Uncertainty Relations in the Framework of Equalities

TL;DR

This paper explores the Schrödinger-Robertson uncertainty relations within an algebraic framework and derives new equalities from specific commutation relations, extending well-known inequalities like Hardy's inequality.

Contribution

It introduces a novel algebraic approach to uncertainty relations and establishes new equality forms of classical inequalities.

Findings

Derived new equality versions of Hardy's inequality

Connected specific commutation relations to novel equalities

Enhanced understanding of uncertainty relations in algebraic terms

Abstract

We study the Schr\"odinger-Robertson uncertainty relations in an algebraic framework. Moreover, we show that some specific commutation relations imply new equalities, which are regarded as equality versions of well-known inequalities such as Hardy's inequality.