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, 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.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, 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.

quant-pharXiv:2601.22697Feb 2, 20260 citations

Hamilton-Jacobi Quantization Of Christ - Lee Model

F. M. Al Saba, Hassan Harb, Sami I. Muslih

مجلة جامعة الأقصى سلسلة العلوم الإنسانيةJan 1, 20030 citations

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, 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.