}$ alongside the continuity equation, which together are equivalent to $i\kappa\partial_t\psi = -\frac{\kappa^{2}}{2m}\nabla^{2}\psi + V\psi$. When $\mathrm{Re}(\kappa) \neq 0$, standard quantum structures—superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born’s rule, and unitary evolution—emerge automatically as consistency requirements, while the classical limit $|\kappa|\to 0$ recovers classical HJ dynamics. The framework further extends to internal degrees of freedom (spin) as an internal fiber and discusses a structural time-asymmetry parameter $\theta = \mathrm{Im}\kappa / \mathrm{Re}\kappa$, highlighting a unified, wave-based representation of classical dynamics with quantum-like features as emergent, not postulated, properties.

Abstract

Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density $ρ= R^{2}$ coupled to Hamilton's principal function $S$. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair $(R,S)$ into a single complex field. Starting from a completely general complex ansatz $ψ= f(R,S) e^{i g(R,S)}$, and imposing two minimal structural requirements, we obtain a unique map $ψ= R e^{iS/κ}$ together with a linear HJS equation whose $|κ| \to 0$ limit reproduces the HJ formulation exactly. Remarkably, when $\mathrm{Re}(κ)\neq 0$, essential features of quantum mechanics, including superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule, and unitary evolution, arise naturally as consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.

Figures (3)

• Figure 1: Schematic summary of the framework introduced here. A single structural theory---the HJS formulation---contains classical mechanics as the limit $|\kappa|\to 0$ and quantum mechanics as the regime where $\mathrm{Re}\,\kappa\neq 0$. Both emerge from the same underlying complexified HJ structure.
• Figure 2: Structural pipeline of the HJS formulation. Classical HJ dynamics and ensemble continuity admit a unique complex embedding. This minimal complexification yields the linear HJS/Schrödinger equation, from which standard quantum structures---superposition, operators, commutators, unitary evolution, and uncertainty---emerge as consistency requirements.
• Figure 3: Visualization of the ensemble dynamics for the one-dimensional harmonic oscillator. Left panel: Time evolution of the position expectation value $\langle q\rangle(t)$ (solid line) together with the width $\langle q\rangle(t)\pm\Delta q(t)$ for several representative values of $\kappa$. The central trajectory is independent of $\kappa$ and coincides with the classical solution, while the width increases with $\kappa$. Right panel: Time evolution of the momentum expectation value $\langle p\rangle(t)$ (solid line) together with the operator-defined dispersion $\langle p\rangle(t)\pm\Delta p_{\mathrm{op}}(t)$. Although $\langle p\rangle(t)$ is again purely classical, the width captures the $\kappa$-dependent momentum fluctuations relevant for the uncertainty relation. The HJ momentum dispersion $\Delta p_{\mathrm{HJ}}(t)$, which vanishes initially and follows the complementary $\sin^2 t$ behavior, is discussed in the Supplementary Information, section HJ versus operator momentum fluctuations, but not shown here. In the numerical illustrations we choose $\varepsilon=\sigma=0.4$ and $p_0=1$, which provide a smooth, localized initial ensemble and a clearly visible classical trajectory.