Uncertainty Relations in the Framework of Equalities

In this paper, we study the Schrödinger-Robertson uncertainty relations as corollaries of equalities in a scalar product space. Moreover, we give a number of characterizations in the case where the associated inequalities are in fact equalities. Our presentation is based exclusively on an algebraic observation on the standard Cauchy-Schwarz inequality and could presumably provide a clear and explicit understanding of uncertainty relations from the point of view of orthogonality. As applications, we show that some specific commutation relations, in the Hilbert space L(R) of square integrable functions on the Euclidean space R of dimensions n, imply new norm equalities in L(R), which are regarded as equality versions of well-known inequalities such as dilation and Hardy type inequalities. In particular, we give a method of recognizing Hardy type inequalities in the framework of commutation relations of operators. This paper is organized as follows. In Section 2, we characterize the Schrödinger-Robertson uncertainty relations in the framework of equalities in a scalar product space. In Section 3, we give a number of examples of uncertainty relations on the basis of equalities in L(R). In the Appendix, we summarize basic theorems on the Cauchy-Schwarz inequality in an algebraic setting. Throughout the paper, H denotes a complex vector space endowed with scalar product ( · | · ) : H ×H ∋ (u, v) 7→ (u|v) ∈ C, which is linear (resp. antilinear) in the first (resp. second) variable. The associated norm is defined by ‖u‖ = (u|u), u ∈ H . There is a large literature on the uncertainty relations. We refer the readers to [7, 9, 5, 14, 24] and references therein.