On the Use of Minimum Volume Ellipsoids and Symplectic Capacities for Studying Classical Uncertainties for Joint Position-Momentum Measurements

TL;DR

This paper explores how minimum volume ellipsoids and symplectic capacities can be used to analyze classical uncertainties in joint position-momentum measurements, revealing parallels with quantum mechanics.

Contribution

It introduces a novel approach linking classical uncertainty measures with symplectic geometry, highlighting similarities to quantum uncertainty relations.

Findings

Classical uncertainties follow relations akin to non-standard quantum mechanics.

Symplectic capacity of covariance ellipsoids serves as an effective uncertainty measure.

Minimum volume ellipsoid estimators provide insights into phase space uncertainties.

Abstract

We study the minimum volume ellipsoid estimator associates to a cloud of points in phase space. Using as a natural measure of uncertainty the symplectic capacity of the covariance ellipsoid we find that classical uncertainties obey relations similar to those found in non-standard quantum mechanics.