Kime-Representation Formulations of Three Open Problems in the Foundations of Classical Mechanics: Uncertainty, Invariant Entropy, and Directional Degrees of Freedom

2026-07-08

AI Summary

This work addresses three foundational problems in classical mechanics within a unified framework based on complex-time (kime) representation: the entropy uncertainty principle for non-canonical multi-degree-of-freedom systems, the existence conditions for coordinate-invariant entropy, and the construction of classical relativistic directional degrees of freedom analogous to spin-1/2 systems. By establishing a symplectic isomorphism between the kime cone and the action-angle variables of single-degree-of-freedom phase space, and by treating kime phase as a circular random variable, the study integrates statistical and dynamical perspectives. Key contributions include deriving a sharp entropy uncertainty relation with a geometric interpretation of its non-canonical correction term, proving the aggregability of multi-degree entropy bounds, uncovering the deep mechanism requiring physical observables to appear in conjugate pairs for invariant entropy, and demonstrating that kime phase diffusion drives entropy to monotonically increase toward the Haar-uniform distribution.

Abstract

We give mathematically self-contained formulations, in the complex-time (kime) representation, of three open problems from the foundations of classical mechanics: (I) the extension of the classical entropic uncertainty principle to non-canonical variables and to multiple degrees of freedom; (II) the characterization of coordinate-invariant measures and entropies, i.e., the question of why continuous physical quantities must be paired for an invariant entropy to exist; and (III) the construction of a classical relativistic directional degree of freedom (a classical analogue of a spin-1/2 system). Throughout, the kime phase is interpreted {statistically as a latent circular random variable whose law Φmodels the intrinsic trial-to-trial variability of repeated, identically controlled experiments indexed by the kime magnitude. The mathematical bridge is an exact symplectic identification of the kime cone with the action-angle chart of a one-degree-of-freedom phase space, under which the kime measure is the Liouville measure and the phase law becomes the angular conditional of a Liouville density. Specifically, we (i) prove a sharp entropic uncertainty relation on the kime cylinder whose extremal family is von Mises x Gaussian, together with a sharp circular Fisher-information inequality saturated exactly by von Mises laws; (ii) prove an exact non-canonical uncertainty relation in which the correction term is the geometric mean of the Poisson bracket, clarifying the conjectured role of the expected bracket; (iii) prove aggregate multi-degree-of-freedom bounds via the Williamson normal form and Fischer's inequality, and isolate the per-degree-of-freedom refinement as a precise open problem of symplectic Schur-Horn type; (iv) prove that diffusion of the kime phase produces monotone entropy growth with the equipartitioned (Haar-uniform) phase law.

Problem

• uncertainty
• invariant entropy
• directional degrees of freedom
• classical mechanics
• non-canonical variables

Innovation

• kime representation
• entropic uncertainty
• symplectic geometry
• invariant entropy
• directional degree of freedom