Uncertainty Relation in Quantum Mechanics with Quantum Group Symmetry

TL;DR

This paper explores how quantum group symmetries modify the uncertainty relations and spectra of position and momentum, revealing minimal uncertainties due to noncommutative geometry, with classical quantum mechanics recovered in certain limits.

Contribution

It introduces a framework for quantum mechanics with quantum group symmetry, showing how noncommutative geometry leads to minimal uncertainties in position and momentum.

Findings

Existence of minimal nonzero uncertainties in position and momentum.

Recovery of standard quantum mechanics in specific limiting cases.

Introduction of length and momentum scales due to noncommutative geometry.

Abstract

We study the commutation relations, uncertainty relations and spectra of position and momentum operators within the framework of quantum group % symmetric Heisenberg algebras and their (Bargmann-) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of minimal nonzero uncertainties in the positions and momenta. The usual quantum mechanical behaviour is recovered as a limiting case for not too small and not too large distances and momenta.