# Uncertainty relations for a non-canonical phase-space noncommutative algebra

## TL;DR

This paper investigates a non-canonical phase-space algebra, demonstrating minimal uncertainties for variable pairs, and showing it violates traditional uncertainty principles, with implications for quantum cosmology and time-frequency analysis.

## Contribution

It introduces a non-canonical algebra with minimal uncertainties and analyzes its stability and violation of standard uncertainty relations.

## Key findings

- Existence of minimal uncertainties for all variable pairs.
- States minimizing uncertainties are ground states of positive operators.
- The algebra violates the standard Heisenberg uncertainty principle.

## Abstract

We consider a non-canonical phase-space deformation of the Heisenberg-Weyl algebra that was recently introduced in the context of quantum cosmology. We prove the existence of minimal uncertainties for all pairs of non-commuting variables. We also show that the states which minimize each uncertainty inequality are ground states of certain positive operators. The algebra is shown to be stable and to violate the usual Heisenberg-Pauli-Weyl inequality for position and momentum. The techniques used are potentially interesting in the context of time-frequency analysis.