Is Quantum Mechanics a proper subset of Classical Mechanics?

TL;DR

This work reframes quantum mechanics as a linearized, decidable projection of a richer classical variational geometry encoded by Hamilton's principal function $S_H$ and its Jacobi reduction $S_J$, arguing that global solvability of $S_H$ is undecidable and that quantum phenomena such as wavefunction collapse, entanglement, and spin arise from representational limitations rather than intrinsic indeterminism. By connecting the Hamilton–Jacobi formalism to QM through semiclassical time evolution, it links undecidability and number-theoretic constraints (KAM/Diophantine) to foundational puzzles and analyzes empirical phenomena like Leggett inequality violations, quantum scars, and chaotic systems (e.g., Hyperion) as residual classical structure within quantum regimes. The paper also proposes concrete experimental tests using lateral double quantum dots and introduces the broader program of Post-Hamiltonian Representation Theory to explore alternative continuations of classical variational principles (e.g., nonlinearity, $p$-adic quantization, sheaf/cohomology frameworks) that could preserve more classical complexity. Overall, it argues that unitarity and separability are emergent within decidable sectors, and that the quantum-classical boundary is governed by logical and computational limits rather than Planck-scale discreteness, with potential implications for quantum technologies and foundational physics.

Abstract

Quantum mechanics is widely regarded as a complete theory, yet we argue it is a tractable projection of a deeper, computationally-inaccessible classical variational structure. By analyzing the coupled partial differential equations of the Hamilton type 1 principal function, we show that classical action-based dynamics are generally undecidable, paralleling spectral gap undecidability in quantum systems. In near Kolmogorov-Arnold-Moser systems, stability hinges on Diophantine conditions that are themselves undecidable, limiting predictability via arithmetic logic rather than randomness. Phenomena like spin 3/2 systems and larger, quantum scars and Leggett inequality violations support this view, naturally explained by time symmetric classical action. This framework offers a principled resolution to the long standing dichotomy between unitarity and entanglement by deriving both as emergent features of a tractable rendering from a fundamentally non-separable classical variational geometry. Collapse and decoherence arise from representational limits, not ontological indeterminism. We propose an explicit experimental test using lateral double quantum dots to detect predicted deviations from standard quantum coherence at the classical chaos threshold. This reframing suggests the classical quantum boundary is set by computability and not by the Planck constant. Implications for quantum computing and quantum encryption are discussed.

Figures (5)

• Figure 1: Ray depiction of various $S_J$ generators with different constant momenta building $S_H$, the HPF.
• Figure 2: Similar to figure \ref{['fig:ray']} but constant momenta now "radiates" out ripples in configuration space than interfere in the shape of the HPF.
• Figure 3: Flowchart illustrating the transition from constants of motion $\alpha$ in Jacobi's type-2 generating function to the wavefunction $\psi(q, t)$ and its probabilistic interpretation via eigenvalue decomposition.
• Figure 4: Conceptual overlap between classical type-2 mechanics, Gottesman-Knill quantum operations, and the limitations of representational frameworks.
• Figure 5: Formation of caustics from intersecting classical trajectories. In the chaotic regime, characteristics cross, causing $S(q,t)$ to become multivalued. Quantum wavefunctions regularize this behavior by encoding phase rather than action directly.

Theorems & Definitions (4)

• Theorem 1
• proof : Proof:
• Theorem 2
• proof