Violation of the Robertson-Schr\"odinger uncertainty principle and non-commutative quantum mechanics

TL;DR

This paper explores how violations of the Robertson-Schr"odinger uncertainty principle can indicate non-commutative extensions of quantum mechanics, showing a deep connection between uncertainty violations and algebraic deformations.

Contribution

It demonstrates that Gaussian states violating the uncertainty principle correspond to states in non-commutative quantum mechanics, establishing a two-way relationship.

Findings

Gaussian states violating the uncertainty principle are quantum states of non-commutative quantum mechanics

All non-commutative quantum mechanics extensions have states that violate the uncertainty principle

Violation of the uncertainty principle signals algebraic deformation in quantum mechanics

Abstract

We show that a possible violation of the Robertson-Schr\"odinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase-space (even if it violates the Robertson-Schr\"odinger uncertainty principle) is always a quantum state of an appropriate non-commutative extension of quantum mechanics. Conversely, all canonical non-commutative extensions of quantum mechanics display states that violate the Robertson-Schr\"odinger uncertainty principle.