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

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 $\rho = R^{2}$ coupled to Hamilton's principal function $S$. Here we develop a complementary formulation, the Hamilton-Jacobi-Schr\"odinger (HJS) theory, by embedding the pair $(R,S)$ into a single complex field. Starting from a completely general complex ansatz $\psi = f(R,S)\, e^{i g(R,S)},$ and imposing two minimal structural requirements, we obtain a unique map $\psi = R\, e^{iS/\kappa}\, $ together with a linear HJS equation whose $|\kappa| \to 0$ limit reproduces the HJ formulation exactly. Remarkably, when $\mathrm{Re}(\kappa)\neq 0$, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.