An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics

TL;DR

This work demonstrates an anti-Gleason phenomenon in classical mechanics by constructing a non-linear median quasi-state on the 2-sphere ${\mathbb{S}}^2$ and using it to derive a robust lower bound on the error of simultaneous measurement of non-commuting classical observables. It provides a concrete geometric construction via Reeb graphs, proves monotonicity and quasi-linearity for commuting observables, and exhibits genuine nonlinearity with explicit examples. A pointer-measurement model yields the bound $\Delta_\infty(F_1,F_2) \ge \tfrac{1}{2}\Pi(F_1,F_2)$, connecting the quasi-state to measurement theory. The discussion links this classical structure to quantum spin through coherent-state quantization and suggests broader generalizations to higher-dimensional symplectic manifolds, highlighting a deep interplay between symplectic topology and topological quasi-states.

Abstract

We report on an "anti-Gleason" phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a non-linear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commutative functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies in the intersection of two seemingly remote mathematical theories - symplectic topology and the theory of topological quasi-states. We use this quasi-state to estimate the error of the simultaneous measurement of non-commuting Hamiltonians.

Figures (2)

• Figure 1: Round sphere. The Reeb graph $\Gamma_F$ is a segment.
• Figure 2: Deformed sphere. The Reeb graph $\Gamma_F$ is a tripod.

Theorems & Definitions (1)

• Theorem 1.1: Gleason