A Generalized Analytical Mechanics in which Quantum Phenomena Appear

TL;DR

This paper introduces a generalized mechanics framework that unifies classical and quantum phenomena by modifying the Hamilton-Jacobi equation to a higher-order form, resulting in quantized energies and testable predictions.

Contribution

It proposes a novel mechanics based on extended diffeomorphism actions, transforming the Hamilton-Jacobi equation into a third-order PDE, capturing quantum effects within a classical-like framework.

Findings

Energy quantization matches quantum mechanics

Particle distributions agree with experiments within uncertainty

Provides a testable alternative to quantum mechanics

Abstract

Currently, dynamics of a massive macroparticle is given by classical analytical mechanics (CM), while that of a massive micro one is given by quantum mechanics (QM). We propose a mechanics effective for both: We transform, under coordinate transformation, the covariant tensor of order two underlying the kinetic energy term of the Hamilton-Jacobi (H-J) eq. of CM, not with the action of the diffeomorphism group, but with an action of an extended diffeomorphism group. Then, the H-J eq., a first-order partial differential eq., is modified to a third-order one. The Euler-Lagrange eq. of CM, a second-order ordinary differential eq., related to the H-J eq. through action integral of the least action principle is accordingly modified to a fourth-order one. We thus obtain a mechanics which, because of the higher-order eqs., accommodates phenomena corresponding to quantum phenomena. Energy of a particle in a confining potential is quantized to the same values as those of QM. Particle distribution in an ensemble disagrees with that of QM, but agrees with experiments within experimental uncertainty. The mechanics therefore is a testable alternative to QM.