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 $\ddot{I}=R^2$ 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 $\ddot{I}ˆ=f(R,S)e^{ig(R,S)},$ and imposing two minimal structural requirements, we obtain a unique map $\ddot{I}ˆ=R{e}^{iS/\hat{I}º}$ together with a linear HJS equation whose $|\hat{I}º|\to 0$ limit reproduces the HJ formulation exactly. Remarkably, when $\mathrm{Re}(\hat{I}º)\ne 0$, 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.

10+6 pages, 4 figures, 1 table. Revised version with improved presentation, clarified Born-rule discussion, a new schematic figure, and a streamlined discussion of the complex-$\hat{I}º$ branch