arXiv:2601.22697

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

Yong Zhang

quant-phhep-thSubmitted Jan 30, 2026• Announced Feb 2, 2026• 0 citations

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.

Comments: 9+5 pages, 3 figures, 1 table

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arXiv:2601.22697

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

Yong Zhang

quant-phhep-thSubmitted Jan 30, 2026• Announced Feb 2, 2026• 0 citations

arXiv HTML PDF

Abstract

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.

Comments: 9+5 pages, 3 figures, 1 table

Related papers

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Frank J. Tipler

I show that the classical Hamilton-Jacobi (H-J) equation can be used as a technique to study quantum mechanical problems. I first show that the the Schrödinger equation is just the classical H-J equation, constrained by a condition that forces the solutions of the H-J equation to be everywhere $C^2$ . That is, quantum mechanics is just classical mechanics constrained to ensure that ``God does not play dice with the universe.'' I show that this condition, which imposes global determinism, strongly suggests that $ψ^*ψ$ measures the density of universes in a multiverse. I show that this interpretation implies the Born Interpretation, and that the function space for $ψ$ is larger than a Hilbert space, with plane waves automatically included. Finally, I use H-J theory to derive the momentum-position uncertainty relation, thus proving that in quantum mechanics, uncertainty arises from the interference of the other universes of the multiverse, not from some intrinsic indeterminism in nature.I show that the classical Hamilton-Jacobi (H-J) equation can be used as a technique to study quantum mechanical problems. I first show that the the Schrödinger equation is just the classical H-J equation, constrained by a condition that forces the solutions of the H-J equation to be everywhere $C^2$. That is, quantum mechanics is just classical mechanics constrained to ensure that ``God does not play dice with the universe.'' I show that this condition, which imposes global determinism, strongly suggests that $ψ^*ψ$ measures the density of universes in a multiverse. I show that this interpretation implies the Born Interpretation, and that the function space for $ψ$ is larger than a Hilbert space, with plane waves automatically included. Finally, I use H-J theory to derive the momentum-position uncertainty relation, thus proving that in quantum mechanics, uncertainty arises from the interference of the other universes of the multiverse, not from some intrinsic indeterminism in nature.

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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

C^2

. That is, quantum mechanics is just classical mechanics constrained to ensure that ``God does not play dice with the universe.'' I show that this condition, which imposes global determinism, strongly suggests that

ψ^*ψ

measures the density of universes in a multiverse. I show that this interpretation implies the Born Interpretation, and that the function space for

ψ

is larger than a Hilbert space, with plane waves automatically included. Finally, I use H-J theory to derive the momentum-position uncertainty relation, thus proving that in quantum mechanics, uncertainty arises from the interference of the other universes of the multiverse, not from some intrinsic indeterminism in nature.

S_p(x)=\langle p,x\rangle

restricted to the mass shell form a complete integral of the Hamilton–Jacobi equation, while their exponential images

\eta_p=\exp\!\left(\frac{i}{\hbar}S_p\right)

constitute a Schwartz basis of the tempered state space. On each spectral fiber, both classical and quantum equations reduce to the same Einstein dispersion relation. We prove that the relativistic Schrödinger equation is precisely the Schwartz–von Neumann S–linear extension of the classical energy relation from certainty momentum states to arbitrary tempered superpositions. In the presence of scalar potentials, the Hamiltonian arises as a mixed (momentum-diagonal and position-diagonal) extension, showing that the extension principle is not restricted to the free case. We further demonstrate that exact quantum dynamics cannot, in general, be represented by a single exponential phase

\exp\!\left(\frac{i}{\hbar}S\right)

unless

S

is affine in space. Instead, quantum evolution is obtained by S–superpositions of the principal exponential family associated with a complete integral of the Hamilton–Jacobi equation. In this sense, classical elimination of parameters is replaced by linear spectral superposition. Geometrically, the exponential mapping transforms the flat affine space of Minkowski generators into a curved manifold of principal waves on which the nonlinear Hamilton–Jacobi flow pushes forward to a linear unitary Schrödinger flow. Through de Broglie–Maxwell isomorphisms, the construction extends to complex electromagnetic-like fields, preserving translation representation, dispersion relations, and polarization geometry. The results suggest that, for translation-invariant systems, quantization may be understood as an infinite-dimensional complex linearization of a classical certainty space rather than as a semiclassical approximation. Within the tempered-distribution setting, relativistic quantum dynamics emerges as the superpositional completion of a classical complete integral.

\phi = S/\hbar

, we interpret physical states as finite-bandwidth phase signals. We demonstrate that the Heisenberg inequality follows immediately from the Fourier localization limit when identifying

p=\hbar k

. Next, we show that the decomposition

\psi = R e^{iS/\hbar}

leads to an equivalent description in terms of a continuity equation and a Hamilton--Jacobi equation modified by a quantum term. The Schrödinger equation then appears as the compact linear form of this dynamics. The result does not constitute a complete microscopic derivation of quantum mechanics, but establishes that its formal structure is compatible with general principles of signal propagation, bandwidth, and phase coherence.