This paper provides mathematically self-contained formulations in the complex-time (kime) representation for three open problems in the foundations of classical mechanics:
- Extension of the classical entropic uncertainty principle to non-canonical variables and multiple degrees of freedom;
- Characterization of coordinate-invariant measures and entropies, addressing why continuous physical quantities must be paired for invariant entropy to exist;
- Construction of a classical relativistic directional degree of freedom (analogous to a spin-1/2 system).
Throughout, the kime phase is statistically interpreted as a latent circular random variable whose law $\Phi$ models 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, along with a sharp circular Fisher-information inequality exactly saturated by von Mises laws; (ii) Prove an exact non-canonical uncertainty relation where 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, isolating 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.
Blogger's Review: This paper introduces the kime representation, offering a novel mathematical framework for addressing several open problems in classical mechanics, particularly regarding entropy uncertainty and multi-degree-of-freedom systems. It showcases profound theoretical implications that warrant further investigation and exploration.