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

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.

💡 Research Summary

The paper introduces a novel reformulation of classical Hamilton‑Jacobi (HJ) theory by embedding the real amplitude R and phase S into a single complex field ψ. Starting from the most general complex ansatz ψ = f(R,S) e^{i g(R,S)} and imposing two minimal structural requirements—(i) the evolution equation must remain first order in time and (ii) nonlinear gradient terms such as (∇ψ/ψ)² must be absent—the authors show that the only admissible embedding is ψ = R e^{iS/κ}, where κ is an arbitrary complex constant. Substituting this map into the coupled HJ‑continuity system yields a deformed HJ equation
∂ₜS + (∇S)²/2m + V + Q