Generalized Analogs of the Heisenberg Uncertainty Inequality

TL;DR

This paper explores generalized Heisenberg uncertainty inequalities within various locally compact groups, extending classical results to new classes like Euclidean motion and specific nilpotent Lie groups.

Contribution

It establishes the validity of a generalized uncertainty inequality for a broad class of locally compact groups, including Euclidean spaces, Euclidean motion groups, and certain nilpotent Lie groups.

Findings

Inequality holds for b^n imes K where K is a unimodular group of type I

Valid for Euclidean motion group

Applicable to several classes of nilpotent Lie groups

Abstract

We investigate locally compact topological groups for which a generalized analogue of Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\mathbb{R}^n \times K$ (where $K$ is a separable unimodular locally compact group of type I), Euclidean Motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, $2$-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups.