The Non-commutative Robertson-Schr\"{o}dinger Uncertainty Principle

TL;DR

This paper explores the non-commutative Robertson-Schrödinger uncertainty principle by analyzing the covariance matrix in non-commutative quantum mechanics, establishing bounds on symplectic capacities, and providing a concrete example.

Contribution

It introduces a novel approach to the uncertainty principle using symplectic capacity in non-commutative phase spaces and derives related inequalities.

Findings

Bounds on symplectic capacities for non-commutative phase spaces

Inequalities between capacities in different non-commutative regimes

A constructive example supporting theoretical predictions

Abstract

We investigate properties of the covariance matrix in the framework of non-commutative quantum mechanics for an one-parameter family of transformations between the familiar Heisenberg-Weyl algebra and a particular extension of it. Employing as a measure of the Robertson-Schr\"{o}dinger uncertainty principle the linear symplectic capacity of the Weyl ellipsoid (and its dual), we determine its corresponding bounds. Inequalities between the capacities for non-commutative phase-spaces are established. We also present a constructive example based on a simple model to justify our theoretical predictions.